*Pages 1--23 from Microsoft Word - 22680.doc*
 On the Substitutability of Local Newspaper, Radio, and Television Advertising in Local Business Sales
 C. Anthony Bush, Ph. D. Chief Economist
 Office of General Counsel
1
 2
 Executive Summary
 A current policy issue is whether there is a single local media market or several distinct local markets for newspaper, radio, and television advertising. An important step in
 defining a market is the estimation of elasticities of demand. In this study we present a model of local business behavior in purchasing advertising for use in sales activities.
 Using reported data on local radio and television advertising revenue contained in the 2001 BIAs Master Access Database and using data from the Newspaper Association of
 America on retail ad expenditures, estimates of elasticities of substitution, ordinary own- and cross- price elasticities are derived for a representative local business establishment.
 The estimated elasticities of substitution and the estimated ordinary cross- price elasticities suggest weak substitutability between local media. In addition, the ordinary
 demand elasticity for a retail ad in a newspaper is approximately unity and negative. Demands for both local radio and local television ads are inelastic. Finally, the data
 support the specified model of local business advertising. An important caveat to these results is that varying degrees of measurement error are associated with local radio
 revenue and revenue from retail ads in newspapers. Due to these limitations inherent in the underlying data, the results of this study cannot be considered conclusive. The
 results, however, are consistent with economic theory.
2
 3
 Introduction
 Section 202 (h) of the Telecommunications Act of 1996 directs the Federal
 Communications Commission to review its broadcast ownership rules every two years in
 light of competitive developments. In conducting such reviews, the Commission may
 need to consider the extent to which established local media compete with each other for
 advertising dollars. The specific policy concern is whether there is a single local media
 market or several distinct local markets for newspaper, radio, and television adverting.
 This study is intended to contribute to that evaluation.

 An important step in defining a market is the estimation of elasticities of demand.
 Therefore, in this study, we estimate elasticities of substitution for local newspaper, local
 radio, and local television advertising media. Ordinary own- and cross- price elasticities
 of demand are also estimated.

 We note that this study evaluates the extent of substitutability of the three largest
 local media  television, radio, and daily newspapers. We do not attempt to address
 whether cable television or other advertising vehicles such as direct mail and outdoor
 advertising compete with local television, radio and daily newspapers for local
 advertising dollars.

 Literature
 The literature on inter- media substitutability in advertising can be divided into
 two broad categories. First, there is national advertising that is associated with national
 sales of product/ services. Second, there is local media advertising by local businesses.
 Several authors have studied national advertising. McCullough and Waldon (1998)
3
 4
 examine the substitutability of network advertising and national spot advertising. They
 assume an aggregate sales function and rely on duality in order to specify the share
 equations of a translog cost function. Three share equations are considered: (1) the share
 of network advertising, (2) the share of national spot television advertising, and (3) and
 the share of a composite of all other advertising forms. Using McCann- Erickson which
 spans the years 1960 to 1994, the authors estimate Morishima elasticities of substitution.
 McCullough and Waldon find that television network advertising and national spot
 television advertising are substitutes.
 Silk, Klein, and Berndt (2001) use a translog model in their evaluation of inter-media
 substitutability. In addition, their analysis permits for the complementary use of
 media in national sales. Their translog model represents the advertising costs of national
 sales. The share equations of the model are: (1) direct mail, magazines, newspapers, spot
 radio, network television, network radio, spot television, and outdoor. The parameters of
 the model are estimated based on national data for the period covering 1960 to 1994.
 Estimates of own- and cross- price elasticities for the various media are presented. Silk,
 Klein, and Berndt conclude that demands for the various media are inelastic. In addition,
 they suggest that weak cross- media effects are consistent with industry practices in which
 media planning is conducted as a multistage decision process. In this process inter- media
 choices are made primarily on strategic and creative grounds. That is initial media buys
 are likely to be a mix of media. Price- sensitivity is shown in subsequent intra- media
 comparisons.
 Sheldon, Jewell, and OBrien (2000) investigate media substitutability and
 economies of scale in advertising. These authors use a translog cost function to model
4
 5
 advertising cost of firms that manufacture beer. The inputs to the sale of beer are print,
 television, and radio. Both the translog function and media share equations are
 simultaneously estimated. Their data are on U. S. beer firms. The data are quarterly and
 cover the period 1983 to 1993. Morishima elasticities of substitution and price
 elasticities of demand are estimated. Sheldon, Jewell, and OBrien find that all
 advertising media are substitutes in the selling of beer. In addition, evidence of
 diseconomies of scale in advertising is presented.
 There is some research on inter- media substitutability in local advertising markets.
 Ekelund (1999) et al. evaluate whether a market for radio services by advertisers exists
 independently of other local advertising markets. Their method for identifying a radio
 market is based on the DOJ Merger Guidelines. Their approach consists of determining
 whether conditions exit such that a hypothetical monopolist could profitably and
 optimally raise price. To this end, Ekelund compares a calculated Lerner Index for radio
 with the reciprocal of the estimated own price elasticity of demand for radio. The issue of
 media substitutability arises in their specification of the functional from of demand. The
 authors estimated a double- log model with the log of revenue as a dependent variable.
 Bona fide explanatory variables are the prices of radio, television, newspaper, and total
 retail sales.
 The unit of observation in this cross sectional study is an Arbitron market from
 BIAs Master Access Database. This BIA database also provided information on total
 radio revenues or expenditures by firms on radio in an Arbitron Market. All data are from
 the year 1995. The price of radio and television are taken from the SQAD database. The
 prices for both radio and television are both cost per rating point ( CPP). Newspaper
5
 6
 advertising prices are based on the price of a one- inch, black- and- white, one day
 advertisement in [Arbitron] markets. These price data are from the Newspaper
 Advertising Source.
 Ekelund et al present estimates of own- price and cross- price elasticities.
 Comparing measures of operating margins, e. g., cash flow, and estimated own- prices
 elasticity of- 2.101, these authors concluded that the radio market constitutes an antitrust
 market. On the issue of inter- media substitutability, demand cross- price elasticities are
 found to be significant at the ten percent level.
 In a separate paper, Ekelund and the same co- authors conduct an analogous
 analysis of television. Again, the 1995 BIA Master Access Database is used to identify
 revenues from Designated Market Areas ( DMA). Both television and radio ad price is
 the cost per rating point from SQAD. A double- log specification is used, where the log of
 television advertising revenue or expenditure is the dependent variable. Explanatory
 variables are the log of retail sales, the price of a television ad, the price of a radio ad, and
 the price of a one- inch, black- and- white, one day advertisement in DMA. Own- and
 cross- price elasticities are estimated. Both the cross- price elasticities of demand for radio
 and newspaper are positive, and the radio cross- price elasticity is positive at a five
 percent level of significance. However, in this analysis their calculated Lerner Index and
 the reciprocal of their estimated elasticity does not suggest that a hypothetical monopolist
 could profitably and optimally increase advertising prices.
 The work of Ekelund et al. is limited in its generality for several reasons. Due to
 data limitations Ekelund could not differentiate between national radio (television) buys
 and local radio (television) buys within an Arbitron (DMA) market. As Silk et al. (2001)
6
 7
 argue, aggregating national and local market demand for media may hide different
 patterns and levels of inter- media substitutability/ complementarity among more finely
 disaggregated components. In addition, our examination of the Newspaper Advertising
 Source suggests that the explanation of newspaper ad price is incomplete. The authors
 do not provide the details on which newspapers are included in the geographic area of
 analysis. Also details on the aggregation of newspaper prices into a single newspaper
 price are not explained. The Newspaper Advertising Source data suggest that numerous
 newspapers could have been included in the construction of the composite price for local
 newspapers. Alternatively, a single newspapers price could have been used.

 Theoretical Framework
 Our focus is on the use of media by local businesses. Following Silk et al. (2001)
 we hypothesize that inter- media buys by local firms are made on strategic grounds and
 that a mix of media is used in advertising to local consumers.
 We began by developing the derived demand for broadcast television, radio, and
 newspaper ads in a local market. Suppose that representative firm A provides
 product/ service z in a local market. For simplicity of analysis, we assume that the
 geographic market for product z or area of geographic distribution of z is the same as
 the geographic market for television, radio, and newspaper ads. That is the geographic
 market for z , the local television ad market, the local radio ad market, and the local
 newspaper ad market all coincide. In these local media markets, firm A is a price taker,
 and, given media ad prices and the firms media budget, firm A employs these local
 media to sell z .
7
 8
 We assume that all goods/ services markets are competitive and that firms are
 Bertrand competitors. Entrepreneurs of firms, e. g., firm A, regard sales, however, as
 random variables. Sales of firms are random because consumers face uncertainty. Given
 perfect information on prices and no transportation costs, consumers must also decide
 where to spend. Consumers choices of firms to patronize are made in accordance with
 consumer convenience in daily routine and with the degree of uncertainty in daily events.
 We assume that all firms advertise under the belief that advertising reduces
 variability in sales and increases expected sales by reminding consumers of the existence
 of the firm. In this model newspaper, radio, and television are complementarily
 substitutable. Firm A uses a mix of media in its outreach to the targeted demographic
 group of potential buyers. If firm A buys television, it also buys some radio and some
 newspaper. As relative media prices change or as the effectiveness of a particular
 medium becomes know, firm A substitutes to a particular medium without necessarily
 abandoning the remaining media.
 Following Veblen (1919), capital and labor employed in advertising (marketing)
 and in manufacturing are embedded in the constant return to scale production function of
 the firm. The unit of output, from the production of good z , is a salable good with
 specific characteristics, including shape, color, material composition, and consumer
 outreach through advertising (marketing) per unit for the period. Since there is constant
 returns to scale in the production of good z , there is a constant unit (marginal) cost,
8
 10
 Empirical Specification and Estimation Technique
 Lesers Transformation of the Linear Expenditure System ( LES) 4 permits us to
 directly estimate elasticities of substitution. Lesers transformation of the LES is

 i ijt o ij it vt y
 _
   +   = ji , where

 o i it it it x p v y  =

 ) ( o i it o j jt o j o i ijt p p p p v w  = 

 jt o j t p x m vt
 4 _

 j=1   =

 An observation, representative firm in a local market, is denoted byt . The price of
 medium i is it p , and expenditure on medium i it isv . The sample mean of expenditures

 on medium i o i isv . The sample mean values of the price of medium i and total
 expenditures are o i p o A o B andm = , respectively. Let o i x be the mean quantity of medium i ,

 and o i o i o i p v x = . The average budget share at this coordinate set is ( o o i o i o i m x p w = ). The
 parameter o ij  is the elasticity of substitution between medium i and j . The parameter i 
 is the marginal expenditure share. In this system, homogeneity, symmetry, and adding up
 restrictions are imposed. Three equations are simultaneously estimated for this derived
 demand system:

 4 Ibid., 62- 66.
10
 11
 Due to singularity of the covariance matrix, which is a result of the adding- up
 restriction, we estimate using the econometric technique of Dhrymes (1984), (1987) and
 (1994) 5 . In this procedure, different variables can appear in different equations, and it not
 necessary to drop an equation from the system. After estimation, own price elasticities
 are calculated at sample means.

 Data
 Revenue data are from the 2001 BIA Master Access Database. Revenues are
 reported for television by DMA. Radio revenues are listed by Arbitron Radio Markets
 which are contained in DMAs. At the time of the work of Ekelund et al. the BIA data
 contained an unknown mix of national and local advertising revenue for broadcast media.
 The 2001 BIA database contains, however, local advertising revenues for both television
 and radio broadcasting 6 .
 We took a random sample of DMAs. The sample contains 45 DMAs, and the
 DMAs are listed in Appendix A. We assume that local radio revenues of Arbitron
 Markets within a DMA are total local radio revenues for the DMA. Spot Quotation and
 Data, Inc. ( SQAD) data are the source of both local radio and television prices. Both
 radio and television prices are in units of cost per point ( CPP) 7 .
 Newspaper data are gathered and constructed from several sources. First, the
 Newspaper Association of America reports categories of newspaper advertising
 expenditures/ revenue for the year 2000. The categories are: (1) national, (2) retail, and (3)

 5 The 1994 paper is the published version of the 1984 working paper.
 6 The share of local revenue is reported for each DMA. Local television revenue is the product of the local

 share and total revenue. Shares of local revenue out of total revenue are provided for each radio market. Local radio revenue is the product of the local share and total radio revenue within the radio market.
11
 12
 classified advertising expenditures 8 . Retail advertising expenditures of $21,409 million
 are allocated across the entire universe of DMAs. Thus, retail advertising expenditure for
 a DMA is the product of $21,409 million and the allocation factor for that DMA. The
 allocation factor for a DMA is the ratio of the population over 16 years of age in a DMA
 to the population of the United States that is over 16 years of age 9 . Second, newspaper
 retail prices are from the Newspaper Advertising Source. Following Ekelund et al.
 (1999), price is the charge for a one- inch, black- and- white, one- day advertisement i. e., a
 Standard Advertising Unit ( SAU). We used Monday through Friday- Saturday prices.
 For every DMA of our sample, we manually collected these price data on all identifiable
 newspapers. Since the number of newspapers varies by DMA, our measure of local retail
 price for a DMA is the mean price for newspapers in the DMA 10 .
 Our theory predicts the use of local advertising by a local business in the sale of
 its goods and/ or services. This implies that we must translate local media revenues in a
 DMA into media expenditures of a representative local business in the DMA. For a given
 DMA, newspaper retail revenue is divided by business establishments in that DMA, and
 the result is local newspaper expenditures by a representative local business 11 .
 Expenditure on local radio advertising by a representative local business is the result of
 dividing local radio advertising revenue by business establishments in that DMA. The

 7 SQAD reports CPP prices for both radio and television by quarter, and we use 4 th quarter 2000 prices.
 Late news rates were used for television and evening rates were used for radio. The rates are for Adults 18+.
 8 Newspaper Association of America at website http:// www. naa. org/ artpage. cfm? AID= 1566& SID= 1002.
 9 The source of the population data is Census 2000, U. S. Census Bureau. Census 2000 population data are

 mapped into DMAs. 10 Newspaper prices are presented by month. We used prices for mid- 4th quarter (November) 2000.
 11 Business establishments in a county are from County Business Patterns, U. S. Census Bureau (1997). We
 mapped county business establishments into DMAs.
12
 13
 expenditure of a representative local business on local television advertising is the local
 television revenue in the DMA divided by the business establishments in the DMA.
 Two dummy variables are created to control for competitive effects that are
 associated with the size or rank of a DMA. A dummy variable is assigned the value of
 one when a DMAs rank is less than or equal to 10 (top 10 DMAs), and, the variable is
 zero, otherwise. A second dummy variable is assigned the value of one when the rank of
 the DMA is between 11 and 50, and, the variable is zero, otherwise.

 Results
 Economic theory predicts that elasticities of substitution can vary from zero to
 infinity. For our function of expected sales, an elasticity of zero would indicate that there
 is no substitutability between two media since an elasticity of substitution 12 is a measure
 of pair- wise substitutability. Alternatively, an elasticity of substitution of infinity
 indicates that two media are perfect substitutes, e. g., television is a perfect substitute for
 radio.
 Our results suggest weak substitutability between local media in the sales
 activities of local businesses. Specifically, with respect to the three media pairs we
 studied, we find: First, the elasticity of substitution between newspaper retail ads and
 local radio ads is 1.16936. 13 Compared to the theoretical limit of infinity for perfect
 substitutability this number is very small but statistically significant. Second, the
 elasticity of substitution between newspaper retail ads and local television ads is 0.91459.

 12 Elasticities of substitution are technical parameters that reveal the shape or curvature of the expected
 sales function. 13 We assumed symmetry, and therefore, the elasticity of substitution between local radio and newspapers

 retail ads is also 1. 16936.
13
 14
 This number is also relatively small but statistically significant. Third, the elasticity of
 substitution between local radio and local television is 0.3094, and this elasticity is not
 statistically different from zero. These results are presented in Table 1 of Chart A.
 Ordinary own- and cross- price elasticities are calculated at sample means. The
 results are presented in Table 2 of Chart A. Ordinary own- price elasticities of demand are
 negative. The ordinary own- price elasticity for a retail ad in a newspaper is -1.0406.
 Demands for both local radio advertising and local television advertising are inelastic.
 Local radios own- price elasticity is -0.8245, and the own- price elasticity for television is
 -0.7960. Moreover, ordinary- cross price elasticities of demand also suggest weak
 substitutability and suggest that several media are gross complements 14 . The cross price
 elasticity between newspaper retail ads and local radio ads is 0.0178, while the cross-price
 elasticity between local radio and newspaper retail ads is 0.0981. The cross- price
 elasticities for newspaper retail ads and local television ads are negative. In addition, the
 cross- price elasticities for television and radio are negative. All ordinary cross- price
 elasticities are presented in Table 2 of Chart A.
 There are certain limitations associated with this study as a result of the
 underlying data that informs our study. Specifically, the study contains varying degrees
 of measurement errors on certain variables. First, local radio revenue within a DMA is
 incomplete because BIA does not report revenue data for all radio stations but only for
 Arbitron markets, which generally are the larger markets. Second, newspaper retail
 expenditure in a DMA results from an allocation process that assigns retail revenues to
 DMAs based on a DMAs share of population 16 years or older. This methodology is

 14 For the Linear Expenditure System all goods should a priori be gross complements. The data support the
 LES specification.
14
 15
 used because there does not appear to be a source for newspaper retail revenues by DMA.
 In addition, the mapping of Census demographic data into DMAs involves some
 judgment because some counties are contained in multiple DMAs. Moreover, the reach
 of some newspapers may extend beyond the assigned DMA, and business establishments
 outside the DMA could have spent advertising resources on newspaper ads which are
 assigned to the DMA. Nonetheless, acknowledging the limitations of the data, the results
 do not appear unreasonable or inconsistent with economic theory.

 Summary and Conclusion
 This study examines the substitutability of local newspaper, radio, and television
 advertising in the sales activities of local businesses. A local business within a DMA is
 hypothesized to maximize expected sales by selecting an optimal mix of newspaper retail
 ads, local radio ads, and local television ads. The expected sales function for goods
 and/ or services of the local business is a scalar multiple of the Stone- Geary utility
 function, and the resulting derived demand for local media is the Linear Expenditure
 System. Lesers transformation of the LES permits the direct and linear estimation of
 elasticities of substitution. Since the LES gives rise to a singular covariance matrix, the
 econometric technique of Dhrymes (1984) and (1994) is used to estimating this singular
 system of equations. A random sample of 45 DMAs is used in the study. Local radio and
 local television data are taken from the BIA Master Access Database. Newspaper retail
 revenues for DMAs are constructed from data published by the Newspaper Association
 of America.
15
 16
 Estimates of elasticities of substitution, ordinary own- and cross- price elasticities
 are derived for a representative local business establishment. The estimated elasticities of
 substitution and the estimated ordinary cross- price elasticities suggest weak
 substitutability between local media. The elasticities of substitution are relatively small,
 and the elasticity of substitution between radio and television is not statistically different
 from zero. The ordinary cross- price elasticities for newspaper retail ads and local radio
 ads are small but positive. The ordinary cross- price elasticities for newspaper retail ads
 and local television ads are negative. In addition, the ordinary cross- price elasticities for
 local radio ads and local television ads are negative. This suggests that local newspaper
 and television ads are complementary inputs in the sales efforts of local businesses. These
 results also suggest that local radio and television ads are also complementary inputs.
 The following caveat must be acknowledged when considering this study in
 communications policy. There are limitations inherent in the underlying data. For
 example, local radio ad expenditures are not total expenditures on radio within a DMA
 because total local radio revenue is not reported. In addition, local newspaper ad
 expenditures are constructed through an allocation process that introduces some degree of
 measurement error. The allocation process used in this paper assigns retail newspaper
 advertising expenditures in each DMA based on a DMAs share of population 16 years or
 older. The estimated elasticities are not, however, inconsistent with economic theory and
 do not appear unreasonable.
16
 17
 References
 BIAs Master Access Database, 2001
 Bush, Clarence A. Generalized Inverse Estimation of Partial Elasticities of Substitution: A Complete Demand System with Exclusion Restrictions and Autoregressive
 Errors. 1994. Working Paper No. 384. Rochester Center for Economic Research, University of Rochester, Rochester, New York.

 Dhrymes, Phoebus J. and Samuel Schwarz. On the Existence of Generalized Inverse Estimators in Singular Systems of Equations. Journal of Forecasting 6 (1987):
 181- 192.
 Dhrymes, Phoebus J. Autoregressive Errors in Singular Systems of Equations. Discussion paper no. 257, Department of Economics, Columbia University, New York. 1984.

 Dhrymes, Phoebus J. Autoregressive Errors in Singular Systems of Equations. Econometric Theory 10 (1994): 254- 285.
 Ekelund, Robert B. Jr., George Ford, and John D. Jackson. Is Radio Advertising a Distinct Local Market? An empirical analysis. Review of Industrial Organization
 14 (1999): 239- 256.
 Ekelund, Robert B. Jr., George Ford, and John D. Jackson. Are Local TV Markets Separate Markets? International Journal of the Economics of Business 7 (2000):
 79- 97.
 Leser, C. E. V. The Pattern of Australian Demand. Economic Record 38 (August 1958): 212- 222.

 Newspaper Association of America. See website http:// www. naa. org/ artpage. cfm? AID= 1556& SID= 1022
 Powell, Alan A. Empirical Analytics of Demand Systems. Massachusetts: Lexington Books, 1974.
 Seldon, Barry J., R. Todd Jewell, Daniel M. Obrien. Media Substitution and Economies of Scale in Advertising. International Journal of Industrial Organization 18
 (2000): 1153- 1180.
 Silk, Alvin, Lisa Klein, and Ernst R. Berndt. Intermedia Substitutability and Market Demand by National Advertisers. Working paper 8624. National Bureau of
 Economic Research, December 2001.
 United States Census Bureau. County Business Patterns, 1997.
17
 18
 United States Census Bureau. Census 2000.
 Veblen, Thorstein. The Vested Interests and the State of Industrial Arts. New York, B. W. Huebsch, 1919.
18
 19
 Chart A
 Advertising in the Generation of Local Business Sales

 Table 1
 Marginal Elasticity Elasticity
 Expenditure of of Dummy Dummy
 Share Substitution Substitution TOP DMA R- SQ
 Local Radio Television 10 DMAs 11- 50
 Newspaper : Retail Ad 0.48494 1.16936 0.91459 44.69234 -286.4512 0.9683
 (0. 01698) (0. 0984) (0. 0644) (417.597) (156.519)
 Radio Advertising 0. 19168 0.3094 282.5515 103.5607 0.9744
 (0. 011508) (. 3087) (266.2748) (98.5364)
 Television Advertising 0. 32337 -327.2439 182.8851
 (0. 01246) (300.0201) (109.2786) 0.9864

 Table 2
 Local Advertising: Ordinary Own- and Cross- Price Elasticities
 Newspaper Radio Television
 Newspaper -1.04062 0.01780 -0. 05823
 Radio 0. 09805 -0. 82450 -0. 22436
 Television -0. 00440 -0. 12399 -0. 79601

 *Standard errors in parentheses
19
 20
 Appendix A: Random s ample of DMAs DMA DMA
 OBS MARKET OBS MARKET 1 Baton Rouge, LA 26 Meridian, MS
 2 Beaumont- Port Arthur, TX 27 Miami - Ft. Lauderdale, FL 3 Billings , MT 28 Nas hville, TN
 4 Bois e, ID 29 New York, NY 5 Boston, MA 30 Oklahoma City, OK
 6 Burlington, VT- Plattsburgh, NY 31 Omaha, NE 7 Casper- Riverton, WY 32 Phoenix, AZ
 8 Chattanooga, TN 33 Pittsburgh, PA 9 Des Moines- Ames, IA 34 Portland- Auburn, ME
 10 Detroit, MI 35 Rapid City, SD 11 Dulut h, MN- Superior, WI 36 Reno, NV
 12 El Pas o, TX 37 Roches t er, NY 13 Ft . Myers -Naples , FL 38 Salt Lake Cit y, UT
 14 Harlingen- Weslaco- McAllen- Brownsville, TX 39 San Diego, CA 15 Houston, TX 40 Springfield- Holyoke, MA
 16 Indianapolis, IN 41 Syracuse, NY 17 Kans as City, KS- MO 42 Terre Haute, IN
 18 Lafayette, IN 43 West Palm Beach- Ft. Pierce, FL 19 Lansing, MI 44 Wheeling, WV- Steubenville, OH
 20 Las Vegas, NV 45 Wichita - Hutchinson, KS 21 Lexingt on, KY
 22 Little Rock- Pine Bluff, AR 23 Louis ville, KY
 24 Madison, WI 25 Medford- Klamath Falls, OR
20
 23
 by Lemma 1.
 Since o ji o ij   = ,

  _
 V y i in o in ij o ij i    +  + +  + = _

 V y j jn o jn ji o ji j    +  + +  + =
 

 which is system (1). q. e. d
23