metadata: authors: 0: author: Smith, J. Barry
1: author: Stelcner, Morton
count: Peru
docna: 0: docna: Modeling economic behavior in Peru's
            informal urban retail sector
docty: Policy Research Working Paper
subtopic: Construction Industry,Common Carriers Industry,Food & Beverage Industry,General Manufacturing,Pulp & Paper Industry,Plastics & Rubber Industry,Wholesale & Retail Trade Industry,Labor Markets,Educational Sciences,International Trade and Trade Rules,Business Cycles and Stabilization Policies
teratopic: Poverty Reduction
abstracts: cdata!: In Lima, Peru, the informal sector makes
            up half the labor force, accounts for 61 percent of the
            hours worked, and generates an astounding 39 percent of GDP.
            More than half the street vendors are women.  In the
            informal sector, the free play of market forces determines
            returns to productive factors, especially labor.  Informal
            enterprises are concentrated in low-income areas of urban
            centers.  The authors analyze Peru's urban informal
            sector - particularly women's role in it -  based on a
            theoretical model of informal retail trade.  They address
            these questions: what factors explain differences in the
            performance of retail busineesses?  If these can be
            identified, what types of policy initiatives might improve
            the performance of firms, especially those run by women?
            The authors recommend : channeling credit to small
            businesses, promoting cooperatives and self-help
            associations, providing technical assistance in basic
            management, making it easier and cheaper to get business
            licenses and facilitating cooperative child care centers.
geo_reg_key: 119228,517191
repnme: repnme: Modeling economic behavior in Peru's
            informal urban retail sector
display_title: Modeling economic behavior in
            Peru's informal urban retail sector
listing_relative_url: /research/1990/08/700200/modeling-economic-behavior-perus-informal-urban-retail-sector
topicv3: Informal Sector m1326258 1168,Manufacturing Sectors m1326521 1281,Trade m1326924 698,Education m1327048 470,Retail Industry m1504598 3778
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Poverty Reduction
origu: Policy, Research & Ect Affairs
no_of_pages: 92
model_extractions: <!-- Start of picture text --> Resear h, and External Affairs<br>LPolicy,<br>Women In Development |<br>WPS 469<br><!-- End of picture text -->

- This paper a product of the Women in Development Division, Population and lHuman Resources Department is part of a larger effort in PRE to determine if and how women's productivity (and thus family welfare) are improved when women are given more access to education, extension, training, credit, health care, and otherpublic resources. Copies are available free from the Wor!d Bank, 1818 H Street NW, Washington DC 20433. Please contact Maria Abundo, room S9-123, extension 36820 (87 pages with diagrams and tables).

The informal sector is a collection of loosely organized, small-scale competitive family businesses that rely little on nonfamily hired labor, use labor-intensive technologies, and operate largely outside of the legal, bureaucratic, and regulatory framework in terms of licenses, taxes, and contractual obligations.

In Lima, Peru, the informal sector makes up half the labor force, accounts for 61 percent of the hours worked, and generates an astounding 39 percent of GDP. More than half the street vendors are women.

Smith and Stelcner analyze Peru's urban informal sectorparticulary women's role in it - based on a theoretical model of informal retail trade (the dominant nonfarm family enterprises), using data from the Peru Living Standards Survey (PLSS).

They address these questions: What factors explain differences in the performance of retail businesses? If these can be identified, what types of policy initiatives might improve the performance of firms, especially those run by women? Among their recommendations: In the informal sector, the free play of market forces determiines returns to productive factors, especially labor. Informal enterprises are concentrated in low-income areas of urban centers, but rural households in Kenya and Peru, among other countries, have joined.

* Channeling crcdit to small businesses.

* Promoting cooperatives and self-help associations, which provide credit, facilitate bulk purchases, and establish markets for entrepreneurs.

The informal sector is an important - if not the sole -- income opportunity for growing numbers of the poor. International aid agencies have explored policies to make informal businesses more profitable. But this surge of interest is not based on much empirical evidence about what determines the firms' performance. Nor is the value of women's entrepreneurial activities reflected in the national accounts.

* Providing technical assistance, such as short-term instruction in basic management.

* Making it easier and cheaper to get business licenses.

* Provide or facilitate cooperative childcare centers, facilities for preparing food, and neighborhood facilities for basic health care to reduce the heavy workload typical for women.

The PRE Working Paper Series disseminates the findings of work undcr way in the Bank's Policy, Rcsearch. and Extemal Affairs Complex. Anobjective of thcseries is to get these findings outquickly, even if prescntations are less than fully polished. The findings, interpretations, and conclusions in these papers do not necessarily represent official Bank policy.

Produced by the PRE Dissemination Center
# **Table of Contents**

|||page|
|---|---|---|
|1.|Introduction .........................................|<br>1|
|2.|Some Stylized Facts on the Informal Sector ....<br>....|...<br>4|
||2.1<br>Magnitudes and other Characteristics ....<br>....|.... 4|
||2.2<br>The Role of Women ..............................|.<br>8|
|3.|Model Formulation and Concepts .......................|.<br>12|
||3.1<br>Assumptions ....................................|.<br>13|
||3.2<br>Specific Aspects of the Model ..................|.<br>15|
||3.3<br>The Revenue Function of the Firm ....<br>.......|..... 18|
||3.4<br>Marginal Revenue Products and Profit||
||Maximization ...................................|.<br>19|
|4.|Description of the Data and Variables ....<br>........|.....<br>22|
|5.|The Empirical Model ..................................|.<br>28|
||5.1<br>Initial Specification ..........................|.<br>29|
||5.2<br>Preliminary Tests for Aggregation and Pooling ...|.<br>29|
||5.3<br>Diagnostic Analysis ............................|.<br>31|
||5.4<br>Analysis of Nonlinearity .......................|.<br>34|
||5.5<br>Distribution of the Errors .....................|.<br>37|
|6.|Empirical Findings and Interpretation ....<br>........|.....<br>38|
||6.1<br>Factor Productivities ..........................|.<br>42|
||6.2<br>Nonparametric Evidence .........................|.<br>59|
||6.3<br>Simulation Experiments .........................|.<br>67|
|7.|Policy Implications ..................................|.<br>70|
||7.1<br>Enterprise-Specific Policies ....................|<br>71|
||7.2<br>Women-Specific Policies .72||
|Appe|ndix A: Nonparametric Analysis .74||
||Nonparametric Modeling .74||
||Defining Conditional Means and Their Properties|75|
||<br>Estimating Conditional Means and Derivatives .76||
|Appe|ndix B: The Rural Model .78||
|Bibl|iography .84||

**J. Barry Smith is a member of the Department of Economics, York University, Toronto, Ontario and Morton Stelcner is in the Department of Economics, Concordia University, Montreal, Quebec. The authors thank the foLlowing individuals: Jorge Castillo, Shahid Khandker, Valerie Kozel, and Marcia Schafgans of the World Bank; Richard Laferriere, Department of Economics, Concordia University; Georges Monette, Department of Mathematics, York University; Jeffrey Racine, Department of Economics, York University; Gwenn B. Hughes, Policy and Strategic Analysis Branch, Labour Canada, Ottawa, and Gail A. Spence, Canadian Pacific Forest Products Limited, Montreal. The authors also benefited from the opportunity to present an earlier version of the paper at a seminar sponsored by the Women in Development Division. Jeffrey Racine performed the nonparametric analysis and wrote Appendix A. The research was funded by the Women in Development Division, the World Bank.**
# **Modeling Economic Behavior in**

# **the Informal Urban Retail Sector of Peru**

J. **Barry Smith and Morton Stelcner**

# 1. **Introduction**

Few topics in the literature on development economics have inspired as much interest, controversy, and rhetoric as the informal sector in developing countries.1 Here the term is intended as a shorthand expression to describe the collection oJf loosely organized (but not necessarily fly-by-night) small-scale competitive family businesses. Such businesses rely little on nonfamily hired labor. Their technologies are labor intensive and they operate largely outside the legal, bureaucratic, and regulatory framework reg.rding such matters as licenses, taxes, and contractual obligations. An important characteristic of the **informal sector is** that the free play of market forces generally determines returns to productive factors, especially labor. The enterprises are usually concentrated in low-income areas of large metropolitan centers, but it is not uncommon to find rural households, for example in Kenya and Peru, that have joined the informal sector.<sup>2</sup> There appears to be a consensus that the informal economy is a sizable and **growing component** of developing economies. It accounts for a substantial fraction 1 For more recent literature see Bromley (1978), Cornia (1987), Hallak and Caillods (1982), Hart (1987), House (1984), IDB (1987a), Mattera (1985), Moser and Marsie-Hazen (1984), Sethuraman (1981), Stewart (1987), and Tokman (1978).

**2** For a discussion see Stelcner and Moock (1988), Moock, Musgrove, and **Stelcner (1989), and** Freeman and Norcliffe (1985).

1
of the labor force, especially in urban areas, and provides an important -- if not the sole - - income opportunity for growing numbers of the poor. But the debate about its role in economic development continues. There is considerable disagreement aoout whether measures should be taken to promote informal activities as an impetus to economic growth and a strategy for improving the earnings of low-income households.

The place of the informal sector in development is all the more imnortant because of the severe economic crisis in most third world economies. As Cornia (1987) discusses, households faced with sharply reduced employment and incom3 prospects in the formal (or modern) sectors -- manufacturing, services, mining, and government - - tend to seek employment and income opportunities in the informal economy. The first to shift are those who have lost jobs in the formal sector. Next, employed formal sector workers, especially government employees, resort to moonlighting activities, most of which are informal. As household incomes decline, married women and children who previously did not work in the market are drawn into informal market activities, and soon new entrants to the labor force begin to find jobs in the informal rather than the formal sector. 3 The rapid growth of the informal sector has led international aid agencies and governments to explore policies to improve the profitability of such businesses. This surge of interest, however, is not based on much empirical evidence about the underlying determinants of the performance of the firms.

Informals perform a remarkable array of activities, ranging from vending foodstuffs and prepared foods to consumer goods and services, including 3 Several recent studies have identified these general patterns. See Cornia (1987), IDB (1987b), PREALC (1985), van der Gaag, Stelcner and Vijverberg (1989), and Tokman (1986).

2
carpentry, tailoring, barbering, shoe-repair, domestic work, vehicle and tool repairs, and transport. In addition, small-scale entrepreneurs manufa _re_ textiles, garments, footwear, household utensils, musical instruments, metal products, furniture and wood products, and leather goods. They process foods and beverages, and recycle junk. Some firms are also involved in such illicit ventures as smuggling and processing alcohol and cocaine.

This research analyzes the informal sector in Peru, particularly the role of women, based on a theoretical model of informal retail trade that uses data from the Peru Living Standards Survey (PLSS). Retailing is the domirnant nonfarm family enterprise. The central questions are: What factors explain differences in the performance of retail businesses? Assuming that these cc.nsiderations can be identified, what types of policy initiatives might improve the performance of firms, particularly those run by women? The analysis is confined to urban areas where most of these bus.nesses are located .<sup>4</sup> 4We attempted to provide an analysis of nonfarm enterprises in both urban and rural areas. But the limited size of the sample in rural areas precluded the estimation of a mod-l that we were confident of using. The resu- ts for rural areas are in appendix L.

3
# 2. **_Some_** Stylized Facts on the Informal **Sector**

## 2.1 Magnitudes and Other Characteristics

Informal economic activity is a mainstay of the Peruvian economy. Its sustained growth stands in marked contrast to formal activity, which has deteriorated in the last decade at an alarmingly rapid rate. There is extensive evidence that a high proportion of the labor force, especially the Lemale component, is in informal activities. As Glewwe and de Tray (1989) and SuarezBerengue]a (1987) discuss, the majority of the bottom socioeconomic strata in urban areas earns a livelihood from self-employment in the informal sector.

5 Peru's shadow economy has recently attracted worldwide attention as a result of the recent publication of El Otro Sendero: La Revolucibn Informal6 by Hernando de Soto, a businessman and president of Lae Instituto Libertad y Democracia (ILD) in Lima. He concludes that the informal sector is the dominant and most dynamic part of the economy, and believes that removing the burdensome obstacles to legitimacy (such as bureaucratic red tape) would considerably improve Peru's economic malaise.

Several other recent studies corroborate de Soto's view ,hat a sizable portion, perhaps a majority, of the labor force is in the informal sector.7 Surveys by the ILD in 1985 and 1986 show that the informal sector in Lima makes up almost half the labor force, accounts for 61 percent of the hours worked and s See The Economist, February 18 and September 23, 1989; and Leaders, March 1989 (12) 1.

> **6** The English translation is The Other Path: The Invisible Revolution in the Third World. (See de Soto 1989).

> 7See Althaus and Morelli (1980), Kafka (1984), Litan and others (1987), Stelcner and Moock (1988), Moock, Musgrove, and Stelcner (1989), Strassmann (1987), Suarez-Berenguela (1987), Vargas Llosa (1987), Webb (1977), and World Bank (1987).

4
generates an astounding 39 percent<sup>of GDP (1984). For such sectors as commerce</sup> mind personal services this share exceeds<sup>60 percent. Litan and others (1987)</sup> report that the official national<sup>accounts estimate of the informal sector's</sup> contribution to GDP it, 3'84 resulted<sup>in an understatement of total GDP of 23</sup> percent. Perhaps even more striking<sup>is the estimate that 439,000 Lima residents</sup> depend on the underground commercial<sup>economy, and almost three fourths (314,000)</sup> of these individuals depend on street<sup>sales. According to de Soto, these</sup> activities generated about $25<sup>million a month in gross sales in 1985 and an</sup> average ret per capita profit of $58 a<sup>month, about 40 percent more than the</sup> legal minimum wage. The 314,000 street<sup>merchants include 91,455 street vendors,</sup> are 42 percent of the Lima work force involved<sup>in commerce.</sup> The ILD surveys show that women make up<sup>54 percent of the street vendors.</sup> Eighty-six percent of the street merchants<sup>occupy curbside sites, while the</sup> remaining 14 percent rove the streets.<sup>Business is also conducted in (illegal)</sup> cooperative markets -- collections<sup>of kiosks, stalls and booths. Of the 331</sup> markets in Lima, 274 were put up illegally;<sup>only 57 were built by the government.</sup> It is estimated that $41 million has<sup>been invested in these illegal markets,</sup> which employ about 125,000 people (including<sup>sc-40,000 vendors). More than 80</sup> percent of the street vendors and 64 percent<sup>of the informal markets are found</sup> in low-income districts.

According to recent studies that used the Peru<sup>Living Standards Survey8,</sup> half of the 5,100 households surveyed owned<sup>at least one nonagricultural family</sup> enterprise. Of the 27,000 individuals in the<sup>survey, 13,600 were in the labor</sup> force and 97 percent were employed.<sup>More than 4,500 worked in nonfarm family</sup> 8 See Stelcner and Moock (1988), and<sup>Moock, Musgrove, and Stelcner (1989).</sup> 5
businesses, and 3,100 worked in family enterprises as their main occupation. About 6,200 worked on family farms.

Metropolitan Lima accounted for 34 percent of n3nfarm family businesses, other urban areas for 44 percent, and rural areas only 22 percenc.. Of course, a large fraction of rural households also operate farms. These proportions correspond closely to the distribution of family workers and households across regions. The average number of enterprises per household is 1.25.

These family businesses are dominated by retail trade, manufacturing (especially textiles), and personal services (mainly in urban areas). Retail trade encompasses small shops, inns and cafes, kiosks, stalls, and street vending. Nontextile manufacturing includes food, beverages, pottery, furniture, toys,novelties, and musical instruments. The textile sector includes spinning, weaving, and tailoring. Personal services range from laundries and hairdresser. and barbers to entertainment, auto and electrical repairs, and cleaning services. Most businesses rely on just one or two family workers; the use of hired labor is negligible.

Many firms do not own any capital or inventory and often have no operating expenses. In the retail sector selling is often on consignment or commission. Large factories, wholesalers, and stores in the formal sector often provide the **goods** and perhaps the cart, stall, or kiosk. The goods are sold either on straight commission, or on consignment: the sellers pay only for what they sell and return the unsold goods. Factories often subcontract textile, clothing, leather and footwear manufacturing to family enterprises, which they provide with materials and equipment. In the labor-intensive personal services sector, little use is made of capital equipment. Thus it is not surprising that many family 6
businesses in the dominant sectors reported<sup>little or no capital, inventories,</sup> or operating expenses.

How much credit do informals use<sup>and obtain? The Peru Living Standards</sup> Survey provided information on<sup>the current debt position of each household, and</sup> on the source and termns of loans obtained in the past year.

Only 10 percent of the households<sup>that operated businesses reported that</sup> they received loans or were in debt.<sup>This is not surprising for several reasons.</sup> First, the PLSS was conducted when<sup>inflation rates were extremely high (June</sup> 1985-July 1986). (During the first<sup>half of 1985 the annual inflationwas200</sup> percent, and in the first half of<sup>1986 monthly inflation was 4 to 5 percent.</sup> Such high rates of inflation are unlikely<sup>to foster a willingness to lend, except</sup> at interest rates so high that few<sup>households would choose to borrow. Second,</sup> given a sensitivity to questions about<sup>indebtedness, the Peru Living Standards</sup> Survey probably underestimated the<sup>debt among respondents. (To preserve good</sup> will, questions on debt and credit<sup>were last in a long questionnaire".</sup> Third, given the uncertain legal position<sup>of family businesses, their ability to obtain</sup> credit from formal lending institutions is very restricted at best. As Carbonetto (1984), Mescher (1985), and Kafka (1984)<sup>document, only a minute fraction of</sup> informal sector firms in Lima borrow<sup>from financial intermediaries</sup> in the formal sector. Households that need to<sup>borrow must depend on loan sharks and</sup> pawnbrokers, or resort to a "pandeiro". This is a revolving fund<sup>to which</sup> members make a weekly contribution. A lottery determines the winner of the<sup>week's</sup> contributions .9 9 See Mescher(1985), World Bank (1989).

7
In Lima, retail trade, personal services, and manufacturing account for more thati 80 percent of the businesses and almost 85 percenc of the family workers (Table 1). The single most important activity is retail trade. The pattern is the same in the rest of Peru. In other urban areas the proportions are 86 percent (retail trade, 52 percent) and 90 percent. In rural areas 80 percent of entrepreneurial families and workers are in retail trade (43 percent) or manufacturing, particularly textiles. These are typical informal sector endeavors in most developing countries. What distinguishes Peru from other countries is the unusually large proportion of women and households in these activities.

# 2.2 The Role of Women

The Peru Living Standards Survey shows that women dominate the informal economy. Schafgans (1989) notes t'at women make up about 45 percent of the labor force. The vast majority work in family-owned firms and farms. Table 2 shows that 82 percent of the 5,952 employed women worked in family-owned nonfarm businesses (28 percent) or on family farms (54 percent). The remaining 18 percent were wage earners, mostly in urban areas. In Lima 45 percent of employed women were wage earners in contrast to 19 percent in other urban areas. In rural areas only 5 percent of employed women worked as salaried employees, usually as schoolteachers and clerks in the public sector. There is practically no labor market for women in rural areas. In Lima, 38 percent of the women were employed in nonfarm businesses and 17 percent in agricultural activities; in other urban areas the corresponding proportions were 44 and 37 percent, and in rural areas 11 and 84 percent.

8
TabTh.

Percendfbgttribuf5 forbEhitysineAqtctofictivity

|Metrbima|OthdirbaAreas|AlUrbabreas|RuraUreas|Peru|
|---|---|---|---|---|
|No.oWDbservationlO45|1387|2432|692|3124|
|Retdilade<br>40.7|51.5|46.8|43.1|46.0|
|Persofekvices 19.8|12.4|15.6|4.5|13.1|
|TextM1mnufacturiI*.9|12.2|11.6|26.4|14.9|
|Othd4anufacturidrW.4|9.7|10.0|11.9|10.4|
|Subtotal<br>81.8|85.8|84.0|85.9|84.4|
|Othdefctors<br>18.2|14.2|16.0|14.1|15.6|

> Note0thdfanufactubtraidoedbvera6irxii OthLactdcswlud6nstruc inanporbdtmdi i:adhnintishfrnhunting. rcorduebis cel1laiieems .

SourcberIiiviSgandaSldviey.

TABLE 2. Percentfbtribuflmpfoy&dbo]Pordvfrypefmployment

|Metrbi|ma|OthdfrbaA|reas|AlUrbaA|reas|Rurarea|s|Peru||
|---|---|---|---|---|---|---|---|---|---|
|Females<br>No.o9Dbservatl8fll|Males<br>1747|Females<br>1978|Males<br>2450|Females<br>3215|Males<br>4197|Females<br>2737|Males<br>3041|Females<br>5952|Males<br>7238|
|NonFarSector 38.2|31.5|44.1|32.5|41.9|32.1|10.6|5.3|27.5|20.8|
|Farigector<br>16.5|3.4|37.0|25.1|29.0|16.1|84.0|77.0|54.4|41.7|
|Wageector<br>45.3|65.1|18.9|42.4|29.1|51.8|5.3|17.7|18.1|37.5|

Notefhemonfaumfarwectae£melf-employhowgsectDf etirdAbor.

SourcBerhiviigandaSdBvey.

9
Of the 7,238 employed men surveyed, 21 percent worked in nonfarm enterprises and 42 percent worked on family farms, while 38 percent worked as wage employees (65 percent in Lima. 42 percent in other urban areas, and 18 percent in rural areas). In Lima about 32 percent of employed men worked in family businesses and 3 percent on farms, while in other urban areas the proportions were 33 and 25 percent, respectively. In rural areas only 5 percent of men worked in nonfarm family businesses; 77 percent worked on family farms.

In the retail food and textile sectors women account for three-fourths of the family workers. In retail nonfood and food processing sectors they make up 60 to '0 percrnt of the work force, and account for about half the workers in urban personal services. The remaining family businesses -- transportation, construction, wood and4 chemical manufacturing, wholesale trade, hunting and fishing, and professional services -- are dominated by men and employ only a small fraction of women. There appears to be a clear division of labor between men and women. The proportion of women employed in the formal sector is much smaller than that of men and the informal sector activities that women pursue are considerably different from those of men.

The role of women in familv enterprises is also highlighted by the large proportion of family businesses in the dominant three sectors that employ exclusively women and children under 20. In urban and rural areas about half the family retail businesses rely only on women and children. About 40 percent of that provide services firms employ only women and children. In textiles, which are largely home-based, oner 66 percent of the rural concerns employ only women and children. In Lima the proportion rises to 70 percent and in other urban areas to 75 percent. These data suggest that women not only make up a high proportion of family workers **but** also operate the family businesses.

10
Despite their importance in these businesses, the value of women's entrepreneurial activities is not adequately reflected -- if at all -- in the national accounts. There have been very few attempts to assess their contribution to the economy or to analyze the relative performance of men and women in the informal economy. This is particularly true in Peru and other Latin American countries where such work has gone largely unnoticed, with the possible exception of domestic work.

Moreover, official Latin America data'° do not give accurate information about women's economic activities. Most of the empirical economic research on family businesses has focused on agricultural activities or on the activities of self-employed urban men. Agricultural research tends to ignore informal nonfarm economic activities in rural areas. And most of the research on the selfemployed analyzes individuals rather than the enterprise, ignoring the contributions to income of capital, nonlabor inputs, and the labor of women and children. These last are typically excluded because most surveys report them as unpaid family workers, while men are usually reported as paid family workers (Chiswick 1983).11 This analysis focuses on the family business rather than the self-employed individual, thereby incorporating enterprise characteristics -- capital,<sup>2</sup> location, nonlabor inputs -- and the labor of all family workers.<sup>1</sup>

**10** See Recchine de Lates and Wainerman 1986; Bunster and Chaney 1985; Babb 1984.

11 Often it is not clear whether these self-employed earnings refer to the value of gross sales or to the value of net production, that is, sales less cost of materials and other inputs.

**12** For related approaches see Blau (1985), Chiswick (1983), Strassmann (1987), Vijverberg (1988), and Teilhet-Waldorf and Waldorf (1983).

11
# 3. Model Formulation **and** Concepts

The analysis of the model of the revenue<sup>process of retail enterprises</sup> includes a theoretical characterization of revenue generation by retailers as well as a basis for estimating economic magnitudes<sup>(such as productivity).</sup> The model incorporates three features of sales revenue -- price, potential<sup>customers,</sup> and the process by which a potential customer<sup>becomes a purchaser, at a given</sup> price.

The traditional economic model of production<sup>is extended to explain the</sup> production process of firms that expend resources<sup>in selling as well as in</sup> producing goods or services. For example, consider<sup>a street vendor who sells</sup> pencils at a given price at a busy intersection.<sup>An important consideration</sup> involves measuring the vendor's output. To argue<sup>that output can be measured in</sup> terms of the number or constant dollar value<sup>of transactions (in this case,</sup> pencils sold) is akin to measuring output by the<sup>value of inputs, and misses a</sup> vital feature of the retailing process. That is,<sup>in every attempt (whether</sup> successful or not) to convince a passerby to<sup>purchase a pencil, the vendor is</sup> also making a sales effort, that includes information<sup>about the product and its</sup> availability. To measure output by the volume or<sup>value of transactions</sup> measures only that part of the vendor's activity that is<sup>successful.</sup> In our model we argue that an enterprise in<sup>the retail sector effectively</sup> 'produces' the probability that a contacted customer<sup>will make a purchase. We</sup> argue further that the firm can adjust its<sup>inputs (including labor, capital,</sup> materials, and inventory) to change the likelihood<sup>that it will make a sale.</sup> Changing inputs may range from increasing inventories<sup>to providing more</sup> information to customers.

12
The model is not explicitly one of profit maximization and not necessarily one where optimization leads to a dual relationship between cost and production. The data (particularly on prices) are not sufficient to estimate such a model. More important, it is not clear whether the textbook model of cost-efficient production and profit maximization is a useful hypothesis. It may be more reasonable to assume simply that firms make efficient use of their inputs and then to test whether observed decisions are consistent with profit maximization. Since our goal is to provide an empirical model of production for retail firms, in our discussion of the theoretical model we will also refer to problems and limitations in the applied work.

# 3.1 Assumptions

Assume that potential shoppers arrive randomly at the location of a vendor. Thus the contacts between buyers and sellers is a random variable. The average number of such contacts will depend upon the characteristics of the firm, including its location and reputation. 1<sup>3</sup> There is no guarantee that a shopper arriving at an enterprise will decide to make a purchase. Of two seemingly identical shoppers, one may decide to buy while the other does not, independently of the characteristics of the enterprise. The fraction of shoppers that makes a purchase is thus a random variable.

We assume that at each point in time and for a given price and type of good, a customer has a random yet rationally determined threshold response level to the vendor's sales effort. Suppose shopper j has threshold level **ti.** The decision by the customer whether or not to make the purchase involves a comparison of **t.** with the variable T. defined as tbh index of sales effort 13 No information is available concerning repeat buyers or customers who purchase more than one unit of a good.

13
produced by firm i. If TL > tj, then the arriving customer will buy from firm i. If customers have their threshold levels ti distributed according to the same (distribution) function F, then a firm with sales effort T. will make a sale to a randomly arriving customer with probability F(Td). This probability, F(T1), can also be thought of as the fraction of arriving customers that buy from the firm. Part of the firms' decision making involves setting the level of TL.

The model described above is similar to stochastic choice models that have lately become quite popular in labor economics. By analogy, the decision to buy is like the decision to enter the labor force and the condition that tj **<** TL is similar to the requirement that the reservation wage at zero work is less than the market wage.

The enterprise can change its operating characteristics to affect the fraction of shoppers that makes a purchase. For example, a business can increase its inventory or stay open longer. Such factors are considered productive if increasing them raises the fraction of potential consumers who make purchases or, equivalently, increases the probability that a given shopper will make a purchase. Since the fraction of shoppers who buy is bounded from above by unity, in the limit for large quantities of factors there must be zero returns at the margin to increasing the level of productive factors. Similarly, while enterprises may adopt different mixes of factors (perhaps due to financing restrictions), labor is a common feature; no retail firm can operate without labor. The fraction of shoppers making purchases will approach zero as the amount of labor input approaches zero. This need not be true, however, for such inputs as capital (for example, a cart or stall) or inventory. Without these factors a customer can still make a purchase. The model is not constrained ex ante to require profit maximizing decisions on the part of firms. We do, however, examine 14
whether the properties of the estimated model are consistent with profit maximization.

In applied work, neither the total number of units sold nor the selling price per unit are generally known. Data sets typically do not contain this information. Similarly, information on the number of customer contacts and the fraction of shoppers who buy is not available. At most, information on revenues, costs, and other characteristics of factors employed by the firm will be available in cross-section or time series data sets. With cross-section data the absence of separate information on price and quantity may cause fewer problems than in a time series setting where constancy of price is difficult to justify as a working assun,ption.

# 3.2 Specific Aspects of the Model

The expected price per unit received by an enterprise is defined as p8. It is assumed that agents treat pB as independent of the decisions of individual enterprises and of customers.

The expected number of shoppers arriving at enterprise **i** is defined as **NE(Xi). NE** is assumed to depend upon (a vector of) characteristics of firm **i, X',** one element of which, for example, would be location. Differences in the expected number of arrivals at firm i versus firm j are assumed to depend only on differences in the characteristics of vectors xL and **x3.** Because the total number of arrivals is subject to random effects, firm **i** will not in general observe arrivals equal to N<sup>1</sup> (XI). In the applied work of subsequent sections it is assumed that the expected number of arrivals to firm **i** can be expressed as: Thus, 15
where XLJ is a measure of the **jth** characteristic in firm **i.** The specification of **NE(X**<sup>**t**</sup> **)** is seen to be linear in its logarithm and introduces the ex ante restriction that the number of arrivals is nonnegative.

The fraction of shoppers that make a purchase or, equivalently, the probability that firm i makes a sale to a randomly arriving customer, is given by F(T(Z')). F depends upon a vector of firm i's characteristics, ZI, which includes labor, materials/expenses, capital, and inventories.

In keeping with the discussion of the previous section, F can be written as a function of T1 where T. is an index of sales effort and output produced by firm i and given by: TL = T(Zi) F will be a nondecreasing function of TL, bounded from below by 0 and from above by 1. Indeed, F is just the cumulative distribution function for the random variable **t** representing individual consumer purchasing thresholds. A drawing of **t** for a given customer j **(tj)** shows the level of (an index of) sales effort needed to guarantee that individual j will make a purchase.

By adopting the above characterization of retailing, we obtain a model whereby something can be produced (sales effort or the probability of a purchase) with no guarantee that any consumer will make a purchase or that a firm will be observed to make a transaction. This will occur when the level of (the index of) sales effort produced by a given firm falls short of the threshold level necessary to convince the customer to buy. In the applied research it is assumed 16
that the cumulative distribution function, F, is given by the logistic function. 14 From (3) it will be recalled that the index TL is a function of a vector of characteristics ZL of firm i. This function must reflect the fact that labor is indispensable to the activity but that other factors are not. If we define ZL as the labor component of Z', the indispensability of labor can be introduced by requiring T, to be an increasing function of the logarithm of z 1L. Introducing the remaining factors affecting TL in a linear fashion leads to the specification: nF TL- bo + bllnzL + E b zL (5) **1 J-2 j J** where nF is the number of factors used in the production of T.. Given that b, > 0, labor will be an indispensable factor with a positive marginal product in terms of increasing the index T.. As labor becomes small, TL decreases without bound and F(T1) approaches zero. Any other factor zi will be productive as long as bi > 0. The parameter bo is the value of the index when labor is equal to one unit and all other factors are zero.It is reasonable to expect bo to be negative and large enough in absoltte value such that exp[-b.] is large and F(TL), the average frequency of sales, is small when almost no factors are allocated to sales.

As a final point, it is possible to extend the production analogy to consider isoquants which, in this case, are isoprobability contours of F. Since " This specification of a logistic probability distribution function has proved valuable in other areas of applied economic research and it has the added benefit that **it** leads to applied models that are somewhat easier to estimate than those based upon the cumulative normal distribution function.

17
F is a monotone increasing transformation of T., isoquants of T1 will coincide with isoquants of F. These isoquants will be straight lines for pairs of inputs, excluding labor. Alternatively, for pairs of inputs one of which is labor measured on the vertical axis, the isoquants are horizontally parallel and intersect the vertical axis. Production processes such as this are called quasilinear.

# 3.3 The Revenue Function of the Firm

The discussion contained in the foregoing sections leads to the specification of a revenue function for a representative firm in the retail sector. This function is comprised of both a deterministic and a stochastic component. The expected revenue of firm i, RE5 , is given by the product of the expected price and the expected number of buyers. The latter quantity is itself given by the product of the expected number of arriving customers and the fraction of customers who make purchases. In terms of the notation introduced above,

# RE = pENE(XI)F(T(ZI))

We assume that the stochastic influence on revenues enters multiplicatively. Thus observed revenues of firm i, R., are given by:

## RL = R' exp[vL]

where vL is a random variable incorporating uncertainties in the price level and the unforecastable factors affecting the number of customer contacts with the firm. We assume that vL is such that E[exp[v,]] = 1. Our applied work will involve estimating the revenue function in logarithmic form. In terms of the 18
specification of **NE** and F(T.) in previous sections, the estimating equation<sup>will</sup> be of the form: A nonlinear least squares algorithm is used to obtain point estimates of the parameters of the model. Since independent information on average price<sup>is</sup> not available, the coefficient estimate of ao will not identify the parameter ao.

# 3.4 Marginal Revenue Products and Profit Maximization

The derivative of the expected revenue function with respect to a righthand side variable (such as labor or capital) can be interpreted as the expected marginal revenue product of the variable. In cases where the unit price of the factor is known, the expected marginal revenue product can be compared with this magnitude to partially assess the efficiency of the firm. For example, if a firm is maximizing expected profits, the factor price and the marginal revenue product should, on average, coincide. In cases where data on factor prices are not available (a common occurrence in the informal sector), the derivative of the expected revenue fur -tion can be considered the shadow price of the factor. This shadow price is the amount of money that would be paid to the factor if the existing situation represented profit-maximizing behavior. In both situations the results lead to interesting insights for policy analysis.

The model is quite flexible with respect to possible relationships between revenue and such productive factors as labor, capital, and expenses. The<sup>fact</sup> that F(T1) is strictly bounded from above and below introduces some features into the relationship between factors that often do not arise in standard models of 19
production. To highlight some of these prope-ties, we present the following example of an expected revenue function.

Suppose that a simplified expected revenue function with two factors (x and y) is given by: R 1/(1 + exp[a - x - y]), a 2 0 (9) where R is (expected) revenue, price and customer effects are fixed (in this example) and (a, -1, -1) are the estimated coefficients of the model. The (expected) marginal revenue product of factor x, R., is the derivative of the right hand side of (9) with respect to x and is given by: Rx =R - R2 The marginal revenue product of x will be positive as long as the right hand side of (10) is positive. Given the definition of R in (9), this will always be the case because R < 1.

The response of the marginal revenue product function to changes, ceteris paribus, in x and y is important for determining the suitability of any profit maximization hypothesis and for determining the relationships between productive factors. The slope of the marginal revenue product function is given by the derivative of the right hand side of (10) with respect to x. Denoting this slope by R., differentiation yields: Eventually, the marginal revenue product curve will slope downwards as R becomes larger than .5. There may be a range of x values where the marginal revenue product curve is upward sloping. This would be the case, for example, 20
if when x - 0, the value of the expression<sup>(a-y) exceeds 0 (and hence, R<.5</sup> when x 0). Thus the slope of the marginal<sup>revenue product curve tor x depends</sup> in part on the quantity of<sup>the other productive factors. Within a profit</sup> maximization setting, the<sup>marginal revenue product curves must be downward</sup> sloping if the optimality conditions<sup>are to be satisfied.</sup> It was noted above that the<sup>levels of other factors affect the slope of</sup> the marginal revenue product<sup>curve for a given factor. The position of the</sup> marginal revenue product curve<sup>for a given factor is influenced</sup> by the quantities of the other factors as well.<sup>This effect can be illustrated by considering the</sup> change in the marginal revenue<sup>product ofxas y changes. Denoting this effect</sup> by R.., the right hand side<sup>of (10) can be differentiated</sup> with respect to y to obtain: where R. is the marginal revenue<sup>product of y. Thus in this simple example, as</sup> long as R < .5, increasing y<sup>makes x more productive. Eventually, though, as y</sup> becomes increasingly large,<sup>more of the factor y will exert a negative effect</sup> on the (marginal) productivity<sup>of x. The explanation lies in the fact that the</sup> probability of making a sale<sup>(in this case 1/(1+</sup> -n[I - x - **y])** is bounded from above by 1. The only way<sup>this condition can be met for increasing values</sup> of y and fixed x is if both factors<sup>are made less productive. If this were not</sup> to happen then x could be increased<sup>over a feasible range of values and the</sup> probability could be made greater<sup>than 1.</sup> 21
# 4. Description of the Data and Variables

The theoretical model is applied to data from the Peru Living Standards Survey (PLSS). The survey results provide informiiati **o** r. a variety of topics:n household composition, demographics, housing conditions, health, education, migration, labor force activities, housework, farm and nonfarm busines-es, and household expenditures (see Grootaert and Arrlagada 1986; INE 1988).

The survey modules on nonfarm family businesses, labor force behavior, and the personal characteristics of household members are particularly useful. For each family enterprise the PLSS gives the labor inputs of family workers, the value of output sold or consumed. by the household, expenditures on purchased inputs, and the value of assets. The survey does not contain information on physical units and prices of output and nonlabor inputs -- only total values. The survey shows how long each business has been open, the type of output, how many months it operated in the past year, and whether the business is in the home, at other fixed premises, or has no fixed premises. The information covers 3,360 businesses: 1,178 in metro Lima, 1,480 in other urban areas, and 702 in rural areas. The most popular activity is retail trade: 425 firms in Lima, 714 in other urban areas, and 298 in rural areas. In the analysis of the retail sector we excluded 51 observations urban because i) only children under 15 were employed (five in Lima, four in other urban areas, seven in rural areas), ii) no sales revenue was reported (nine in Lima, eight in other urban areas, three in rural areas), and iii) values of capital or expenses were more than 100,000 intis (six in Lima and nine in other urban areas). This reduced the sample to 1,386 firms: 405 in Lima, 693 in other urban areas, and 288 in rural areas.

A novel feature of this study is that it explicitly addresses the predominate role of women in the retail sector by distinguishing among three 22
types of businesses: 1) female.-only firms (possibly with children), 2) male-only firms (perhaps with children), and 3) mixed firms. Most retail firms employ only one or two family workers and do not use hired laborers. (See Table 3) We use the follr.wing variables in the empirical analysis. The dependent variable is the logarithm of monthly gross revenues, which is the value of output. The set of regressors can be grouped into two categories -- those that describe customer arrivals and those chat affect the probability of a purchase. The former group includes the age of the enterprise in years, which can be interpreted as a reflection of the reputation of the firm and perhaps as a predictor of learning by doing in attracting clientq. Also included is the place - of operation as a proxy for ease of access by customers. Two dummy variables - 'in the home' and 'at a fixed location' (a kiosk or stall) -- incorporate the site information. Itinerant operations with no fixed location (such as peddling and street hawking) are excluded.

The following variables are deemed to affect the probability of a purchase. First, the value of capital, which includes land, buildings, furniture, tools, machinery, equipment and vehicles. Second, the value of inventory stocks and third, monthly operating expenses which measure the cost of goods purchased for resale, raw materials, and such items as repairs, utilities, and fuel. The timeframe of this variable corresponds to that of the dependent variable. The fourth variable is labor input, measured by the logarithm of monthly hours devoted by all family workers in the ent._rprise. The annual hours of farming labor are divided by the months of operation. Other aspects of managerial or sales skills are described by two proxies: 1) total work experience (in years) of the most experienced adult family worker in the firm and 2) the level of educational attainment of the most educated adult family worker in the firm. Work 23
experience is entered with linear and quadratic (scaled by 100) terms. The effects of educational attainment are entered in two alternative ways. First, - we use three dummy variables primary school completed, secondary school completed, and postsecondary school completed; the excluded category is less than - primary schoolirg. Second, we use three splines years of primary school (zero to five), secondary school (six to ten) and postsecondary school (more than 10 years). Finally, the effects of vocational training are reflected by a dummy variable that takes a value of unity if any adult family worker in the enterprise had vocational training, or a value of zero otherwise.

24
**TABLE 3.**

**Pescriptjte Statistics of Urban Retail Sector**

|**T fFi**|<br>|METRO LIMA<br>|<br>||**OTH**<br>|**ERURBAN**<br>|**ARFAS**<br>||**ALL**|URBAN**AR**|**EAS**||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|**ype orm**<br>Rumber<br>**of**<br>**Firms**|Female<br>**203**|**Male**<br>**98**|**Mixed**<br>**104**|**Total**<br>**405**|Female<br>**375**|Male<br>**132**|**Mixed**<br>**188**|Total<br>**693**|**Female**<br>**576**|male<br>**230**|**mixed**<br>**292**|Total<br>1098|
|PLACE OF OPERATION<br>|S<br>||||||||||||
|Home<br>Fid i|25.1<br>|8.2<br>|24.0<br>|20.7<br>|37.5<br>|18.2|34.6|33.0|33.2|13.9|30.8|28.5|
|xe locaton<br> <br>|21.2<br>|18.4<br>|37.5<br>|24.7<br>|25.5|25.8|31.9|27.3|24.0|22.6|33.9|26.3|
|Itinerant<br>(Streets)|''.7|73.5|38.5|54.6|37.0|56.1|33.5|39.7|42.9|63.5|35.3|45.2|
|MONTHS OPERATED DURING Y|EAR<br>8.8<br>|9.2<br>|10.0<br>|9.2<br>|9.7|9.6|10.1|9.8|9.4|9.4|10.1|9.6|
||(3.9)|(3.8)|(3.0)|(3.7)|(3.6)|(3.8)|(2.9)|(3.4)|(3.7)|(3.8)|(3.0)|(3.6)|
|1-<br>6 months<br>6<br>9|X<br>28.1<br>|27.6<br>|14.4<br>|24.4<br>|22.0<br>|22.7|11.2|19.2|24.1|24.8|12.3|21.1|
|-<br>months<br>9-12 th|16.3<br>|9.2<br>|17.3<br>|14.8<br>|9.1<br>|8.3<br>|18.1<br>|11.4|11.6|8.7|17.8|12.7|
|mons|55.7|63.3|68.3|60.7|68.9|68.9|70.7|69.4|64.2|66.5|69.9|66.2|
|AGE**OF**FIRM<br>**(Tears)**|6.0<br>|7.8<br>|8.8<br>|7.1<br>|7.7<br>|9.3<br>|10.8<br>|8.9|7.1<br>|8.7|10.1|8.2|
||(8.8)|(10.1)|(9.1)|(9.3)|(9.8)|(11.5)|(11.8)|(10.8)|(9.5)|(10.9)|(10.9)|(10.3)|
|Lessthan<br>4 months<br> <br>|Z<br>12.3<br>|8.2<br>|4.8|9.4|9.4|9.8|4.3|8.1|10.4|9.1|4.5|8.6|
|4months<br>-<br>1<br>year<br>1<br><br>3<br>|14.3<br>|13.3<br>|9.6<br>|12.8<br>|8.8<br>|7.6<br>|3.2<br>|7.1|10.8|10.0|5.5|9.2|
|<br>-<br> <br>years<br> <br> <br>|27.1<br>|24.5<br>|19.2<br>|24.4<br>|23.1<br>|18.2|14.4|19.8|24.5|20.9|16.1|21.5|
|3<br>-<br>5<br>years<br> <br> <br>|14.3<br>|17.3<br>|15.4<br>|15.3<br>|14.5<br>|14.4|15.4|14.7|14.4|15.7|15.4|14.9|
|5<br>-<br>10<br>years<br>Over 10<br>|11.8<br>|7.1<br>|17.3<br>|12.1<br>|16.9<br>|17.4<br>|22.9<br>|18.6<br>|15.1<br>|13.0|20.9|16.2|
|<br>years|20.2|29.6|33.7|25.9|27.3|32.6|39.9|31.7|24.8|31.3|37.7|29.6|
|**MONTHLY**REVENUES|**$**<br>2731<br>|4328<br>|8300<br>|4548<br>|i8 <sup>8 9</sup><br>|5961<br>|4801<br>|3455<br>|2186|5266|'148|3858|
||(5489)|(6110)|(12138)|(8178)|(4262)|(21147)|(6337)|(10406)|(4743)|(16501)|(8;o&8)|(9655)|
|$ 1<br>-<br>500<br>Revenues<br>500<br>1000|X<br>24.6<br>|20.4<br>|2.9<br>|18.0<br>|32.7<br>|28.0<br>|12.2<br>|26.3|29.9|24.8|8.9|23.2|
|<br>-<br> <br>1000<br>2000|19.7<br>|16.3<br>|9.6<br>|16.3<br>|19.8<br>|12.9<br>|12.2<br>|16.5|19.8|14.3|11.3|16.4|
|<br>-<br> <br>2000<br><br>4000|21.2<br>|14.3<br>|11.5<br>|17.0<br>|22.5<br>|12.9<br>|14.9<br>|18.6<br>|22.0<br>|13.5|13.7|18.0|
|<br>-<br> <br>More<br>than<br>4000|17.7<br>16|20.4<br>|25.0<br>|20.2<br>|15.5<br>|18.9<br>|25.0<br>|18.8<br>|16.3<br>|19.6|25.0|19.3|
|<br> <br>|.7|28.6|51.0|28.4|9.4|27.3|35.6|19.9|12.0|27.8|41.1|23.0|
|**Log**of Revenues|S<br>7.0791<br>|7.4453<br>|8.3448<br>|7.4927<br>|6.7389<br>|7.2857<br>|7.7911<br>|7.1285|6.8588|7.3537|7.9883|7.2629|
||(1.2895)|(1.5110)|(1.1663)|(1.4142)|(1.2733)|(1.6530)|(1.2595)|(1.4224)|(1.2882)|(1.5926)|(1.2536)|(1.4296)|
|**MONTHLY**EXPENSES|$ 1874<br>|2491<br>|5633<br>|2988<br>|1294|3399|4376|2531|1498|3012|4824|2700|
||(3394)|(4177)|(8076)|(5391)|(3552)|(8032)|(8927)|(6513)|(3505)|(6671)|(8640)|(6125)|
|Spline 2 Expenses<br>|$ 1331<br>|1945<br>|4919<br>|2401<br>|819<br>|2883<br>|3696|1993|1000|2483|4132|2143|
|> 830|(3265)|(4029)|(8000)|(5273)|(3455)|(7910)|(8860)|(6415)|(3395)|(6548)|(8570)|(6020)|
|Spline 3 Expenses<br>|$ 960<br>|1489<br>|4133<br>|1903<br>|550<br>|2343|3013|1560|695|1979|3412|1686|
|> 2000|(2977)|(3715)|(7780)|(4994)|(3276)|(7699)|(8676)|(6211)|(3177)|(6320)|(8372)|(5792)|
|$ 0<br>Expenses<br> <br>|X<br>1.5|5.1|1.0|2.2|2.7|9.1|0.5|3.3|2.3|7.4|0.7|2.9|
|1 -<br>500<br> <br>|42.4<br>|38.8<br>|15.4<br>|34.6<br>|48.8|32.6|20.2|38.0|46.5|35.2|18.5|36.7|
|500-<br>1000<br> <br>|18.2<br>|9.2<br>|11.5<br>|14.3|19.8|7.6|11.7|15.3|19.3|8.3|11.6|14.9|
|1000<br>-<br>2000<br>2000<br>4000|14.8<br>|19.4<br>|11.5<br>|15.1<br>|13.1<br>|13.6<br>|23.9<br>|16.2|13.7|16.1|19.5|15.8|
|<br>-<br> <br> <br>|10.3<br>|10.2<br>|25.0|14.1|9.1|19.7|18.6|13.7|9.5|15.7|20.9|13.8|
|Over<br>4000<br> <br> <br>|12.8<br> <br>|17.3<br>|35.6<br>|19.8<br>|6.4|17.4|25.0|13.6|8.7|17.4|28.8|15.8|
|$ 0-<br>830<br>Over<br>830|X<br>57.6<br>|51.0<br>|23.1<br>|47.2<br>|66.8<br>|47.0<br>|28.7<br>|52.7|63.5|48.7|26.7|50.6|
|<br> <br>O<br>2000|42.4<br>|49.0<br>|76.9<br>|52.8<br>|33.2<br>|53.0<br>|71.3|47.3|36.5|51.3|73.3|49.4|
|ver<br>|23.2|27.6|60.6|33.8|15.5|37.1|43.6|27.3|18.2|33.0|49.7|29.7|

Notes: All monetary values in the table **are** in June 1985 Intis. **The** exchange rate **vas** $1.00 US = 11 Intis. Standard deviations are in parentheses.
. 3.

|<br>Type**of**<br>**Finr**<br>**Female**<br>Number<br>of<br>**Firms**<br>203<br> <br>|METRO LIK<br>**Kale**<br>98|A<br>**Mized**<br>104|**Total**<br>**405**|OTH<br>**Female**<br>**_375_**|ER URBAN<br>**Kale**<br>132|AREAS<br>Mixed<br>**188**|**Total**<br>693|ALL<br>**Female**<br>576|URBAN A<br>**Male**<br>230|REAS<br>**Mized**<br>292|**Total**<br>1098|
|---|---|---|---|---|---|---|---|---|---|---|---|
|TOTAL CAPITAL<br>$ 2191<br>(7733)<br> <br> <br>|8603<br>(21303)|10523<br>(18407)|5882<br>(15474)|3264<br>(8173)|8175<br>(19969)|9195<br>(13732)|5808<br>(13040)|2886<br>8030|8358<br>20504|9668<br>15541|5836<br>13980|
|Spline 2 Total Capital<br>$ 1693<br>> 1000<br>(7610)|8110<br>(21103)|9683<br>(18320)|5298<br>(15325)|2715<br>(8011)|7555<br>(19819)|8327<br>13649|5159<br>|2355<br>|7792<br>|8810<br>|5210|
|<br> <br> <br>||||||()|(12893)|7880|20333|15457|13833|
|**$**<br>0<br>Total Capital<br>X<br>15.3<br>1-<br>250<br>261|23.5<br>204|0.0<br>106|13.3<br>207|6.4<br>|12.1<br>|2.1<br>|6.3<br>|9.5<br>|17.0|1.4|8.9|
|.<br>250-<br>500<br>138|.<br>112|.<br>67|.<br>114|29.8<br>|19.7<br>|7.4<br>|21.8<br>|28.5|20.0|8.6|21.4|
|.<br>500-<br>2000<br>24.1|.<br>173|.<br>221|.<br>220|14.5<br>198|9.8<br>|4.3<br>|10.8<br>|14.2<br>|10.4|5.1|11.0|
|<br>2000-<br>4000<br>10.8|.<br>5.1|.<br>15.4|.<br>10.6|.<br>113|14.4<br>136|17.0<br>202|18.0<br>141|21.4<br>|15.7<br>|18.8<br>|19.5<br>|
|4000-<br>12000<br>59|82|212|<br>|.<br>|.<br>|.<br>|.|11.1|10.0|18.5|12.8|
|.<br>Over<br>12000<br>3.9<br>|.<br>14.3|.<br>24.0|10.4<br>116|10.7<br>75|13.6<br>167|26.1<br>|15.4<br>|9.0<br>|11.3<br>|24.3<br>|13.6|
|$ 0-<br>1000<br>2<br>655|643|<br>2|.<br>|.<br>|.<br>|22.9<br>|13.4|6.3|15.7|23.3|12.8|
|<br> <br> <br>.<br>Over<br>1000<br>345|.<br>357|7.9<br>721|55.6<br>|60.6<br>|47.0<br>|20.2<br>|47.0<br>|62.3|54.3|22.9|50.2|
|.<br> <br> <br>|.|.|44.4|39.4|53.0|79.8|53.0|37.7|45.7|77.1|49.8|
|TOTAL_CAPITAL_<br>EXCL. STOCK$ 1559<br>(5386)|5750<br>(17125)|7050<br>(14833)|3983<br>(12130)|2004<br>5657|4459<br>|5044<br>|3296<br>|1847<br>|5009|5759|3550|
|<br> <br> <br>||||()|(13907)|(10553)|(9268)|(5563)|(15338)|(12262)|(10416)|
|STOCK (Inventory)<br>$ 632<br>(2874)<br> <br>|2853<br>(8422)|3473<br>(6634)|1899<br>(5835)|1260<br>(3913)|3716<br>(10140)|4151<br>(7048)|2512<br>(6555)|1039<br>(3592)|3349<br>(9436)|3910<br>(6899)|2286<br>(6303)|
|MONTHLY PROFITS<br>$ 858<br>(3475)<br> <br> <br>|1838<br>(4172)|2667<br>(7519)|1560<br>(5021)|595<br>(2198)|2562<br>(19426)|425<br>(8850)|924<br>(9790)|688<br>(2717)|2253<br>(14946)|1224<br>(8456)|1158<br>(8357)|
|Profits>0<br>2<br>84.7<br>|88.8|84.6|85.7|79.1|80.3|73.4|77.8|81.1|83.9|77.4|80.7|
|FAMILY WORKERS<br>1.4<br>(09)|1.2<br>(06)|2.8<br>(13)|1.7<br>11|1.5<br>|1.3<br>|2.7<br>|1.8<br>|1.5|1.3|2.8|1.8|
|.<br> <br> <br>|.|.|(.)|(0.9)|(0.6)|(1.0)|(1.1)|(0.9)|(0.6)|(1.1)|(1.1)|
|1 worker<br>%<br>72.9<br>2 workers<br>167|85.7<br>92|0.0<br>577|57.3<br>254|68.1<br>|75.0<br>|0.0<br>|50.9<br>|69.8|79.6|0.0|53.3|
|.<br>3 workers<br>5.4|.<br>41|.<br>231|.<br>96|19.6<br>70|18.2<br>|56.9<br>|29.4<br>|18.6<br>|14.3<br>|57.2|28.0|
|<br>4 or more workers<br>4.9|.<br>10|.<br>192|.<br>77|.<br>54|6.1<br>|23.9<br>|11.4<br>|6.4<br>|5.2<br>|23.6|10.7|
|<br>|.|.|.|.|0.8|19.1|8.2|5.2|0.9|19.2|8.0|
|Adult Male Workers Only<br>X<br>0.0<br>Male Workers Only<br>00|92.9<br>969|0.0<br>00|22.5<br>235|0.0<br>|81.8<br>|0.0<br>|15.6<br>|0.0<br>|86.5|0.0|18.1|
|<br>.<br>Adult Female Workers Only<br>76.4|.<br>00|.<br>00|.<br>383|0.0<br>735|94.7<br>|0.0<br>|18.0<br>|0.0<br>|95.7<br>|0.0|20.0|
|<br> <br>Female Workers Only<br>89.2|.<br>0.0|.<br>00|.<br>447|.<br>882|0.0<br>00|0.0<br>|39.5<br>|74.5<br>|0.0<br>|0.0|39.1|
|<br>Adult<br>Workers Only<br>76.4|<br>92.9|.<br>70.2|.<br>78.8|.<br>735|.<br>818|0.0<br>691|47.5<br>739|88.5<br>|0.0<br>|0.0<br>|46.4<br>|
|||||.|.|.|.|74.5|86.5|69.5|75.7|
|MONTHLY HOURS OF LABOR<br>200.3<br>(1720)|227.6<br>(1865)|499.5<br>2891|283.7<br>|210.1<br>|227.6<br>|459.0<br>|281.0<br>|206.7|227.6|473.4|282.0|
|.<br>1<br> <br> <br> <br>|.<br>|(.)|(246.4)|(162.7)|(172.3)|(259.2)|(223.3)|(166.0)|(178.1)|(270.4)|(232.0)|
|-<br>100<br>Hours<br>2<br>32.5<br>|25.5|1.0|22.7|28.4|25.8|27|209|299||||
|100-<br>200<br>Hours<br>28.1|235|96|222||<br>|.<br>|.<br>|.<br>|25.7|2.1|21.6|
|<br>200-<br>300<br>Hours<br>15.8|.<br>20.4|.<br>173|.<br>173|27.3<br>206|19.7<br>250|12.8<br>|21.9<br>|27.6<br>|21.3<br>|11.6|22.0|
|300-<br>400<br>Hours<br>11.8<br> <br>|<br>19.4|.<br>16.3|.<br>14.8|.<br>11.5|.<br>18.2|12.8<br>15.4|19.3<br>139|18.9<br>116|23.0<br>187|14.4<br>|18.6<br>|
|400-<br>500<br>Hours<br>4.9|71|125|74|||<br>|.<br>|.<br>|.|15.8|14.2|
|<br>Over<br>500<br>Hours<br>6.9|.<br>4.1|.<br>43.3|.<br>156|5.9<br>62|5.3<br>61|17.6<br>|8.9<br>|5.6<br>|6.1<br>|15.8<br>|8.4|
||||.|.|.|38.8|15.0|6.4|5.2|40.4|15.2|
|Log Monthly Hours<br>4.8406<br>(11192)<br>|5.0002<br>11016|6.0544<br> <br>|5.1909<br>|4.9867<br>|5.0871<br>|5.9532<br>|5.2680<br>|4.9352<br>|5.0501<br>|5.9893<br>|5.2396<br>|
|.<br> <br> <br> <br>|(.)<br>|0.5902)<br>|(1.1265)|(0.9735)|(0.9464)|(0.6361)|(0.9822)|(1.0286)|(1.0140)|(0.6210)|(1.0379)|
|Adult Male/Total<br>Hours 2<br>0.0<br>Adult Female/Total<br>Hours**X**<br>91.6<br>|95.8<br>0.0|48.3<br>46.9|35.6<br>58.0|0.0<br>89.8|91.6<br>0.0|46.3<br>463|30.0<br>609|0.0<br>905|93.4<br>00|47.0<br>|32.1<br>|
|Child HourslTotal<br>Hours X<br>8.4|4.2|4.8|6.4|10.2|8.4|.<br>7.4|.<br>9.1|.<br>9.5|.<br>6.6|46.5<br>6.5|59.8<br>8.1|
TABLE 3. (Coutd)

|**Type**<br>**of**<br>**Ftrm**<br>**Fe**|**M**<br>**male**|**ETRO LIMA**<br>**Male**|<br>**Mixed**|**Total**|**OTE**<br>**Female**|**URBAN**<br>**Kl**|**AREAS**<br>**Mi**||**ALL **<br>|**URBANAR**<br>|**RAS**<br>||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|Number of Firms|203|98|<br>104|<br>405|<br>375|**ae**<br>132|**xed**<br>**188**|**Total**<br>693|**Female**<br>576|**Male**<br>230|**Mixed**<br>292|Total<br>1098|
|EDUCATION<br>|||||||||||||
|<br>(Most Educated Worke<br>Years of Schoolin|r)<br>||||||||||||
|g<br>Sline 0<br>5|6.2<br>(3.8)<br>|7.6<br>(3.6)<br>|9.1<br>(3.3)<br>|7.3<br>(3.8)|5.7<br>(4.0)|7.2<br>(3.9)|8.6<br>(3.6)|6.8<br>(4.1)|5.9<br>(4.0)|7.3<br>(3.8)|8.8<br>3.5|6.9<br>4.0|
|p-<br>years of school<br>Slin 6 10|4.0<br>(1.7)<br>|4.5<br>(1.1)<br>|4.9<br>(0.5)|4.4<br>(1.4)|3.7<br>(1.8)|4.3<br>(1.4)|4.7<br>(0.8)|4.1<br>(1.6)|3.8<br>(1.8)|4.4<br>(1.3)|4.8<br>(0.7)|4.2<br>(1.5)|
|pe - years of school<br>Sline 10+<br>f|1.9<br>(2.2)<br>|2.6<br>(2.3)<br>|3.5<br>(2.1)<br>|2.5<br>(2.3)|1.6<br>(2.2)|2.3<br>(2.3)|3.2<br>(2.2)|2.2<br>(2.3)|1.7<br>(2.2)|2.4<br>(2.3)|3.3<br>(2.2)|2.3<br>(2.3)|
|p <br>years o school<br>Hihest Edti<br>|0.2<br>(1.0)<br>|0.5<br>(1.3)|0.7<br>(1.6)|0.4<br>(1.3)|0.3<br>(1.1)|0.5<br>(1.3)|0.7<br>(1.5)|0.5<br>(1.3)|0.3<br>(1.1)|0.5<br>(1.3)|0.7<br>(1.6)|0.4<br>(1.3)|
|g ucaon<br>Level Comple<br>None<br>%|ted<br>8.9|2.0|00|49|123||||||||
|Primary|42.9|<br>37.8|.<br>23.1|.<br>365|.<br>477|3.8<br>394|0.5<br>287|7.5<br>|11.1<br>|3.0<br>|0.3<br>|6.6|
|Secondary|40.9|44.9|<br>57.7|.<br>462|.<br>311|.<br>409|.<br>|41.0<br>|46.0<br>|38.7<br>|26.7|39.3|
|Post-secondary|7.4|15.3|<br>192|.<br>123|.<br>88|.<br>1|47.9<br>|37.5<br>|34.5<br>|42.6<br>|51.4|40.7|
|**2 OF**FAMILY WORKERS|||.|.|.|5.9|22.9|14.0|8.3|15.7|21.6|13.4|
|WHO:<br>Attended Public Schools|77.6|85.9|807|804|798||||||||
|Have Vocational<br>Training|32.8|<br>27.7|.<br>34.8|.<br>321|.<br>226|87.6<br>135|86.5<br>178|83.1<br>|79.0<br>|86.9<br>|84.4<br>|82.1|
|ArLEAST 1 FAMILY||||.|.|.|.|19.6|26.2|19.6|23.9|24.2|
|WORKER<br>HAS VOCATIONAL<br>TRAINING<br>X<br>AGE OF OLDEST|39.4<br>|28.6<br>|60.6<br>|42.2|27.9|15.2|37.8|28.1|31.9|20.9|45.9|33.3|
|WORKER (Years)<br> <br>**JOB**EXPERIENCE<br>|39.1<br>(11.8)<br> <br>|40.7<br>(15.4)<br>|44.0<br>(11.8)|40.7<br>(12.9)|41.8<br>(13.2)|42.2<br>(15.8)|44.1<br>(12.6)|42.5<br>(13.6)|40.9<br>(12.8)|41.5<br>(15.6)|44.0<br>(12.3)|41.8<br>(13.4)|
|<br>(MOST EX2ERIENC<br>Years|D<br>WORK<br>9.7|ER)<br>14.8|18.6|132|151|194|212||||||
||(9.9)|(13.3)|<br>(113)|.<br>(118)|.<br>(142)|.<br>148|.<br>|17.6<br>|13.2<br>|17.4<br>|20.3|16.0|
||<br>|<br>|.<br>|.|.|(.)|(12.7)|(14.1)|(13.1)|(14.3)|(12.3)|(13.5)|
|Years squared/100|1.9<br>(3.3)|4.0<br>(6.3)|4.7<br>(5.2)|3.1<br>(4.8)|4.3<br>(6.9)|5.9<br>(7.3)|6.1<br>(7.2)|5.1<br>(7.1)|3.5<br>(6.0)|5.1<br>(6.9)|5.6<br>(6.6)|4.4<br>(6.4)|
# 5. **The Empirical Model**

This section reports the specification and estimation of the revenue model for the informal retail sector. In contrast to popular approaches to estimating the properties of production technologies, we introduce no assumptions about the optimizing behavior of agents. One reason for this is that there are seldom welldeveloped markets for the factors employed by these firms and thus no way to construct independent measures of the opportunity cost necessary for optimizing models.

Our work uses the revenue function specified in equation (8). Ultimately, though, the statistical process of specification, estimation, diagnostic analysis, and nonlinearity analysis that we employ is iterative. In our case there were iterations with respect to both model specification and inclusion/exclusion of data points in the sample reflecting the inflow of information from the battery of diagnostic tests to which the model and data were subjected. Since our model is nonlinear in some parameters it was necessary to estimate the extent of this nonlinearity to determine whether the local diagnostic analysis, based on a version of the model linearized about the least squares optimum and other measures of goodness of fit and precision, retained their traditional meaning. It is known, for example, that as the measured degree of model/parameter nonlinearity increases,traditional confidence ellipsoids may become distorted, with the result that traditional measures of (joint) significance of parameters lose their validity.

We describe below the iterations of testing and diagnostic analysis that separate the initial model from the final model for which parameter estimates are reported. The approach led to a model that is extremely robust, provides an excellent fit of the data, and is very close to the initial model in both 28
specification and sample. Two versions of the final model are reported: the difference between the two lies in the measurement of education -- dummy or splined variables. Finally, as a further test of validity, we analyzed the data using nonparametric techniques (see Appendix B). The results are discussed below.

# 5.1 Initial Specification

The initial model was given by: lnRL - **ao +** alLOCTION_HOME **+** a2LOCATION_FIXED **+** a3FIRM_AGE - ln(l + e **T)** where:

- TL - **bo +** bjEXPENSES **+** b2CAPITAL **+** b3STOCK **+** b,ln(LABOR) **+** b 5 TRAINING **+** b 6 SCHOOL_PRIMARY **+** b7 SCHOOL_SECONDARY **+** b 8 SCHOOL_POSTSEC **+** bgEXPERIENCE **+** b2OEXPERIENCE<sup>2</sup> (14)

The model was fitted to three subsamples of the data: Lima, other urban areas, and rural areas. All the first round point estimates of the parameters (obtained by a nonlinear least squares Gauss-Newton algorithm) had the correct signs and between 50 and 60 percent of the variance in the logarithm of revenues was explained.

# 5.2 **Preliminary Tests for Aggregation** and Pooling

The initial results showed similar patterns in the estimated parameters of the Lima and other urban areas models. Some parameters from the rural model were similar to their counterparts for Lima and for other urban areas. These results led to the hypothesis that an aggregate model based on a pooled sample could be estimated. At the same time it was necessary to determine whether we were justified in pooling data among female, male, and mixed enterprises. This question was particularly important given that one of our goals was to estimate 29
women's productivity. Finally,<sup>regardless of the outcome of these initial</sup> hypothesis tests, all the tests<sup>would have to be redone since the model</sup> specification and the data set might change as<sup>a resultofinformation obtained</sup> from the diagnostic analysis<sup>(see below). All hypotheses were tested using the</sup> sample likelihood ratio statistic<sup>compared with its 1 percent critical value.</sup> Table 4 gives the test results.'5 We could not reject the hypothesis of equality of parameters or structure<sup>in Lima and other urban areas. All pooling</sup> that included rural data, however,<sup>was rejected. That is, the structure of rural</sup> firms was found to be different<sup>from that of urban firms. Finally, we could not</sup> reject the hypothesis of equal<sup>parameters in the three classes of enterprises:</sup> male, female, and mixed."6 For these reasons our urban model is estimated on a sample pooled with respect to enterprise<sup>and location (Lima and other urban</sup> - areas). Tables Bl B3 in appendix<sup>B give the results for the rural model.</sup>

**Tabl4.**

**Summarqestifool1iagtructure**

|HYPOTHESIS|TESTSTATISTIC<br>|<br>CRITICAQALUE<br>DECISION|
|---|---|---|
|l.Lima1thdfrbaArea;i&ur<br>%reas<br>calbeaggregated.|<br>83.040|56.123<br>REJECT|
|2.LimaandthdirbaAreas<br>calbeaggregated.|25.041|33.409<br>DO NOT REJECT|
|3.FoiLimandDthd1rbaAreas,<br>Femalftlean&ixegirms<br>caibeaggregated.|52.800|56.123<br>DO NOTREJECT|

# Note.TestwbaseAilikelihabeoatistics.

15 The results correspond to the final<sup>estimated model where education is</sup> represented by dummy variables.<sup>The pattern of results is similar to that</sup> obtained with the initial version<sup>of the model.</sup>

**16** This hypothesis was tested on<sup>the pooled Lima and other urban areas samples.</sup>

30
# 5.3 **Diagnostic Analysis**

Our analysis involved assessing the sensitivity of parameter estimates to what we term "data problems." Although the PLSS is an unusually clean data set, where much effort was devoted to correcting anomalies, it is still open to a variety of impurities due to, inter alia, measurement errors, data entry, and inaccurate reporting by respondents and enumerators. Advances in statistical research have made it possible to implement a set of data diagnostic tests to reveal statistical problems arising from imperfect data or highly influential sets of observations. These tests provide another useful way to assess the reliability of a given model."7 The diagnostic techniques involve searching the data for single observations or sets of observations that differ significantly from the 'average' data point and may have an excessive influence on the regression results. Parameter estimates that are highly dependent on the properties of small subsamples of the data should be treated with caution. For this reason the isolation and careful study of influential observations is an important task in applied modeling. The theory behind these tests for linear models has been developed extensively in the statistics literature. For our nonlinear model, the tests were performed on a version of the model linearized about the nonlinear least squares optimum.

Two possible sources of influential observations are high leverage points and outliers. A high leverage point is an observation for which the vector of independent variables is "far" from the rest of the data. In the leverage analysis, the data were searched for points that were farthest from the center " For an excellent discussion of these issues, see Belsley, Kuh, and Welsch (1980), and Chatterjee and Hadi (1988).

31
of the remaining data. Since leverage points need not be influential points, these observations were iteratively dropped and the model reestimated to see if the points were particularly influential in determining the overall fit. The second test involved plotting residuals against leverage values to identify outliers, that is, those observations where the residuals were large. We isolated observations where the fitted values of the model were farthest from the actual values of the dependent variables. These observations were removed from the data set and the model was reestimated to evaluate their influence on the regression results. This process was repeated several times because the removal cf any one high leverage point can, and generally does, cause a change in the set of high leverage points.

Finally, we examined the plots of the studentized residuals against the independent variables and against the fitted values of the dependent variable. The first set of plots was studied for patterns (for example, positive residuals for large values of the variables) that might indicate that the specification was not robust. The second plots provided information about possible nonlinear relationships in the residuals.

Ultimately the analysis showed that 15 data points were influential. That is, their removal from the sample led to significant changes in the parameter estimates for capital, stock, and expenses. All the influential observations we isolated had extremely large values for capital, stock, or expenses (more than 100,000 intis). We chose not to include these 15 data points in the estimation of subsequent models.

In strdying the stability of the estimated parameters, we were particularly concerned about their sensitivity to restrictions on the capital, stock, and expenses variables. Our analysis involved setting critical values for 32
these variables and dropping observations in excess of these values. When we reestimated the model for successively smaller critical values, we found that the parameters associated with capital, stock, and expenses were quite sensitive to these restrictions, but that other parameters were stable. The parameter estimates for capital, stock, and expenses tended to increase as the independent variables were restricted.

These results suggest that a model with constant coefficients for capital, stock, and expenses might not be appropriate. The coefficients should be given the flexibility to decrease as the variables increase.18 The economic interpretation of these results was that there were variable returns to these factors beyond what the original specification of the model could encompass.

We reestimated the model with piecewise linear splines for capital, stock, and expenses. Because the nonlinear nature of the model combined wich the large sample size made estimation somewhat expensive, we undertook only limited experimentation on determining the knots of the splines. Up to three segments appeared necessary to remove a large amount of the instability of the coefficients.

The parameter instability that remained appeared to involve a trade-off in the values of the parameters of the stock the capital variables similar to a multicollinearity problem. Independent of the choice of knot points, the spline coefficients for the stock variables were never significant and typically had t-statistics less than 0.5. The estimated coefficients were of the wrong sign as well. But if the added spline variables for stock were removed, an unstable 18 We also estimated the model with quadratic terms for capital, stock, and expenses, but the parameter instability remained. Box-Cox transformations were not possible because these variables often had values of zero.

33
but statistically significant coefficient of the correct sign arose for the remaining stock variable.

We resolved this problem by aggregating the stock and capital variables into a single variable called total capital. We reasoned that statistical testing based on the likelihood ratio test provided no clear evidence against the aggregation decision. Whether or not the test rejected or did not reject aggregation depended on the number of spline variables introduced. In no case was aggregation as strongly rejected as the competing hypothesis that the stock variable should simply be dropped from the model. Second, the model with capital and stock aggregated was characterized by stable and significant parameter estimates of the correct sign. Third, prior to aggregation the nonlinearity tests (see below) indicated that the model was highly nonlinear in terms of the curvature properties of the estimated revenue function. After aggregation this problem disappeared.

# 5.4 Analysis of Nonlinearity

The diagnostic analysis is based on the assumption that the underlying regression model is linear in the parameters. Other statistics, such as confidence intervals about the point estimates of the parameters, assume that the model is linear. This linearity assumption is violated in the strict sense, but that there will be a linearized version of the model around the nonlinear least squares optimum. The important question to raise is whether the linearized version of the nonlinear model is accurate over a sufficiently large range of the parameter space so as to include confidence intervals measured in the standard way. Alternatively, the model may be so nonlinear that the linear approximation model becomes unacceptably inaccurate within the range of the traditionally (linearly) measured confidence intervals.

34
Current research in the statistics literature has been aimed at resolving such questions using differential geometry. This branch of mathematics has welldeveloped notions and measures of curvature and nonlinearity. Nonlinearity of statistical models has beer, reduced, in part, to the study of the radius (of curvature) of the largest approximating ball "covered" by the estimated model. Intuitively, the larger the radius of curvature, the better will be the local linear approximation to the model and the more confident one can feel about results based on the linearized model. The total curvature of a model can be decomposed to three parts: one representing the intrinsic curvature of the model (and about which nothing can be done short of respecification) and the other two representing parameter effects curvature (which can, to some degree, be mitigated by reparametrization of the model without distorting the specification). Statistical tests of the extent of curvature relative to the distance of the estimated model from the dependent variable can be performed and the significance of deviations from linearity can be assessed (see Bates and Watts 1980).

Table 5 shows the nonlinearity analysis arising from the last diagnostic iteration. All the parameters of the model fell within acceptable limits for nonlinearity. Those parameters for which the curvature measures were greatest were associated with the spline variables for capital and expenses. These results confirm our finding that it was inappropriate to specify constant coefficients for the aggregate expenses and capital variables. They also suggest that the spline approach leads to a model with a sufficiently accurate linear approximation around the nonlinear least squares optimum. The nonlinearity analysis increased our confidence in the quality of the estimated model and in the validity of applying traditional statistical tests to the model.

35
# TABLE 5.

# ANALYSIS OF NONLINEARIrY

# CurvaMn Type

Pmn etr C omnskd

|Pmn etr C omnskd|Tota|kltrkis2|GeodC s|A<br>oakn|
|---|---|---|---|---|
|CONSTANT2|.036|.020|.014|.027|
|EXPEN SES_1|.286|.119|.180|.187|
|EXPEN SES_2|.385|.172|.216|,269|
|EXPENSES_3|_551_|.271|247|.41'|
|CAPITAL_1|.311|.199|.088|.223|
|CAPIrAL_2|.327|.211|.080|.236|
|LABOR|.023|.006|.015|.016|
|SCHOOL_PRHM4ARY|.022|.013|.010|.015|
|SCHOOL_SECONDARY|.034|.017|.015|.025|
|SCHOOL_POSTSEC|.065|.041|.019|.047|
|TRA_IN ING_|.018|.003|.012|.012|
|EX PER IEN CE<br>|.030|.018|.015|.019|
|EXPERIENCE<sup>2</sup>|.041|.030|.016|.023|

No':

1. Cu r& nbe riould be onpaId tD: lanbda, - F(17,1081; a)f. W bsn a = £5, knbt= .778; whEn a = .01, nbcl= .7C6.

2. Cue atn hnmbc inpv ttdid cmE 36
# 5.5 **Distribution** of the Errors

The specified form of the (logarithmic)<sup>model in (13) contains the</sup> representative error term **v..** The error term<sup>is unobserved but the regression</sup> residuals (that is, the differences between the<sup>dependent variable,</sup><sup>**lnRL,**and</sup> its fitted value, lnR L) provide information<sup>about the distributionofthe error</sup> terms. Using the standard Shapiro-Wilk<sup>test (based upon order-statistics),</sup> we 9 could not reject the hypothesis that<sup>the residuals were normally distributed.'</sup> This in turn suggests that the multiplicative error term for total revenues given by exp[v1I in equation (7) is lognormally<sup>distribt'ted.</sup> There are two implications of these distribution<sup>results. First, because</sup> the error term appears to be normally<sup>distributed in the</sup> garithmic model, the least squares parameter estimates are also maximum<sup>likelihood estimates.</sup> The second point is technical but important<sup>for the simulation analy'As.</sup> In some of the simulation work it is necessary<sup>to construct an estimate of R</sup> (the expected value of revenue for the<sup>**ith**firm). Given that the model is</sup> estimated with the logarithm of revenues<sup>as the dependent variable: lnR-h(x,)</sup> **+** vL, then, becau:e the **v.** appear to<sup>be normally distributed, the appropriate</sup> estimate for RE1 is given by:

## **RE-** exp[h(xL)]exp[o<sup>2</sup> /2]

where CA<sup>2</sup> iS the estimated variance of v1. The second<sup>exponential term is a</sup> scaling factor arising in the transition from<sup>a normally distributed random</sup> variable to one that is lognormally distributed.<sup>In the empirical work we found</sup> that **o 2** was about equal to 0.773 and<sup>thus that the scaling factor was about</sup> 1.47. If this scaling factor were ignored,<sup>the estimate of expected revenue would</sup> be biased downward by approximately 47 percent.

**9** An examination of the normal probability plot of the residuals as well as the shape of the plotted density function for the residuals<sup>confirms this finding.</sup>

**3,**
# **6. Empirical Findings and Interpretation**

We first present regression results for the final models for urban areas and comment on some (ex post) tests for aggregation and pooling. Second, we discuss the factor productivity of labor, expenses, and capital, and explain the distributions of productivity overall and by type of enterprise. Finally, we report some simulation experiments in which we provide loans to selected groups of firms. We use information from the estimated revenue model is to assess the firms' ability to repay these loans. We also consider simulations that change the level of schooling and the number of labor hours available to the firm.

The exact form of the final regression model is given by: where:

- **bo + b1 EXPENSES_1 + b2EXPENSES_2 + b3EXPENSES_3 + b4CAPITAL_1 + bcCAPITAL_2 + b61n(LABOR) + b7TRAINING +** b8SCHOOL_PRIMARY **+ b9SCHOOL_SECONDARY + bloSCHOOL_POSTSEC + b.11 XPERIENCE + b12EXPERIENCE2 (16)**

**The** introduction of spline variables increased the number of expenses **variables to** three and the number of capital variables to two. The capital variable now **includes** the values of both physical capital and stock. Finally, we consider two versions of the model: one where the schooling variables are **measured** as splines, and the other where binary variables are used to distinguish **levels of schooling.**

Tables **6** and 7 present the regression results. The t-statistics suggest that most of the parameters are significantly different from 0 at the 1 percent level. In addition, each model explains about 60 percent of the -'--iation in the **dependent variable.** This is high by cross-section standards.

38
TABLE 6. REGRESSION RESULTS **ALL FIRMS (Binary SchoolLng Variables)**

|CONSTANT1(ao)<br>9.703<br>(57. 818)|
|---|
|LOCATION_HOME<br>-0.360|
|(-5.180)|
|LOCATION_FIXED<br>-0.033|
|(-0.449)|
|FIRM AGE<br>0.006|
|(2.079)|
|CONSTANT2(bo)<br>-5.289<br>(-21. 478)|
|EXPENSES 1<br>1419.004<br>(10. 044)|
|EXPENSES 2<br>-1102. 338|
|(-4.926)|
|EXPENSES_3<br>-82.269|
|(-0.654)|
|CAPITAL_1<br>474.017<br>(5.269)|
|CAPITAL 2<br>-456.014|
|(-5.013)|
|LABOR<br>0.215|
|(6.494)|
|SCHOOL PRIMARY*<br>0.194|
|(1.696)|
|SCHOOL_SECONDARY<br>0.356|
|(2. 759)|
|SCHOOL POSTSEC<br>0.402|
|(2. 659)|
|TRAINING<br>0.113|
|(1.521)|
|EXPERIENCE<br>0.022|
|(3. 025)|
|EXPERIENCE <sup>2 </sup>1100<br>-0.058|
|(-4.049)|
|LLF<br>-1419.360|
|SSR<br>852.963|
|Adjusted R-squared<br>0.61614|
|MeanDependent Variable<br>7.26285<br>St. Dev.<br>1.42960|

# **Notes: t-statLetics are in parenthease.**

* Excluded category is less than prLmary schooling completed.

39
TABLE **7. REGRESSION YtESULTS** (Spline Schooling Variables)

|Type<br>of Ylra<br>Number of**Firms**|Total<br>1098|Female<br>576|Male<br>230|**mixed**<br>292|
|---|---|---|---|---|
|CONSTANTl(ao)|9,720|10.035|9.406|9.835|
||(57.079)|(22.131)|(32.539)|(40.607)|
|LOCATION HOME|-0.363|-0.302|-0.464|-0.431|
||(-5.224)|(-3.329)|(-2.072)|(-3.292)|
|LOCATION_FIXED|-0.030|-0.074|0.176|-0.147|
||(-0.419)|(-0.737)|(0.929)|(-1.123)|
|FIRM AGE|0.006|0.014|0.003|-0.007|
||(2.003)|( 3.179)|( 0.439)|(-0.136)|
|CONSTANT2(bo)|-5.281|-5.338|-5.806|-5.839|
||(-22.362)|(-10.989)|(-9.556)|(-7.458)|
|EXPENSES_1|1422.047|1498.644|1861.097|823.518|
||(10.108)|(8.858)|(4.515)|(2.545)|
|EXPENSES_2|-1105.146|-1072.956|-1987.134|-354.705|
||(-4.962)|(-4.022)|(-2.893)|(-0.736)|
|EXPENSES 3|-88.862|-263.425|505.992|-233.278|
||(-0.713)|(-1.745)|(1.262)|(-0.991)|
|CAPITAL_1|468.985|303.434|522.262|844.269|
||(5.243)|(2.691)|(2.225)|(3.963)|
|CAPITAL_2|-451.513|-269.499|-500.477|-837.998|
||(-4.992)|(-2.332)|(-2.106)|(-3.912)|
|LABOR|0.210|0.160|0.323|0.282|
||(6.361)|(3.977)|(3.911)|(2.598)|
|SCHOOL PRIMARY|0.046|0.051|0.053|0.016|
||(2.007)|(2.015)|(0.719)|(0.165)|
|SCHOOL-SECONDARY|0.041<br>|0.007|0.093|0.087|
||;<sup>2</sup>.<sup>2 </sup>5<sup>8</sup>)|(0.309)|(1.867)|(2.231)|
|SCHOOL POSTSEC|-0.005|0.036|-0.109|-0.010|
||(-0.157)|(0.848)|(-1.376)|(-0.215)|
|TRAINING|0.080|0.057|0.200|0.227|
||(1.078)|(0.615)|(0.844)|(1.580)|
|EXPERIENCE|0.022|0.014|0.032|0.041|
||(3.096)|(1.587)|(1.484)|(2.565)|
|EXPFRIENCE<br>2/100|-0.058|-0.048|-0.065|-0.085|
||(-4.058)|(-2.640)|(-1.516)|(-2.914)|
|LLP|-1416.23|-710.608|-320.041|-356.472|
|SSR|848.114|397.669|217.710|196.461|
|Adjusted R-squared|0.61614|0.57135|0.59712|0.54541|
|**Mean**Dependent Variable|7.26285|6.85879|7.35373|7.98832|
|St. Dev.<br>**Notes:**<br>t-statiatics<br>**are**<br>**i**|1.42960<br>**n**parentheses.|1.28823|1.59263|1.25358|

40
The pattern of signs and relative<sup>magnitudes of the estimated parameters</sup> is quite reasonable. All<sup>levels of schooling and vocational training have</sup> positive impacts on revenues<sup>in the binary variable model. The results suggest</sup> there are diminishing benefits<sup>st the margin for postsecondary schooling. The</sup> spline model tells an identical<sup>story. The fact that the third spline coefficient</sup> is not significantly different<sup>from zero suggests that there is no important</sup> difference (in terms of revenues) between secondary and postsecondary education.

As the sign of the linear<sup>term suggests, there are positive returns to</sup> work experience. While<sup>these returns may initially increase, they ultimately</sup> diminish. The fact that the<sup>quadratic term is negative implies that diminishing</sup> returns set in more quickly.<sup>There appears to be a distinct disadvantage to</sup> operating a business from<sup>the home as opposed to being itinerant. Finally, the</sup> older the firm, the greater<sup>are the estimated revenues.</sup> Labor, expenses, and capital<sup>variables continue the reasonable pattern</sup> of results. The labor parameter<sup>has the correct sign and is highly significant.</sup> With respect to the expenses<sup>parameters, their pattern and relative sizes are</sup> sensible. The two spline<sup>knots for expenses were at 830 and 2,000 intis</sup> respectively. Thus a firm<sup>with less than 830 intis in expenses will have an</sup> expenses coefficient of 1419.004.<sup>This drops to 316.666 for a firm with between</sup> 830 and 2,000 intis of<sup>expenses, and to 234.397 for a firm with more than 2,000</sup> intis of expenses. The<sup>fact that the coefficients decline, however, does not</sup> guarantee that there<sup>is everywhere diminishing marginal productivity for</sup> expenses. Eventually though,<sup>the marginal products will decline. The same is true</sup> for capital. Here the spline<sup>knot is at 1,000 intis. Thus, a firm with less than</sup> 1,000 intis in stock and<sup>capital has a coefficient of 474.017 for aggregate</sup> capital, and when the firm<sup>has aggregate capital in excess of 1,000 intis, the</sup> 41
coefficient is 18.003. There will eventually be diminishing returns to this factor for all firms. Some firms in the sample, though, may be operating in a range of increasing marginal productivity for capital.

The (ex post) tests for aggregation and pooling are identical to the ex ante results. The hypothesis of equal structures in Lima and OUAs could not be rejected. Similarly, the hypothesis that male, female, and mixed enterprises have similar structures could not be rejected for all urban areas.20 An important implication of the last test is that productivity does not differ by type of firm. A female firm with the same factor endowments and other characteristics as a male or mixed firm will have the same revenues, on average, as other businesses.

# 6.1 Factor Productivities

The estimated revenue function provides some insights into the behavior of factor productivity, particularly the marginal revenue products of factors. The derivative of the revenue function with respect to a right-hand side variable is the marginal revenue product of that variable. These derivatives will differ from observation to observation and thus we can determine their distribution separately for all firms, female-only firms, male-only firms, and mixed firms. The derivatives are the product of the (unknown) price and the marginal product of the factor. Despite the fact that the marginal products of these factors remain unobservable, some interesting related results can be obtained for the expected marginal revenue products.

The marginal revenue products can be thought of as shadow input prices. A profit maximizing firm in a competitive setting would attempt to equate the 20 In **all** cases rural firms had significantly different structures. (See Table 4).

42
marginal cost of factors (prices) with corresponding marginal revenue products. The shadow price for a factor is deduced fr-m the marginal revenue product as the amount the firm would be paying to the factor if the firm were being observed at a profit-maximizing equilibrium.21 For labor, expenses, and capital we can make some conjectures about the size of the marginal revenue products. For example, one might expect the marginal revenue product of labor in the informal sector to be less than the wage in the formal sector. With respect -o expenses, we might expect the shadow price to be close to unity. Except for that part of current expenses going to purchase stock,<sup>2 2</sup> we would not expect any firm to spend one more inti on expenses unless it also expected to recoup it in revenues.

For aggregate capital the case is less clear. It is difficult to deduce dynamic effects from a static model. Nonetheless the marginal revenue product of capital, for example, can be thought of as the rate of return to having one more unit (inti) of, say, stock in the firm. This is over and above the price that a firm will get when it sells the unit currently in stock. Table 8 shows the measured marginal revenue products.

> 21 A further restriction for these results to hold is that the marginal revenue product curve is downward sloping at the profit maximizing equilibrium. If this condition is v<sup>4</sup> olated then the firm is not maximizing profits.

22 Unsold goods purchased with current expense money enter the inventories of firms. Our data set does not contain any information about inventory policies.

43
TABLE 8.

# D1STRIBUTDN OF MARGINAL REVENUE PRODUCTS<sup>OF</sup> LABOR (), EXPENSES E) AND CAPIrAL (C)

|_TYPE_**OF**FiM||TOTA|L||FEM A|LE|M|ALE<br>1D|--<br>-||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|DECLE|L|E|K|L|E|K|L|E|K|L|E|K|
|1|.054|490|.021|.057|.468|.019|.057|.492|.019|.049|.601|.096|
|2|.087|.638|.038|.088|.584|.037|.093|.629|.038|.079|.763|.037|
|3|.115|.751|.057|.116|445|.062|.138|.713|.066|.101|.884|.050|
|4|.147|.860|.078|.150|.814|.132|.110|.817|.094|.125|.985|.060|
|5|.190|.992|.116|200|.925|.232|375|.957|.178|.147|1.091|.069|
|6|.245|1.127|.253|.263|1.067|.315|.321|1.067|.266|.185|1 .204|.082|
|7|322|1.300|.391|.360|1.266|.438|.415|1.225|.404|.226|1.352|.100|
|8|.456|1.536|.590|.498|1.494|.603|.645|1470|.848|.306|1.614|.398|
|9|.916|1.945|1.069|1.188|1.882|1.010|1.601|1.765|1.367|.431|1h418|1.183|
|10|67.053|7.247|3550|67.053|4.542|3202|7.835|7247|3.550|6.980|6.396|3.363|
|M EAN|.535|1.170|.385|.646|1.096|.385<br>|.632<br>|1.154<br>|.467<br>|.249<br>|1.327<br>|.322<br>|
|ST DEV|2253|.801|568|3.000|.642|.488|1.152|.927|.676|.497|.944|.614|
|OBS|1098|1098|1098|576|576|576|230|230|230|292|292|292|

A feeling for the results can be<sup>obtained by considering the row of means.</sup> For the total sample of firms,<sup>the mean marginal revenue product for expenses</sup> is 1.170 intis and is slightly<sup>in excess of the value of one inti suggested</sup> above.<sup>23</sup> The mean rate of return to<sup>capital is38.5percent and the shadow wage</sup> for one hour of labor is .535 inti.<sup>Before considering comparative returns in</sup> different types of enterprises,<sup>it is useful to determine the extent to which</sup> the mean values are informative. Table 8 also shows the cumulative<sup>distribution</sup> (by decile) of the marginal revenue<sup>products for each factor and by each type</sup> 23 As discussed earlier, these estimates<sup>are unbiased with respect to the</sup> expected revenue component.

44
of enterprise. It is clear that all of the distributions are highly skewed and that the mean value is typically not encountered until the seventh or eighth decile. Thus for policy purposes the mean is not ar. informative statistic. Moreover, before policy questions can be answered, it is necessary to determine which firms are in the tails of the distributions (high versus low returns at the margin), their characteristics, and the extent to which having a high (or low) return for one variable coincides with a high (low) return for another variable.

We analyzed the distribution of the marginal revenue products of capital, expenses, and labor for the four sample groups: (1) all firms, (2) female-only firms,(3) male-only firms, and (4) mixed firms. We also examined the relationship of the marginal revenue products of factors by firm to the levels of all productive factors. The results are presented in a set of 12 diagrams labelled Cl, C2, C3, C4; El, E2, E3, E4; and Ll, L2, L3, L4. The letter refers the productive factor -- capital, expenses, and labor; the number refers to the relevant subgroup of firms above. Thus Diagram L2 shows the marginal revenue product of labor for women-only firms.

In a representative diagram the information is: In diagram Cl, the relevant sample is all firms, ordered from largest to smallest according to the size of the estimated marginal revenue product of the relevant factor (in this case, capital). Thus the firm with the largest marginal revenue product of capital becomes the first firm and the firm with the smallest marginal revenue product becomes the 1,098 firm. The marginal revenue product of capital is measured on the left-hand vertical axis and, by construction, the plotted values of the marginal revenue product of capital will decrease as higher-numbered firms are encountered. The median firm is the $ 5 4 9 th and for this firm the marginal 45
# DIAGRAM C1

MARGINAL REVENUE<sup>PRODUCT OF CAPITAL</sup> SAMPLE VALUES:<sup>ALL ENTERPRISES</sup> <!-- Start of picture text --> Cf0.0129<br>!~~~~~~~~~~~~~~~~~~~~~~~~~~<br>I 0.0099<br>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C.:C<br>1le 7X j<br>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I<br>~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.0079<br>A* f O.D069<br>/<br>4<br>j<br>:.rr X / _ __- "- 0.0039<br>_o----8zH ~ ~ ~ ~ ~ ~- - O.Ot<br>- - ~~~~~~~~~~~~~~~~~~~~0.0019<br>100 200 300 400 500 600 700 800 900 1000 1100<br>ENTERPRISES ORDERED SY NAP Of CAPITAL<br><!-- End of picture text -->
# DIAGRAM C2

# MARGINAL REV7ENUE PRODUCT OF CAPITAL SAMPLE VALUES: FEMALE ENTERPRISES

<!-- Start of picture text --> o.o1o7<br>. 0.0097<br>0.J0T7<br>!<br>c .<br>o~ i1 - ./ .:os<br>141.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I<br>- j \ X t Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~.0003<br>{.s.3 g / t ~~~~~~~~~~~~~~~~~~~~~~~~0.002t<br>9 I<br>r~~<br>} / . ~~~~~~~~~~~~~~~~~~~~~~~~'-0.0003<br>o 4, _____ - -- - 1-- ' , ' -T-- 0 ' >0.0013<br>O io0 zoo 300 40 500SO<br>EXtERPRIME ORDERED By URP Of CAPITAL<br><!-- End of picture text -->
<!-- Start of picture text --> DIAGiA.iN,/ C3<br>-MARGINAL REVENUE PRODUCT OF CAPITAL<br>SA.NMIPLE VALUES: MALE ENTERPRISES<br>5 J ' ~~~~~~~~~~~~~~~~~~~~~~~~~~~0.0196<br>{ * ' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.020S_<br>0.016<br>l<br>I F~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0<br>0196<br>r 0.016<br>N~~ ~ ~ .. F°0i':^<br>0.0076<br>r 0.0076<br>/0.0106<br>--. | , / L t 0.0106<br>26<br>P0.00<br>0.0106<br>A /t 08^6S0<br>d \0.0076 A~~~~~~~~~~~~~~~~~~~~~ 0 .0016<br>/\ - ~~~~~~~~~~~~~~~~~~~~~~~~~0.0006<br>/ < '0~~~~~~~~~~~~~~~~~~~~~~~0.0004<br>/ \ '*~~~~~~~~~~~~~~~~~~~~~~~~0.0014<br>/ * -~~~~~~~~~~~~~~~~~~~~~~~~~0.0024<br>/ , '-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0. 0034<br>0 100 200 300<br>ENTERPRISES OROERED BY NRP OF CAPITAL<br>84<br><!-- End of picture text -->
# DIAGR AM C4

# MARGINAL REVENUE PRODUCT OF CAPITAL SAMPLE VALUES: MIXED ENTERPRISES

<!-- Start of picture text --> f4 5 f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.0153<br>1 t 0.0133<br>.. .. .. . .. .. .. . .. .. .. . .. .... . .. .. -..... .. .... -... .. ......... ..........- L<br>0.0103<br>A~~~~~~~~~~~~~~~~~<br>l 0 / k:~~~~~~~~~~~~~~~~~~~~~~~~~~.0093<br>.01~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I<br>Pl \/ 0.0073<br>o A t ~~~~~~~~~~~~~~~~~~~~~0.01033<br>Iz X ,,,,-- t~~~~~~~~~~~~~~~~~~~~~~0.0063<br>0.00.3<br>I~ ~~<br>0 -, - - , - - - -, ' T - 0.0023<br>0 loo 200 300<br>ENTERPRISES OROECRED SY KRP OF CAPITAL<br><!-- End of picture text -->
# DIAGRAAM El

# MARGINAL REVENUE<sup>PRODUCT OF EXPENSE'</sup> SAMPLE VALUES:<sup>ALL ENTERPRISES</sup>

<!-- Start of picture text --> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.01&7<br>I<br>0.0107<br>a r , 0.0087 0.0067<br>L0.007<br>0 J<br>-t 3. \<br> j<br>,, I \ t0.0067<br>S [~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.005?<br>0.0027<br>100 200 300 400 500 600 700 800 900 1000 1100<br>EUtERPRISES ORDERED BY MAP Of EXPENSES<br><!-- End of picture text --> <!-- Start of picture text --> 0-<br><!-- End of picture text -->
# DIAGRAM E2

# MARGINAL REVENUE<sup>PRODUCT OF EXPENSES</sup> SAMPLE VALUES:<sup>FEMALE ENTERPRISES</sup>

<!-- Start of picture text --> l g<br>*1<br>r 0.0102<br>~~~~~~~* 1 [~~~~~~~~~~~~~~~~~~~~~~~~~~'0.006<br>.1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~ ~~~~~0.012<br>3 ',''''"'- L ,°°°;0.002<br>L o.oc<br>r 1<br>I.00<br>IIITERPRISES ORDERED BYtXRP OF EXPENSES<br>.~~~~~~~~~~~~~~~~~~~~~~.0<br>L~~<br>1.5<br><!-- End of picture text -->
# DIAGRAM E3

# MARGINAL REVENUE PRODUCT OF EXPENSE.' SAMPLE VALUES: MALE ENTERPRISES

<!-- Start of picture text --> s 0.00.9<br><!-- End of picture text --> <!-- Start of picture text --> i0 | ,,,,,, [ 0.0ca<br>t 11 ; s. ~~~~~~~~~~~~~~~~~~~~~~~~~ \ ~~~0.0073 .O °°<br>t3.n \<br>:1 0.003<br>O - / v ' 0 .0008.0033<br>s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br>0 ~~~~~~~~~~~~ ~0,0.0008<br>0 100 200 300<br>ENTERPRISES OROERED BY NRP Of EXPENSES<br><!-- End of picture text -->
<!-- Start of picture text --> DIAGRAM E4<br>MARGINAL REVENUE PRODUCT OF EXPENSES<br>SAMPLE VALUES: MIXED ENTERPRISES<br>.... ~ f' ..<br>.0 1 2 2<br>E0<br>of r 0.0112<br>. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / 0.0102<br>0.0092<br>^ r c O . D~~~~~~~~~~~~~~~~~~~~~~~~~~092<br>Oo<br>/<br>w I<br>F , 0.0022<br>//<br>x J \<br>0.0052<br>3 ,,<br>/ 0.0002<br>30.o03z<br>200.00<br>la0<br>O<br>ORDERED OY XRP Of EXPENSES<br>EUIERPRISES<br>S 3<br><!-- End of picture text --> w I
# DIAGRAM Ll

<!-- Start of picture text --> MIARGINAL REVENUE PRODUCT OF LABOUR<br>SAMPLE VALUES: ALL ENTERPRISES<br>3.0 i al S 1 0.0130<br>, 0.0129<br>I ~~~~~~~~~~~~~~~~~~~~~ ~~~~~~0.0109<br>F°.°1°<br>1 . ..<br>Ml - 0.00o9<br>1 \ ...<br>_ < t~~~~~~~~~~~~~~~~~~~~~~~~~~ 0.00L9<br>0 100 200 300 400 Soo 600 700 goo 900 1000 1 t00<br>ENTERPRISES OROERED er PtP oF EABOUJR<br><!-- End of picture text -->
# DIAGRAM L2

MARGINAL REVENUE PRODUCT OF LABOUR SAIPLE VALUES: FEMALE ENTERPRISES <!-- Start of picture text --> .0 1L<br>V O 0<br>-<br>}_0 _ wR =_ C Si ~~~~~~~~~~~~~~~~~~~O.OQ3<br>-Ce- -e - --------<br>L ,o<br>aU~=<br>==<br>Ltoo<br>tS,R9RIStS 0OERtt as gP OF I,D6BW<br><!-- End of picture text -->
# DIAGRAM L3

# MARGINAL REVENUE PRODUCT OF LABOUR SAMPLE VALUES: MALE ENTERPRISES

<!-- Start of picture text --> O .013Z<br>0.0102<br>J~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br>F-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ J~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br>p<br>N I I / i. 0 7<br>a * K' 0.00S<br>0.0 _- 0<br>EE<br>0.0032<br>0 100 200 300<br>ENTERPRISES ORDERED BY KRP Of LABOWR<br><!-- End of picture text --> <!-- Start of picture text --> ('(9<br><!-- End of picture text -->
# DIAGRAM L4

# MARGINAL REVENUE PRODUCT OF LABOUR SAMPLE VALUES: MIXED ENTERPRISES

<!-- Start of picture text --> I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I<br>' ,,,,,,_,,,.-J-"''''"''''''''" ~~~~~~~~~~~~~00122<br>0.0t22<br>A~~~~~~~~<br>0 1.5 . , 0112<br>p.~~~~~~~~~~~~~~~~~~~~~~~~~0<br>.L o 0.0052<br>N '-/ 0.002<br>0 1~~~~~~~~.0<br>.00,~~~~~~~~~~~~~~~~~~~~~~~~~~008<br>/// ~~~~~~~~~~~~~~~~~~~~~~~0.0032<br>0.0022<br>0 . _ 100 200 300<br>ENTERPRISES ORDERED BY HRP OF LA8LUR<br>'7<br><!-- End of picture text -->
revenue product is about .15 inti. Notice how quickly the marginal revenue product of capital by firm declines as we consider the first 200 firms.

Next consider the relationship of the marginal revenue products to the levels of the productive factors. Capital, expenses, and labor variables are all measured on the right-hand vertical axis. The variables were scaled to all fit on these axes. The lines C, E and L represent the least squares regression lines of the capital, expenses, and labor variables (respectively) of firm i on the rank (1 to 1,098) of firm i. The rank of a firm continues to be determined by the size of its marginal revenue product. An asterisk in the label of a regression line indicates that the slope of the regression line is significantly different from zero.

These regression lines provide a considerable amount of additional information that explains the distributions of the marginal revenue products. In diagram Cl we see that firms with high marginal revenue products of capital also had significantly lower endowments of capital. They also tended to have significantly lower quantities of expenses and labor. Extending the analysis to capital input for female enterprises (diagram C2), we see that firms with high marginal revenue products of capital had significantly lower endowments of capital. There is, however, no significant relationship between the distribution of the marginal revenue products and either expenses or labor.

Two patterns can be isolated in the marginal revenue product diagrams. First, for all groups of firms and for all productive factors there is a significant relationship between the marginal revenue product of a factor and the amount of the factor. Firms with smaller amounts of a given factor tend to have higher marginal revenue products for the factor. Second, in about 75 percent of the cases where a significant relationship exists, there is an inverse 58
relationship between the marginal revenue product of one factor and the quantities of the other factors. This suggests that small firms are the most productive in terms of measured marginal revenue products.

These results have important implications for the competitive structure of the informal urban retail sector. The sector's rapid growth may well signal a greater degree of competitiveness in retail trade, and large incumbent firms may be feeling the pressure of both inefficient size and falling (real) prices. Leaner and more aggressive small firms appear to be making significant inroads. Some supporting evidence for this view comes from examining the location structure of firms in the upper and lower 20 percent tails. Firms with low returns to labor and capital are much more likely to operate out of homes and fixed locations while those with high returns are mobile. For the capital variable, firms in the upper tail are twice as likely to be mobile as firms in the lower tail. And female-only firms tend to have lower capital, labor, and expenses than male-only and mixed firms. It appears that women are good candidates for the high rates of return obtained by smaller firms. It may be, however, that these returns at the margin can be further increased. Female firms may be operating where marginal revenue product curves have a positive slope. In this case growth would tend to make these firms even more productive.

# 6.2 **Nonparametric Evidence**

These estimates are based on parametric methods. The results below are based on a nonparametric approach (described in appendix B). That estimates marginal revenue products and revenue without having to specify (parametric) model. In the nonparametric analysis all the variables appearing in the parametric model were included except dummy variables. Since the nonparametric 59
technique is robust to data transformations, the capital and expenses variables were not transformed as splines. Graphs Bi to B6 show that the order of magnitude and the direction of the estimates from the two approaches are consistent. These results support the validity of the parametric specification. Diagrams B1 and B2 show plots of the fitted values of the dependent variable against the sample values. The parametric model (Bi) on average does not over- or underestimate. The nonparametric model (B2) tends to overestimate for smaller values of the dependent variable and underestimate for larger values. At first the cloud of points lies mostly on the 45-degree line initially and then drops below it. The mean value of the dependent variable is about 7.2.

A plot of the residuals front the two models (B3) shows that they to move in the same direction. The residuals are plotted as pairs for the same enterprise. Positive (negative) errors in the parametric model coincided with positive (negative) errors in the nonparametric model. This suggests that the approaches estimated similar relationships.

Diagrams B4, B5, and B6 show plots, by enterprise, of marginal revenue products from the two model. Using the 45-degree line as a reference point, the marginal revenue product of expenses tends to be higher in the parametric model, while those of labor tend to be lower. The marginal revenue product for capital in the nonparametric model is initially greater and then smaller than those in the parametric model.

These findings come from a first application of the nonparametric technique. Nonetheless we conclude that the directions and orders of magnitudes of the parametric model estimates are supported.

60
# DIAGRAM Bi

# PARAMETRIC FIT

<!-- Start of picture text --> 10 ++ +<br>+ +++<br>E9']<br>F ~ ~++4 . +<br>T<br>ET *+ + ++<br>D 7<br>+ *+~~~~~~~<br>V + ++: :~+<br>A<br>L 6 ++ 4t 4 +** + ~<br>s + ~+~ F+<br>4++p<br>T 44 5 6 9 1<br>1 8~ ~ ~~~6<br>3<br>3 4 5 6 7 8 10<br>ACTUAL VALUES<br><!-- End of picture text -->
# DIAGRAM B2

# NONPARAMETRIC FIT

<!-- Start of picture text --> T + ++<br>T +<br>D 7 *<br>' 8<br>F + +t<br>6 + + ? 8 9 10<br>4<br>3<br>3 a 9 10<br>ACTUAL VALUES<br><!-- End of picture text -->
# DIAGRAM B3

# COMPARISON OF MODELS RESIDUALS

<!-- Start of picture text --> 3<br>N +<br>r:1 N +' + + se + + + +<br>A +<br>A , + + ++<br>E + + +<br>T +<br>R<br>L + C +IA +<br>D 6<br>c + a+$ + 4.<br>A<br>-2 ~ ~1 0 1 23<br>PARAMETRIC RESIDUALS<br><!-- End of picture text -->
# DIAGRAM B4

# COMPARISON OF MODELS

# EXPENSE MARGINAL PRODUCTS

<!-- Start of picture text --> 2.0 ' +<br>N ~~~~~~~~~++<br>m + +<br>0 +<br>N 1.8-+ +/<br>p + +<br>A +<br>R + + +*4<br>A + + + .+<br>E 1.4 + + +<br>T ++<br>R + + ++<br>1 1.2 + + +<br>C + +<br>M1.0 + +<br>0 .+ +<br>0 + + +<br>E 0.8 ++++<br>L *~~~~~+ +* + +<br>V 0.6 ++ ++<br>U 0.4<br>+<br>0.2 +<br>0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0<br>PARAM4ETRIC MOOEL VALUES<br><!-- End of picture text -->
# DIAGRAM B5

# COMPARISON OF MODELS LABOUR MARGINAL PRODUCTS

<!-- Start of picture text --> N 1.2 + + +<br>0 + ++<br>N + + + ++<br>p ++<br>A 1.0 +T +<br>00 +F . + +<br>A 4-;+ + 4+ + + +<br>N +~~~++ + +~-1++ ++<br>R 0.0 +~~-* 0 . ,4 + + + *++f + 0<br>I 4 ~ 1+ ~ 4 yi- +* + +<br>C ++R Y +<br>0.66<br>0<br>0 +<br>0. + + ++<br>0V +<br>A<br>L<br>U +<br>+~~~~~~~<br>0.0<br>0.0 0.2 0.4 0.6 0.8 1.0 1.2<br>PARAMETRIC MODEL VALUES<br><!-- End of picture text -->
# DIAGRAM B6

# COMPARISON OF MODELS CAPITAL MARGINAL PRODUCTS

<!-- Start of picture text -->
1.2<br>N<br>a  +<br>N.<br>p<br>A  1.0<br>A  A  +/ .<br>N  ~+<br>E  +  +<br>+  ~~~~~~~~+<br>C  +  /  +<br>N 0.6<br>+<br>E  ~+<br>D  /<br>0.4<br>V  +X<br>UD  /<br>SE0.2<br>0.0  0.2  0.4  0.6  0.8  1.0  1.2<br>PARAMETRIC MWEL VALUES<br><!-- End of picture text -->

_b_
# 6.3 **Simulation Experiments**

The estimated revenue model can address a variety of other questions. For example, **it** can assess the impact of policies that affect Lhe amount, quality, and distribution of productive factors such as loans (either unrestricted<sup>or</sup> **factor-specific)** to a given subgroup. Evaluating such a credit policy<sup>would</sup> invclve estimating the effects on expected revenues and determining the payback **period for** the loan. Policymakers can also assess the revenue impact of social **policies** that raise education levels or allow family members to substitute work **in** the family business for housework (perhaps through child care programs).

Table 9 shows the effects on expected revenues, averaged over firms, of seven different policy experiments. The first four experiments are 'simple' in that only one underlying variable is changed. The other four experiments are 'compound' in that more than one variable changes. For example, in the<sup>first</sup> experiment, the most educat-d individual in the firm is given two more years of **schooling if** the number of years of schooling is less than nine. The<sup>group of</sup> firms receiving this education changes from all firms, to the bottom 50 percent (in terms of education), and then to the bot om 25 percent (in terms of **education) and** finally, for comparison purposes, to the top 25 percent.<sup>Within</sup> **each group,** the effects on all, female firms, male firms, and mixed firms are **calculated** separately. The effect of formal schooling on revenues is 100 to 200 **intis a** month.

The second simulation increases the amount of labor by 100<sup>hours a month.</sup> **This could be** accomplished by policies that reduce the costs<sup>of noncompliance,</sup> **encourage bulk** purchasing by cooperatives, and offer training in management. **The time devoted** to business activities by women could be<sup>increased by providing</sup> 67
Table 9. SIMULATION RESULTS - EFFECTS ON REVENUES

|INCREASE|IN<br>BY<br>|I<br>(1|)<br>|1<br>(2|)<br>|1<br>|(3)<br>|1<br>(|4)<br>|1<br>(|5)<br>|1<br>|(6<br>|1<br>|17<br>l|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|Years of|Schooling||2||||||||||||2|
|Monthly<br>H|ours||||100||||||||100||100|
|Expenses|S||||||1000||||500||500||500|
|Total Cap|ital<br>$||||.I||||1000||500||500||500|
|_||<br>INobs|-I<br>S|<br>Nobs|S|Nobs|S|.<br>Nobs|S|Nobs|S|Nobs|S|Nobs|_S_|
|**Glven to**|AllFLrmsb<br>**Y**|||||||||||||||
|All En|terprises<br>|11098|103|1098|187|1098|1079|1098|<br>3141|1098|843|1098|1100|1098|1245|
|Female|Enterprises|576|97|576|182|<br>576|988|576|<br>323|576|828|576|1101|576|<br>1250|
|Male|Enterprises|230|112|230|237|<br>230|1136|230|<br>399|230|907|<br>230|1255|230|1377|
|Mixed|Enterprises|292|109|292|157|<br>292|1213|292|<br>231|292|820|292|1001|292|<br>1130|
|GLven to|Bottom 50 Z of Firms|||||||||||||||
|All E|nterprises|528|157|546|239|<br>549|1279|549|<br>578|755|872|845|1001|926|1039|
|Female|Enterprises|343|126|366|218|<br>361|1102|358|<br>495|471|810|514|977|538|1057|
|Male|Enterprises|100|175|127|299|<br>111|1394|125|<br>690|165|908|195|1030|209|1074|
|Mixed|<br>Enterprises|85|2601|53|243|<br>77|1944|66|<br>816|119|10701|136|1052|179|943|
|Given to|Bottom 251of Flrms|||I||l||||||||||
|All En|terprises|242|1441|274|284|<br>274|1409|274|641|446|6691|541|737|637|695|
|Female|Enterprises|180|119||197|257|<br>163|1176|182|588|281|5671|347|650|411|6151|
|Male|Enterprises|40|199|67|363|<br>71|1449|72|671|109|7291|131|843|145|8321|
|Mixed|Enterprises|22|248|10|267|<br>40|2285|20|1011|56|1068|63|998|81|853|
|Givento|Top<br>25 2 of Firms||||||||||I||||I|
|All En|terprises|146|42|274|115|274|978|274|58|421|3341|522|330|577|309||
|Female|Enterprises|47|32|75|104|81|973|82|49|138|296|181|270|210|239|
|Male|Enterprises|36|40|27|114|61|988|59|58|96|328|108|321|121|2991|
|Mixed|Enterprises|63|<br>50|172|120|132|<br>975[|133|<br>641|187|3641|233|3821|246|<br>3741|

Notes: The bottom 50X refers to those firms that have less than the median value of the variables) of interest for all firms. For example, if $1000 of expenses is provided, then only those firms with less than the median value of expenses ($806) receive the $1000. The same applies to the bottom 25X and the top 252. The median and quartile values are:

||Median|Bnttom 25X|Top 252|
|---|---|---|---|
|Years of Schooling|<br>7.0|4.0|10.0|
|Monthly Hours|234.0|113.6|386.3|
|Expenses|$ 805.6|247.2|2478.3|
|Total Capital|$ 967.7|170.5|4337.8|

cooperative child care and facilities for meals. This experiment suggests that revenues will increase by a respectable 200 to 300 intis a month, especially among firms in the bottom 50 and 25 percent (in terms of labor).

The third simulation lends 1,000 intis to selected firms for operating expenses. The results are impressive. If all firms receive the loan, on average 68
they will be able to repay<sup>the loan, retaining 79 intis as net profit for the</sup> month. Alternatively, because<sup>they are more productive, when only the lowest</sup> quartile (in terms of expenses)<sup>receive the loans, the average net profit is 409</sup> intis. The small number (40 firms)<sup>of mixed enterprises in this quartile would</sup> be the most successful, with an average net profit of 1,285<sup>intis for the month,</sup> compared to 176 intis for the<sup>163 female businesses, and 449 intis for the 71</sup> male firms.

When interpreting the fourth experiment,<sup>(each firm receives 1,000 intis</sup> for capital expansion), the fact<sup>that the average return is less than 1,000 intis</sup> does not mean the policies are not<sup>successful.</sup> The funds would be used to acquire capital that would not completely depreciate<sup>in a month. In fact, the ratio of</sup> 1,000 intis to the average monthly<sup>increase in revenues is the average payback</sup> period for the loan in months assuming<sup>zero economic depreciation.</sup> For example, all enterprises would take just over three months<sup>(1,000/314)</sup> on average to repay the loan. It should be noted, however,<sup>that firms with low capital endowments,</sup> mostly female firms, tend to perform<sup>better than those with high endowments. Two-</sup> thirds of the firms in the lowest<sup>quartile are female businesses, with an</sup> estimated payback period of less than<sup>two months (1,000/588).</sup> The payback period for firms in the top quartile is<sup>about 18 months (1,000/58).</sup> The remaining 'compound' experiments<sup>(five, six, and seven)largely</sup> conifirm the 'simple' ones. The effects on predicted<sup>revenues of firms with lower</sup> endowments of expenses, capital,<sup>and labor are generally higher than three of</sup> better-endowed firms.

Finally, in comparing the simulated<sup>increases in revenues it is important</sup> to examine more than just the<sup>average increases. There are disttibutions</sup> associated with these point estimates.<sup>Standard statistical testing shows few</sup> 69
significant differences among the three types of firms, but this was not the case among firms in different quartiles.

The simulation results in Table 9 are quite similar to those obtained in the analysis of factor productivity discussed earlier. These results are very helpful in terms of quantifying the effects of policies for the informal sector. It appears a policy of loans to poorly endowed firms, most of which are female businesses, could offer significant social gains.

# 7. Policy Implications

The extent to which estimated productivity differs among male, female, and mixed firms appears to be due more to the distribution of factors across enterprises than to any inherent advantage or disadvantage in the type of firm. On average, the productivity of female firms is neither better nor worse than that of male or mixed firms with the same factor endowments (capital, expenses, and labor). The influence of these endowments, however, affects productivity more than the gender or education of the entrepreneurs. Firms with smaller endowments tended to have higher factor productivity. Thus the process of retailing is the same among firms, but the endowment effect makes them different. Put another way, firms with less capital, expenses, or labor usually have higher returns at the margin but typically lower revenues. As seen in Table 3 or Table 9, a high proportion of female firms, but a much smaller proportion of male and mixed firms, have low amounts of capital, expenses, and labor. In our sample about a fourth of the businesses had little or no capital and a third carried no inventories -- most of these were female firms. The empirical analysis suggests that proactive policies would increase substantially the productivity of small 70
businesses. Providing assistance<sup>to small businesses makes good economic sense-</sup> - particularly if directed to those<sup>with low factor endowments.</sup> The ultimate objective of policy measures<sup>is to alleviate poverty and</sup> improve household welfare. Improving<sup>productivity in the informal sector helps</sup> to achieve this goal. While we do not<sup>elaborate on implementation,</sup> two directions will be effective. First enterprise-specific policies that directly affect productivity, and second, set of<sup>policies aimed at women that address the</sup> competing demands on their time.

# 7.1 Enterprise-Specific Policies

Governments and international donors<sup>have primarily assisted small</sup> businesses by providing low-cost credit<sup>to firms that do not have access to</sup> formal financial markets. Loans<sup>are small--usually less than $500--but there</sup> are many beneficiaries. Results are good and the repayment<sup>rate is often more</sup> than 90 percent. Our estimates of<sup>productivity confirm that channeling credit</sup> to small businesses is effective in raising<sup>productivity. Examples of different</sup> credit programs include the Grameen<sup>Bank in Bangladesh,</sup> the Self-Employed Women's Association in India, the Banco Popular<sup>in Costa Rica, the Badan Kredit Kecamatan</sup> in Indonesia, the Northeast Union of Assistance<sup>to Small Businesses in Brazil,</sup> the Institute for the Development of<sup>the Informal Sector in Peru, Fondo de</sup> Fomento para la Pequena Industria y la<sup>Artesania in Ecuador, and the Small</sup> Projects Program of the Inter-American Development Bank.<sup>24</sup> Another way to help small businesses<sup>is by promoting cooperatives and</sup> self-help associations. In addition to<sup>providing credit these organizations can</sup> 24 See Cornia (1987), Everett and Savara (1986), Herz (1989), Hossain (1987), IDB (1988), Tendler (1983), United<sup>Nations (1985), and World Bank (1989).</sup> 71
make bulk purchases from wholesalers, provide storage facilities for perishable or nonperishable commodities, and establish markets for entrepreneurs (see Bunster and Chaney 1985, de Soto 1989).

Technical assistance programs are a third option, offering short-term instruction in basic management, including keeping records, marketing, purchasing, and dealing with municipal authorities and other formal sector institutions. Inter-American Development Bank (1988) notes that instruction can be provided informally at neighborhood markets or at the business site, Most family businesses in Peru are clandestine operations that generally avoid complying with regulations. The performance o2 these businesses could be -mproved by reducing the costs of non compliance. This would mean eliminating various forms of harassment by local authorities, and simplifying the process of conforming to the laws (for example, by making it easier and less costly to obtain a business license).2 5 7.2 Women-Specific Policies Although enterprise-specific policies can improve the productivity, they may have undesirable side-effects in those instances where women operate or work in the business. Most of these women also have household responsibilities and children, end inevitably these obligations entail conflicting demands.

Enterprise-specific policies may inadvertently encourage women to devote more time to the firm, leaving them with less time for activities related to health, nutrition, education, and child care. Hence these policies may have negative effects on household welfare in general and child welfare in particular (Cornia 1987, Jolly and Cornia 1984).

> 25 For discussions see Bunster and Chaney (1985), de Soto (1989), Mescher (1985), Tinker (1987), and World Bank (1987).

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These negative effects can be mitigated by a complementary set of womenspecific policies directed toward reducing the workload associated with home activities. These may enhance the firm's productivity by allowing women to devote more time to the business. Cornia (1987) discusses such programs in Latin America. Coopetative child care centers ("clubes de madres"), facilities for the preparation of food ("comodores populares"), and neighborhood facilities for basic health care will improve the firm's productivity.

This study suggests that proactive policies and projects to help small firms will result in social gains, contribute to the alleviation of economic hardship, and enable disadvantaged -- especially women -- to become effective agents in the development process.

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# **Appendix A: Nonparametric Analysis**

A nonparametric approach provides an alternative to the parametric model presented in the paper. The specification of functional forms is not required in nonparametric modeling, while in parametric modeling a functional form is explicitly defined. Also, under certain regularity conditions, the nonparametric estimates are consistent. This contrasts with the least squares approach where any misspecification of the functional form, however minor, will lead to inconsistent parameter estimates. It is not possible to say that nonparametric methods are more or less powerful than parametric methods -- the technique may provide a different explanation of the data. When parametric and nonparametric models provide similar results, it is reasonable to hold a stronger belief in the validity of the parametric specification.

## **Nonparametric** Modeling

Consider the model:

**Yi -g(X) +** ui i **=** l. ,n

where YL is the **ith** observation of the dependent variable Y and **X.** is the **ith** row of the matrix of explanatory variables, X. There are k explanatory variables and n observations on each dependent and independent variable. The unobserved error term for the **ith** observation is u.

The parametric approach to estimating this model involves specifying a functional form for **go** and then applying a least squares or maximum likelihood algorithm to obtain point estimates of the parameters. The results of such an approach were presented in the paper.

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The nonparametric approach to estimat..ng (17) starts with the observation that the model could be written equivalently as: Y- g(XL) + U-

> - E(YJIXd) + UL

where g(X) - E(Yi1Xi) is the expectation of Y1 conditional upon **X..** This is also called the conditional mean of Y. The researcher estimates the model in (17) or (18) and its properties by estimating the conditional mean and how the conditional mean depends upon the conditioning varirbles. Technioues exist for estimating the conditional mean without specifying a particular functional form for **go.** Of course, different distribution and regularity assumptions are introduced. In particular, the independent variables (drawings from which form the conditioning matrix X) are typically assumed, but not required, to be Jointly distributed random variables. As well, continuity and differentiability assumptions are often implicit in the estimation of the conditional mean function.

The following is a brief description of the nonparametric approach. The reader is referred to Ullah (1988) for a more complete -discussion of the development and use of nonparametric techniques in economics. **Defining Conditional** Means and Their Properties Suppose that the vector of random variables (Y, X) has a joint density function given formally by:

**dProbl(Y,X)** < (y,x)]

- f(y,x) (19)

# **d(y,x)**

The marginal density function for X is defined by:

> **fl(x)** .ff(y,x)dy (20)

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Finally, the conditional density function of Y given X is defined by:

- f(ylx) V.x) fl(Z')

Using these definitions, the conditional mean introduced in (18) is given by :

- E(ylx) fyf(ylx)dy

In a similar fashion, the vector of changes or responses of the conditional mean to a change in the vector of realizations of X. is given by: The parametric analogue of the vector of derivatives of the conditional mean with respect to the realizations of the explanatory variables is the vector of derivatives of the dependent variable with respect to the independent variables. In terms of our revenue model, they are the marginal revenue products of the factors.

Estimating Conditional Means and Derivatives Equations (22) and (23) show that the only information required to estimate the conditional means and the derivatives is the joint density function f(y,x). Once this is available, the marginal density can be computed (as in (20)) and all the right hand side of (22) is known. As (21) shows, the derivative on the right hand side of (23) can be computed and the integral can then be evaluated.

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There are many ways to estimate a multivariate density function. For example it is possible to generalize the one-dimensional technique for obtaining histograms. However, the resulting density estimate will not be differentiable. More popular techniques involve kernel estimators. Formally, the estimate of the multivariate density f(w) = f(y,x), where W. = (Yt,X 1 ) is given by: A n - - f(w) (1/nhq)E **K((W1** w)/h) (24) **i-1** where: h is the window width, n is the number of observations, q is the number of elements in any W. (equal to the number of explanatory variables, k, plus one for the dependent variable) and K() is the kernel function. The kernel function is typically taken from the set of many times differentiable multivariate density'functions. One popular kernel function is the multivariate normal or Gaussian density. Note, however, that the choice of a Gaussian kernel in no way restricts the estimated density. Indeed, it would appear that density estimates are much less sensitive to the kernel function than they are to the window width, h. The choice of the window width determines, in part, the degree of smoothing of the densicy estimate. Recent work by Racine (1989) has examined the optimal window width, h, in the sense of integrated mean square error. The results for the Peru Living Standards Survey data make use of this optimal window width and a Gaussian kernel.

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