metadata: authors: 0: author: Eden, Maya
1: author: Kraay, Aart
count: World
docna: 0: docna: "Crowding in" and the returns to
            government investment in low-income countries
docty: Policy Research Working Paper
theme: Analysis of economic growth,Debt management and fiscal sustainability,Macroeconomic management,Other economic management
majtheme: Economic management
subtopic: Macro-Fiscal Policy,Consumption,Fiscal & Monetary Policy,Taxation & Subsidies,Economic Adjustment and Lending,Social Analysis,Quality of Life & Leisure,National Governance,Government Policies,Public Sector Economics,Infrastructure Regulation,Infrastructure Finance,Youth and Governance,Infrastructure Economics,Public Finance Decentralization and
            Poverty Reduction,Infrastructure and Law
teratopic: Macroeconomics and Economic Growth,Finance and Financial Sector Development,Private Sector Development
abstracts: cdata!: This paper estimates the effect of
            government investment on private investment in a sample of
            39 low-income countries. Fluctuations in a predetermined
            component of disbursements on loans from official creditors
            to developing country governments are used as an instrument
            for fluctuations in public investment. The analysis finds
            evidence of "crowding in": an extra dollar of
            government investment raises private investment by roughly
            two dollars, and output by 1.5 dollars. To understand the
            implications for the return to public investment, a CES
            production function with public and private capital as
            inputs is calibrated. For most countries in the sample, the
            returns to government investment exceed the world interest
            rate. However, for some countries that already have high
            government investment rates, the return to further
            investment is below the world interest rate.
repnme: repnme: "Crowding in" and the returns to
            government investment in low-income countries
display_title: "Crowding in" and the
            returns to government investment in low-income countries
listing_relative_url: /research/2014/02/19005742/crowding-returns-government-investment-low-income-countries
topicv3: Infrastructure and Growth m1326244 1083,Political Economy of Reform m1326263 836,Macroeconomic and Structural Policies
            m1327380 439,Taxation m1327615 1121,Population, Aging and Health m1327736 640
publishtoextweb_dt: 2014-02-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Finance and Financial Sector Development,Private Sector Development
no_of_pages: 35
model_extractions: WPS6781 Policy Research Working Paper 6781 “Crowding in” and the Returns to Government Investment in Low-Income Countries _Maya Eden Aart Kraay_ The World Bank Development Research Group Macroeconomics and Growth Team February 2014
Policy Research Working Paper 6781

# **Abstract**

This paper estimates the effect of government investment on private investment in a sample of 39 low-income countries. Fluctuations in a predetermined component of disbursements on loans from official creditors to developing country governments are used as an instrument for fluctuations in public investment. The analysis finds evidence of “crowding in": an extra dollar of government investment raises private investment by roughly two dollars, and output by 1.5 dollars. To understand the implications for the return to public investment, a CES production function with public and private capital as inputs is calibrated. For most countries in the sample, the returns to government investment exceed the world interest rate. However, for some countries that already have high government investment rates, the return to further investment is below the world interest rate.

This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at meden@worldbank.org and akraay@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# “Crowding in” and the Returns to Government Investment in Low-Income Countries<sup>_∗_</sup>

Maya Eden Aart Kraay World Bank World Bank February 17, 2014

**JEL Classification:** E22, E23, H11

**Keywords:** Public investment, crowding out, crowding in, complementarity between government and private capital

> _∗_ We would like to thank the World Bank’s Ethiopia country team for supporting this work, and particularly Lars Moller for helpful discussions. We also thank Luis Serven for helpful comments. This paper reflects our own views and not necessarily those of the World Bank, its Executive Directors or the countries they represent. Please send comments to meden@worldbank.org and akraay@worldbank.org.
# **1 Introduction**

This paper studies the effects of government investment on private investment in developing economies, and draws implications for the optimal rate of government investment. We use fluctuations in past approvals of loans from official creditors to developing country governments to isolate a predetermined source of variation in government investment, and use this as an instrument to estimate the response of private investment to government investment. The analysis finds evidence of “crowding in”: an additional dollar of government investment causes private investment to increase by around 2 dollars, and output by around 1.5 dollars.

The increase in private investment in response to government investment is not, however, a good indicator of the desirability of government investment, which instead depends on the return to public capital. We use a constant elasticity of substitution (CES) production function, with government and private capital inputs, to illustrate the relationship between the return to government capital and the extent of crowding in. Not surprisingly, this relationship depends critically on the elasticity of substitution between the two types of capital. Consider for example the case of extreme complementarity between government and private capital, where the two are required in fixed proportions to produce output. In this case, we should expect to see very strong crowding in of private investment as private capital responds equiproportionately to an increase in government capital. If, however, there are diminishing returns to aggregate capital, this tells us nothing about whether the level of government capital is too high or too low, i.e. whether more government investment is desirable. If, on the other hand, government capital and private capital are substitutes, public investment might “crowd out” private investment even though the return to public capital is high.

We show how our estimates of the effects of government investment on private investment and output can be used to calibrate a production function with a constant elasticity of substitution between government and private 1
capital. Our calibrations suggest a strong degree of complementarity between government and private capital. Nevertheless, we find that the return to government investment in most countries in our sample is well above the world interest rate, with excess returns between 5-170%. However, for some countries where public investment is high relative to private investment, we find negative excess returns to public investment.

Our paper contributes to a large literature on the optimal scale of public investment. A fundamental challenge in this literature is the difficulty in directly measuring the rate of return on government investments. Many government investment projects are aimed at providing public goods, which are provided free of charge. Absent a price mechanism, measuring the direct economic benefits of these public goods is a challenge. The difficulty in measuring the returns to government investment at the micro level has directed researchers towards a structural macroeconomic approach, that aims to identify the returns to government investment through the equilibrium dynamics associated with government investment shocks. Mostly, the emphasis has been on studying the effects of government investment on private investment.<sup>1</sup> The conceptual framework motivating this focus was introduced by Aschauer [1989], who considers a closed economy model in which government capital increases total factor productivity (TFP).<sup>2</sup> In the long run, gains from government investment should be reflected in changes in the marginal return to private capital, and hence in changes in private investment. _Within this_

> 1A complementary line of work attempts to quantify the social and economic benefits associated with certain types of capital stocks that are often (but not exclusively) publicly provided, such as infrastructure. For example, Calderon et al. [2011] estimate a positive relationship between infrastructure and output using a panel time-series approach; Calderon and Serven [2004] use a similar methodology to establish a positive relationship between infrastructure and growth and infrastructure and equality; Leduc and Wilson [2013] study the short and long term effects of infrastructure investment in the US, and find a positive short-run relationship but no long run effects.

> 2See also Baxter and King [1993] who consider a similar specification in which the production function is Cobb-Douglas in public and private capital.

2
_conceptual framework_ , the extent to which government investment “crowds in” private investment in the long run is a sufficient statistic to back out the optimal scope of government investment.<sup>3</sup> The empirical literature that followed focused on trying to quantify the equilibrium responses of private investment to changes in government investment. The literature has mostly relied on time-series and cross-country correlations, finding mixed results. In the context of the US, Aschauer [1989] (and a large literature that followed) uses time series estimates of a positive correlation between public and private investment to infer high returns to government investment. However, these estimates remain highly controversial due to plausible endogeneity issues, summarized in Gramlich [1994]. In later work, Leduc and Wilson [2013] consider the effects of exogenous shocks to federal highway spending across US states, and find short-run effects on GDP but no long-run effects. In the context of developing economies, Cavallo and Daude [2011] use panel analysis to explore the empirical correlations between government and private investment, and find a predominantly negative relationship. Using data from India, Serven [1996] finds that, while infrastructure investment crowds in private capital in the long run, other types of government investment (that compete more directly with the private sector) crowd out long run private investment.

There are two shortcomings of this literature which we seek to address in this paper. First, at the conceptual level, most of this literature has failed to take into account the role of the elasticity of substitution between public and private capital when using estimates of crowding in to draw conclusions regarding the desirability of public investment.<sup>4</sup> As noted above, a large

> 3In this model, “crowding out” can only result from short-run dynamics, as agents try to inter-temporally smooth the increase in their future income by dissaving.

> 4One notable exception is Fisher and Turnovsky [1998], whose theoretical contribution notes that the equilibrium response of private capital to public investment depends on the elasticity of substitution between government and private capital, as well as congestion effects and distortionary taxation, from which we abstract here. We are unaware of any empirical studies that attempt to calibrate the elasticity of substitution between

3
crowding in effect may result either from high complementarity between government and private capital, or from high marginal returns to government investment. For a given amount of “crowding in”, under-estimating the degree of complementarity will overestimate the return to government investment.

We illustrate this point more precisely in the following sections, where we assume a production function in which output is produced using labor and a CES aggregate of government and private capital. We use this framework to show the relationship between the extent of crowding in and the returns to government capital, and how this relationship depends on the elasticity of substitution between the two types of capital. In addition, we show how to retrieve the elasticity of substitution from the observed responses of output and private investment to increases in government investment, which in turn allows us to specify the relationship between the extent of crowding in and the optimality of government investment.

The second difficulty is empirical, and is well-understood in the existing literature. Estimating the equilibrium relationship between government and private investment is complicated by unobservable circumstances that affect both government and private investment decisions. For example, a positive correlation between the two might be due to factors that raise the returns to both private and government investment, such as low world interest rates, low oil prices, or even high domestic productivity. On the other hand, a negative correlation may be reflective of countercyclical government spending, with government investment increasing during periods where private investment is low. In either case, the observed simple correlations are not convincing evidence of a causal effect of government investment on private investment.

To address this difficulty, we use a novel instrumental variable approach introduced in Kraay [2012] and Kraay [2013] to estimate of the relationship between government and private investment, in the context of developgovernment and private capital, as we do here.

4
ing economies. Specifically, we exploit the fact that (a) loans from official creditors (such as the World Bank, and other multilateral and bilateral aid agencies) finance a significant fraction of government spending, particularly in the low-income countries we study, and (b) disbursements of these loans, and the spending they finance, are spread out over many years following the approval of the loans. This permits the isolation of a predetermined source of variation in government investment that reflects only loan approval decisions made in previous years, and well before the associated spending takes place. Under the plausible identifying assumption that loan approval decisions made by official creditors do not anticipate future macroeconomic shocks that matter for private investment and output, this predetermined component of spending can be used as an instrument for government investment.

Using data on government and private investment in a panel of 39 lowincome countries, we find that government investment crowds in private investment: an extra dollar of government investment leads to 2 additional dollars of private investment, and 1.5 additional dollars of output in the subsequent period (though the 95% confidence intervals for both estimates are large). We then use these estimates to calibrate the CES production function. In contrast with the existing literature, which has mostly focused on the Cobb-Douglas case, we find that the data suggest a substantially higher degree of complementarity between government and private capital.

This higher degree of complementarity tends to imply more crowding in. The key point, however, is that for a given amount of crowding in, the implied rate of return on government investment is decreasing with the degree of complementarity. We use our estimated functional form to calibrate rates of return on government investment. Even with high complementarity, we find that, for most countries in our sample, the return to government investment exceeds the world interest rate, with excess returns between 5- 170%. However, this conclusion does not apply to countries in our sample 5
that already have a high ratio of government to private capital stocks.

In the next section we lay out a simple model that can be used to interpret the relationship between estimates of crowding in, the elasticity of substitution, and the return to government investment. Section 3 contains our empirical evidence on the responses of private investment and output to changes in government investment, while Section 4 uses these estimates to calibrate the model of Section 3. The final section offers concluding remarks.

# **2 Model**

Consider a model in which the per capita production function depends on a CES aggregate of government and private capital: where _kg_ is the per capita government capital stock, _kp_ is the per capita private capital stock, _α_ is the aggregate capital share, and _A_ represents aggregate productivity. The parameter _γ ∈_ [0 _,_ 1] is a share parameter, and the parameter _σ ≤_ 1 represents the degree of substitutability between government and private capital. Lower values of _σ_ indicate greater complementarity: the limit _σ →−∞_ is the Leontief case where government and private capital are required in fixed proportions. At the other extreme, when _σ_ = 1, government and private capital are perfectly substitutable. The particular case of _σ_ = 0 corresponds to the Cobb-Douglas case which is most widely used to motivate and interpret crowding out regressions.<sup>5</sup> Both private and government capital accumulation follow:

> 5Note that the specification _Y_ = _A_ ( _kg_ ) _kpα_<sup>,inwhichproductivitydependsonpublic</sup> capital, is essentially the Cobb-Douglas case if _A_ ( _kg_ ) = _kg_<sup>_η_).Inthecase</sup><sup>_σ_=0,output</sup> depends on _kg_<sup>_γk_</sup> _p_<sup>1</sup><sup>_−γ_</sup> .

6
where _δ_ is the depreciation rate and _ii,t_ is investment of type i=(government, private). We assume a small open economy framework: the interest rate is given exogenously by _r_<sup>_∗_</sup> . We also abstract from population growth.

Regardless of the elasticity of substitution, _σ_ , it is optimal to equate the marginal return to government capital with the marginal return to private capital, and to equate both with the rate of return in global markets. To see this, consider the following optimization problem: The first order conditions of this problem yield: These two first-order conditions imply that the optimal ratio of private and public capital is given by: Before continuing, it is worth emphasizing that our assumption that the private and government sectors face the same world interest rate is a simplifying one, and abstracts from two potentially important considerations. First, it is likely that governments face lower financing costs than the private sector, which is typically more credit constrained. In our framework, this wedge would tend to imply a lower cost of capital and higher excess returns to government investment. Second, we are implicitly assuming non-distortionary taxation: the government’s cost of funds is given by the world interest rate, and there are no distortions associated with raising those funds. In practice, part of the costs of financing public investment is the distortionary costs of taxation, either for direct financing of government investment or for repayment of government debt in subsequent periods. In our framework this 7
would tend to imply a higher cost of capital and a lower excess return to government investment.

In reality, the rate of government investment and the public capital stock are unlikely to be chosen according to this optimality condition, but rather reflect a variety of institutional and political economy factors outside the model: there may be over- or under- investment on the part of the government relative to the optimal benchmark. In particular, the excess return to government investment, i.e. _∂k∂Yg_<sup>_−_(</sup><sup>_r∗_+</sup><sup>_δ_),isagoodindicatoroftheextent</sup> of over- or under-investment: if<sup>_<u>∂Y</u>r∗_+</sup><sup>_δ_,moregovernmentinvestment</sup> _∂kg_<sup>_>_</sup> is desirable; however, if _∂k_<sup>_<u>∂Y</u>_</sup> _g_<sup>_< r∗_+</sup><sup>_δ_, less government investment is desirable.</sup> The excess return on government investment, defined as _∂k_<sup>_<u>∂Y</u>_</sup> _g_<sup>_−_(</sup><sup>_r∗_+</sup><sup>_δ_),sum-</sup> marizes the extent to which there is over- or under- investment in government capital.

We do, however, assume that the private sector chooses its investment and capital stock optimally given the behavior of the government. In particular, the second equality in equation 4 describes the optimal behavior of the private sector for a given choice of the government capital stock. This first-order condition for the private sector implicitly defines a relationship between government and private capital, which in turn implies a relationship between government and private investment, i.e. the extent of crowding in or crowding out.

The key conceptual point here is that the extent of crowding in or crowding out of private capital by government capital is, in general, not a sufficient statistic for the return to government investment: simply observing crowding in of private investment alone is not enough to conclude whether or not more government investment is desirable. To see this, we derive expressions for the response of private investment and output to government investment, and show what they tell us about the return to government investment.

Consider an arbitrary benchmark<sup> ̄</sup> _ip,t_ ,<sup> ̄</sup> _ig,t k_<sup> ̄</sup> _p,t_ , _k_<sup> ̄</sup> _g,t_ and _Y_<sup> ̄</sup> _t_ , which represents the economy’s predicted levels of investment, capital and output for time _t_ .

8
Consider a small shock to the rate of change in _ig,t_ (∆ _ig,t_ = ( _ig,t − ig,t−_ 1) _−_ (<sup> ̄</sup> _ig,t −_<sup> ̄</sup> _ig,t−_ 1) _>_ 0). In the empirical work that follows, we will capture the benchmark changes<sup> ̄</sup> _ig,t −_<sup> ̄</sup> _ig,t−_ 1 and<sup> ̄</sup> _ip,t −_<sup> ̄</sup> _ip,t−_ 1 using country and year fixed effects. The first-order Taylor expansion of the equilibrium condition _∂k_<sup>_<u>∂Y</u>_</sup> _p_<sup>=</sup> _r_<sup>_∗_</sup> + _δ_ implies that the change in private investment ∆ _ip,t_ must satisfy: This condition states that the adjustment in private investment must be such that the marginal product of private capital at _t_ + 1 leaves private investors indifferent with respect to investing another unit. Manipulating the above equation yields: Similarly, the first-order Taylor expansion for output yields: where ∆ _Yt_ +1 = ( _Yt_ +1 _− Yt_ ) _−_ ( _Y_<sup> ̄</sup> _t_ +1 _− Y_<sup> ̄</sup> _t_ ).

**Lemma 1** _Given the functional form in equation 1, equations 7 and 8 can be written as:_

9
_where λ_ =<sup>_kpTheexcessreturnongovernmentinvestmentisgivenby:_</sup> _kg_<sup>_._</sup> The proof of this lemma is detailed in the appendix, together with other omitted proofs.

Equation 9 captures the extent to which government investment crowds in or crowds out private investment, which naturally depends on the elasticity of substitution between government and private capital. For example, when _σ_ = 1, corresponding to perfect substitutability, increases in government investment always crowd out private investment. On the other hand, when _σ_ is very negative, corresponding to strong complementarity, private investment is always crowded in by government investment.

It is also clear from comparing Equations 9 and 11 that the extent of crowding in is not informative about the return to government investment. Rather, both depend on the two unknown parameters of the production function, _γ_ and _σ_ , as well as the ratio of private to government capital, _λ_ .

To see the importance of this, consider how the previous literature assumes a Cobb-Douglas production function and uses estimates of “crowding in” to back out the marginal return to government investment. This practice is not innocuous, and tends to over-estimate the returns to government investment (assuming that the actual degree of complementarity is higher), as can be seen from the following lemma:

**Lemma 2** _Let β denote an estimated effect of government investment on private investment. For a given estimate_ 0 _< β < λ, the excess return on government investment is increasing in σ; overestimating the substitutability between government and private capital results in overestimating the excess return to government investment._

Assume, hypothetically, that we estimate that an additional unit of government investment leads to _β_ additional units of private investment - what 10
are the implications for the excess return to government capital, depending on our prior for _σ_ ? Figure 1 presents the implied marginal product of government capital, for different values of _β_ and different values of _σ_ . The figure illustrates the claim in Lemma 2 (which is proved in the appendix): for positive values of _β_ , assuming a higher _σ_ tends to imply higher rates of return for government investment.

Intuitively, a “crowding in” parameter of _β >_ 0 simultaneously captures two effects: first, additional government capital raises the marginal product of private capital through increasing overall productivity - this channel would imply that more “crowding in” is associated with higher marginal products of government capital, as illustrated in the figure. The second effect is that additional government capital raises the marginal product of private capital through complementarity - thus, “crowding in” is not necessarily reflective of overall productivity gains, but of higher returns to private capital resulting from complementarity. Naturally, the elasticity of substitution determines the relative importance of these two effects: assuming a high degree of complementarity implies low marginal returns to government investment, because most of response of private capital is attributed to the complementarity channel. In contrast, when government and private capital are more substitutable, more “crowding in” is indicative of higher marginal returns to public investment and overall productivity increases.

The figure also illustrates that not every value of _β_ is consistent with every value of _σ_ . For example, for _σ_ = 1, crowding out ( _β ≤_ 0) is _necessary_ : when government and private capital are perfect substitutes, government capital necessarily crowds out private capital. Intuitively, for _σ_ = 1, more crowding out (a lower _β_ ) implies higher marginal returns to public capital, since more crowding out is indicative of relatively more productive public capital.

In order to make progress in interpreting estimates of crowding in, we need to somehow estimate _σ_ and _γ_ . Unfortunately, conventional approaches to estimating the elasticity of substitution are difficult to apply in the case of 11
<!-- Start of picture text --> Marginal Product of Government Capital<br>0.35<br>0.3<br>0.25<br>0.2<br>mpkg, sigma=0<br>mpkg, sigma=1<br>0.15<br>mpkg, sigma=-1<br>0.1 mpkg, sigma=-2<br>0.05<br>0<br>-1 -0.5 0 0.5 1 1.5 2 2.5 3<br>Estimate of beta<br><!-- End of picture text --> Figure 1: The implied marginal product of public capital for different values of _σ_ , as a function of _β_ .

government and private capital.<sup>6</sup> The standard approach to estimating the elasticity of substitution between two factors of production is to gather data on their relative use, and data on their relative prices, and then to estimate the proportional response of relative factor use to relative factor prices, which defines the elasticity of substitution. In the case of government capital, it is very difficult to directly observe its price, making this approach infeasible.<sup>7</sup> Instead, we note that the response of private capital (in Equation 9) and the response of output (in Equation 10) are jointly sufficient to identify _γ_ and _σ_ , given empirical estimates of these two parameters. In particular, let _β_ and _βy_ denote empirical estimates of the response of private investment and output to government investment. Given _λ_ (which we calibrate based

> 6See, for example, Antras [2004] on how to estimate the elasticity of substitution between capital and labor using aggregate data for the United States.

7Indeed, if one takes the simple framework here literally, the relative price of government and private capital is always one, and so there is no variation in the relative price that could be used to identify the elasticity of substitution.

12
on the average ratio of private and government investment) and _r_<sup>_∗_</sup> + _δ_ (for which we plug in standard values), we can back out _γ_ and _σ_ by solving: Then, we use the calibrated parameters for _γ_ and _σ_ to calculate the excess return on government investment using equation 11. In the next section of the paper we develop empirical estimates of _β_ and _βy_ that are required to implement this calibration exercise.

# **3 Empirical analysis**

In this section we discuss our empirical strategy for estimating the responses of private investment and output to government investment, i.e. _β_ and _βy_ in Equations 12 and 13. Our goal is to use the empirical estimates of _β_ and _βy_ to retrieve estimates for _γ_ and _σ_ , which in turn will allow us to draw implications for the returns to government investment.

We estimate a series of regressions of changes in private investment and changes in output on changes in government investment: where _i_ and _t_ subscripts denote country and time; _IP,i,t_ , _IG,i,t_ and _Yi,t_ are private investment, government investment and output (respectively), all measured in constant local currency units; _μi_ and _θi_ are country fixed effects; _ζt_ and _ηt_ are year fixed effects; and _εi,t_ and _νi,t_ are error terms capturing all other 13
shocks to private investment and output, respectively. The key parameters of interest are _β_ and _βy_ . Note that, since changes in private and government investment and changes in output are all normalized by the same variable (the lagged level of output) these parameters capture the change in private investment and output due to a unit change in government investment.

The standard difficulty in statistically identifying _β_ and _βy_ is that changes in government investment may be correlated with other contemporaneous shocks to private investment and output captured in the two error terms, so that OLS estimation of Equations 14 and 15 will be inconsistent. Moreover, the direction of the bias in the OLS estimates is ambiguous. For example, if government investment increases as a countercyclical policy response during bad times when private investment and output are low, then changes in government investment will be negatively correlated with the error terms and OLS estimates of _β_ and _βy_ will be biased downwards. If, on the other hand, government investment is procyclical and falls during bad times (when the government cannot borrow to finance spending), then OLS estimates of _β_ and _βy_ would be biased upwards. Another pedestrian, but likely very important concern is measurement error in government investment, which will bias estimates of _β_ and _βy_ towards zero. Moreover, this downward bias will be even stronger for the private investment equation, since private investment is measured as total investment from the national accounts less government investment. This implies that when measured government investment is artificially high, measured private investment will be artificially low, inducing a spurious negative correlation between the two.

We address this endogeneity problem using the identification strategy suggested by Kraay [2012] and Kraay [2013] in the context of estimating government spending multipliers. The goal here, as well as in those papers, is to find a predetermined source of variation in government spending that plausibly is uncorrelated with contemporaneous macroeconomic shocks that affect output and investment, and that can be used as an instrument for 14
changes in government investment. We achieve this by exploiting the fact that (a) a significant fraction of government spending in developing countries is financed by loans from official creditors, and (b) these loans typically disburse over a period of many years following the initial approval of the loan. Drawing on a unique loan-level dataset on nearly 60,000 individual loans from official creditors to developing country governments, Kraay [2013] constructs a measure of predicted disbursements on loans from official creditors that reflects only fluctuations in loan approval decisions made in previous years, but not the current year.<sup>8</sup> The key identifying assumption is that loan approval decisions in a given year do not anticipate shocks to private investment or output in subsequent years. Given this identifying assumption, subsequent predicted disbursements are uncorrelated with contemporaneous macroeconomic shocks, and are a valid instrument for changes in government investment. Moreover, in countries where official creditor financing of government investment is large, predicted disbursements will also be a strong instrument for changes in government investment.<sup>9</sup>

> 8The predicted disbursements instrument is constructed as follows. We start with a set of approximately 60,000 loans from official creditors to developing country governments, as recorded in the Debtor Reporting System database of the World Bank. For each loan we have the year of the original commitment, and the schedule of subsequent annual actual disbursements. For each loan we construct a 10-year disbursement profile, i.e. the fraction of the original commitment that is disbursed in year 0, 1,2,...,10, with year zero corresponding to the approval year of the loan. We next sort loans into creditor-decaderegion-specific bins and average disbursement profiles across loans within bins. We then apply these typical disbursement profiles to individual loans within each bin, and construct predicted disbursements for each loan, i.e. the disbursements that would have occurred had the loan disbursed at the typical rate for all loans in the same bin. The rationale for this step is that it removes any country-specific source of variation in actual disbursements on previously-approved loans, which might respond endogenously to contemporaneous macroeconomic shocks. Finally, we aggregate the predicted disbursements across all loans for each country-year observation, but excluding loans approved in the same year. By construction, the only country-specific variation in this predicted disbursements measure reflects loan approval decisions from previous years, which we assume to be uncorrelated with contemporaneous macroeconomic shocks. For details, see Sections 2 and 3 of Kraay [2013].

> 9In addition, we are also implicitly assuming that the multiplier effect of government consumption expenditures on output and on private investment is zero. This assumption is

15
We obtain data on government investment and total investment from the IMF’s World Economic Outlook (WEO) database, measured in current local currency units.<sup>10</sup> We convert both to constant local currency units using the overall GDP deflator. Data on GDP in constant local currency units comes from the World Bank’s World Development Indicators. Data on predicted disbursements on loans from official creditors are taken directly from Kraay [2013].

We focus on a set of 39 low-income countries eligible for concessional lending from the International Development Association (IDA, the soft-loan window of the World Bank). We restrict attention to those countries for which at least 15 annual observations on investment are available. These countries are shown in Figure 2, which plots the time-average of government investment and private investment as a fraction of GDP for each country in the sample. Government investment rates vary widely across countries, ranging from as low as 2.5 percent of GDP in Cameroon to nearly 15 percent of GDP in Guyana. There is also a weak negative correlation across countries, with countries with higher government investment rates also having lower private investment rates.

We focus on these low-income countries because of the importance of official creditor financing of government spending in this sample. To see this, consider the over-time average of total disbursements on loans from official creditors as a fraction of total government spending, and as a fraction of total government investment. Among these 39 low-income countries, the median values of these ratios are 18 and 63 percent, respectively, suggesting important because disbursements on loans from official creditors finance both government consumption and government investment expenditures, and so our predicted disbursements instrument is correlated with fluctuations in both types of spending. If government consumption spending raised investment or output, this would violate the exclusion restriction. See Kraay [2012], Section 6 for a more complete discussion.

10 We are grateful to Toh Kuan of the WEO team for kindly making this data available to us. The government investment series is not available in the public WEO database, but is maintained by the WEO team as part of their larger internal database. This is to our knowledge the largest existing dataset for public investment.

16
<!-- Start of picture text --> GUY<br>MOZ<br>ETH<br>MWI<br>DJI<br>MDGRWA MLI BDI<br>COM BENBFAKHMSEN BGD NPL HND NIC<br>CAF NERBOLTGO PAKYEMGHAUGA COGUZBMRT<br>CIVSLE GIN TZAKEN LKAARMMDA<br>SDN<br>CMR<br>.05 .1 .15 .2 .25 .3<br>Private Investment/GDP<br>.15<br>.1<br>.05<br>0<br><!-- End of picture text --> Figure 2: This graph plots the over-time average of government investment as a fraction of GDP (on the vertical axis) against the over-time average of private investment as a fraction of GDP (on the vertical axis). Total investment and total government investment are drawn from the IMF’s World Economic Outlook database. Private investment is the difference between the two. Annual averages are calculated over all years between 1980 and 2012 for which data are available.

17
a very substantial role for official creditor financing of government spending. This will be important for the strength of the first-stage relationship between changes in predicted disbursements on loans from official creditors and changes in government investment.<sup>11</sup> Our main empirical results are in Tables 1 and 2, which report estimates of _β_ and for _βy_ obtained through a number of different variants on our basic empirical specification. Consider first Table 1, which presents estimates of the response of private investment to government investment. The first three columns report results for the full set of 916 annual country-year observations, while the second three columns report results for the subset of these countries located in Sub-Saharan Africa. Panel A reports ordinary least squares (OLS) estimates, Panel B reports two-stage least squares (2SLS) estimates, and Panel C reports the first-stage regression of changes in government investment on changes in the predicted disbursement instrument.

In the full sample of observations and in the Sub-Saharan Africa sample, we find OLS estimates of _β_ that are negative, small, and insignificantly different from zero (columns (1) and (4)). The 2SLS estimates are substantially larger, in the vicinity of 2. As expected given the importance of official creditor financing of government investment, the first-stage relationship between changes in government investment and changes in predicted disbursements is positive, and highly significant, particularly in the full sample where the first-stage F-statistic is 13.9. On the other hand, the relationship is weaker in the Sub-Saharan Africa sample, possibly due to greater measurement error in the government investment data, with the first-stage F-statistic of 7.3 falling short of the rule of thumb of 10. We therefore base inferences on weak-instrument consistent confidence sets reported at the bottom of the table. These are unfortunately quite wide, but exclude zero, suggesting a

> 11Interpreting these ratios is complicated by the fact that we do not observe the extent to which individual loans finance government investment versus government consumption. However, the scale of disbursements on these loans clearly is large relative to overall government spending.

18
Table 1: The response of private investment to government investment.

||**All IDA Countrie**|**s**|**IDA Cou**|**ntries in Sub-Saha**|**ran Africa**|
|---|---|---|---|---|---|
|||Control for<br>||||
||Excluding|Lagged||Excluding|Control for Lagged|
||All Observations<br> <br>Influential<br>Observations|<br>Dependent<br>Variable|All Observations|<br>Influential<br>Observations|<br>Dependent<br>Variable|
|_Panel A: Ordinary Least Squares(De_<br>|_pendent Variable is Change in Private_<br><br>|_Investment)_<br>||||
|Change in Government Investment|-0.0858<br>-0.113|-0.0780|-0.0900|-0.108|-0.0778|
||(0.125)<br>(0.116)|(0.126)|(0.167)|(0.148)|(0.166)|
|_Panel B: Two-Stage Least Squares(D_|_ependent Variable is Change in Priva_|_te Investment)_||||
|Change in Government Investment|1.881*<br>2.298*|1.892*|1.891|1.668|1.939|
||(1.066)<br>(1.297)|(1.059)|(1.526)|(1.204)|(1.557)|
|_Panel C: First-Stage Regression(Dep_|_endent Variable is Change in Govern_|_ment Investment)_||||
|Change in Predicted Disbursements|0.290***<br>0.270***|0.290***|0.248**|0.326***|0.246**|
||(0.0778)<br>(0.0750)|(0.0775)|(0.0918)|(0.0732)|(0.0915)|
|First-Stage F-Statistic|13.94<br>12.98|14.00|7.30|19.88|7.24|
|Weak Instrument Consistent 95%||||||
|Confidence Interval for|[0.583,4.863]<br>[0.772,6.275]|[0.594,4.878]|[0.137,9.526]|[0.292,5.219]|[0.180,9.854]|
|Number of Observations<br>Notes: This table reports the results<br>constant local currency units and are<br>Africa (second three columns). Pane<br>regressions. Weak instrument-consi<br>loans from official creditors are used<br>clustered at the country level are ind|916<br>908<br>from a series of regressions of chan<br>scaled by lagged GDP. The sample c<br>l A reports OLS estimates, Panel B re<br>stent confidence intervals are based<br>as an instrument for changes in gov<br>icated in parentheses. * (**) (***) i|916<br>ges in private investm<br>onsists of IDA-eligible<br>ports 2SLS estimates,<br>on the Moreira Likeli<br>ernment investment i<br>ndicates significance a|611<br>ent on changes in publi<br>countries (first three c<br>and Panel C reports the<br>hood Ratio statistic. Ch<br>n Panel B. Heterosked<br>t the 10 (5) (1) percent|607<br>c investment. All<br>olumns) and IDA-e<br>corresponding fir<br>anges in predicted<br>asticity-consistent<br>level.|611<br>changes are in<br>ligible countries in<br>st-stage<br>disbursements on<br>standard errors|

19
Table 2: The response of output to government investment.

|||<br>**All IDA Countries**|Control for<br>|**IDA Cou**|<br>**ntries in Sub-Saha**|<br>**ran Africa**|
|---|---|---|---|---|---|---|
|||Excluding|Lagged||Excluding|Control for Lagged|
||All Observations|<br>Influential<br>Observations|<br>Dependent<br>Variable|All Observations|<br>Influential<br>Observations|<br>Dependent<br>Variable|
|_Panel A: Ordinary Least Squares(Dep_<br>|_endent Variable is_<br>|_Change in Output)_<br>|||||
|Change in Government Investment|0.190*|0.148*|0.0911|0.184|0.127|0.0982|
||(0.0994)|(0.0757)|(0.0830)|(0.122)|(0.0928)|(0.107)|
|_Panel B: Two-Stage Least Squares(D_|_ependent Variable i_|_s Change in Output_|_)_||||
|Change in Government Investment|1.418*|1.248|1.267|2.580**|1.684*|2.439*|
||(0.797)|(0.819)|(0.783)|(1.177)|(0.831)|(1.305)|
|_Panel C: First-Stage Regression(Dep_|_endent Variable is C_|_hange in Output)_|||||
|Change in Predicted Disbursements|0.290***|0.255***|0.269***|0.248**|0.308***|0.211**|
||(0.0778)|(0.0720)|(0.0786)|(0.0918)|(0.0663)|(0.0967)|
|First-Stage F-Statistic|13.94|12.57|11.71|7.3|21.59|4.74|
|Weak Instrument Consistent 95%<br>Confidence Interval for|[-0.046,3.933]|[-0.336,4.160]|[-0.293,4.035]|[0.346,11.943]|[0.109,5.713]|[-0.168,25.653]|
|Number of Observations<br>Notes: This table reports the results<br>currency units and are scaled by lagg<br>three columns). Panel A reports OLS<br>instrument-consistent confidence in<br>creditors are used as an instrument f<br>country level are indicated in parent|916<br>from a series of reg<br>ed GDP. The sampl<br>estimates, Panel B<br>tervals are based o<br>or changes in gover<br>heses. * (**) (***)|907<br>ressions of change<br>e consists of IDA-el<br>reports 2SLS estim<br>n the Moreira Likel<br>nment investment<br>indicates significan|916<br>s in real GDP on chan<br>igible countries (firs<br>ates, and Panel C rep<br>ihood Ratio statistic.<br>in Panel B. Heteros<br>ce at the 10 (5) (1) pe|611<br>ges in public investme<br>t three columns) and ID<br>orts the corresponding<br>Changes in predicted<br>kedasticity-consistent s<br>rcent level.|606<br>nt. All changes ar<br>A-eligible countri<br>first-stage regres<br>disbursements on l<br>tandard errors clu|611<br>e in constant local<br>es in Africa (second<br>sions. Weak<br>oans from official<br>stered at the|

20
significant positive effect of government investment on private investment, i.e. crowding in. The fact that the 2SLS estimates are much larger than the OLS estimates is consistent with a downward bias in the former, due to some combination of (a) measurement error in government investment, and (b) countercyclical behavior of government investment.

In columns (2) and (5) we explore the sensitivity of our results to the presence of influential observations. We use the procedure suggested by Hadi [1992] to identify influential observations in the first-stage and reduced-form regressions, and then re-estimate the OLS, 2SLS and first-stage regressions in the slightly-reduced sample that excludes these observations. Removing influential observations has little effect on our findings in the full sample, but it dramatically improves the strength of the first-stage relationship in the Sub-Saharan Africa sample, where the first-stage F-statistic jumps to 19.9. As a result, the confidence set for the response of private investment to government investment shrinks considerably, although it is still uncomfortably wide. The point estimate of the effect falls slightly to 1.67.

In columns (3) and (6) we perform an important robustness check that addresses concerns about the validity of the exclusion restriction that justifies the instrument. Our identifying assumption is that loan approval decisions made by official creditors in a given year are not correlated with future macroeconomic shocks that matter for private investment and output. However, loan approval decisions might very well respond to contemporaneous macroeconomic shocks in the borrowing country. If these shocks are correlated over time, then this would violate the exclusion restriction. To address this concern, we include lagged private investment as a control variable to pick up any persistence in shocks to private investment. To conserve space we do not report the coefficient on the lagged dependent variable, which generally is not significantly different from zero. Consistent with this, our estimates of the response of private investment to government investment do not change much, and remain in the vicinity of 2.

21
Table 2 reports our estimates of the response of output to government investment, _βy_ , and has the same structure as Table 1. Across all specifications, the OLS estimates are small and positive, ranging from 0.1 to 0.2, and are weakly significantly different from zero in the full sample. The first-stage relationship between changes in government investment and changes in predicted disbursements are of course identical to those in Table 1 in the full sample (columns (1) and (4)), and only slightly different in the no-influential observations sample and controlling for the lagged dependent variable. The 2SLS estimates of the output response to government investment vary somewhat across specifications, and range from 1.2 to 2.6. However, the weakinstrument consistent confidence sets are quite wide in several specifications, and include zero in four out of six cases. Overall, we can summarize these results as suggesting a (rather imprecise) estimate in the vicinity of about 1.5.<sup>12</sup>

# **4 Structural interpretation**

In this section, we use our empirical estimates of _β_ and _βy_ to calibrate our CES production function using equations 12 and 13, and draw implications for the rates of return on government investment. To back out the parameters _γ_ and _σ_ from the coefficients _β_ and _βy_ , we use _α_ = 0 _._ 33, _r_<sup>_∗_</sup> + _δ_ = 0 _._ 13 (which

> 12It is useful to note that these quite large effects of government investment on output are consistent with considerably smaller estimates of overall government spending multipliers. Kraay [2012] and Kraay [2013] reports estimates of overall government spending multipliers in low-income countries that are in the vicinity of 0.5. Under the assumption that government consumption expenditures have no effect on output, it is straightforward to show that the estimated overall government spending multiplier is _γβiy_ + _<u>γγic</u>_<sup>,where</sup><sup>_γi_</sup> and _γc_ are the slopes of the first-stage regressions of changes in government investment and government consumption on changes in the predicted disbursements instrument. As long as disbursements on loans from official creditors raise both government consumption and investment expenditures, i.e. and are both positive, the overall government spending multiplier will be smaller than the effect of government investment on output that we estimate here. In our full sample of observations, _γi_ = 0 _._ 29 and _γc_ = 0 _._ 36, consistent with an estimated overall spending multiplier less than half of our estimate of 1.5 for _βy_ .

22
can be thought of as a 10% annual rate of depreciation and a 3% world interest rate), and _λ_ = _kkgp_<sup>=3,whichistheaverageratioofprivateto</sup> government investment in our sample.<sup>13</sup> Given that our estimates of _β_ and _βy_ are quite imprecise, we use a range of estimates that roughly fall within the confidence intervals for these parameters. The theory also restricts the range of admissible estimates of _β_ and _βy_ , as some values are inconsistent with _γ ∈_ (0 _,_ 1) and _σ ≤_ 1. The admissible range for _β_ requires _β < λ_ ; thus, we consider values of _β_ between 1 and 2.5, and values of _βy_ between 0.5 and 2.

Table 3 presents the calibration results for different combinations of _β_ and _βy_ . The third and fourth columns present the calibrated values of _σ_ and _γ_ , respectively, given the values of _β_ and _βy_ . Recall that the Cobb-Douglas specification, which is typically used in the literature, is nested within the CES framework with the parameter _σ_ = 0. Our estimates suggest a high degree of complementarily ( _σ <_ 0). This has important implications for the mapping between the extent of “crowding in” and the implied rate of return on government investment.

The fifth and sixth columns use the calibrated values of _σ_ and _γ_ to calibrate the excess returns to government investment using equation 11, for different values of _λ_ : the average value in the sample ( _λ_ = 3), and the 10th percentile ( _λ_ = 1). As evident from equation 11, the excess return depends on the ratio of government and private capital stocks, _λ_ . As could be expected, the excess return to government investment is always higher in countries with a higher _λ_ (i.e., where government capital is relatively scarce). The calibration suggests that for the “average” country, the excess returns to government investment range from 5% to 170%. These estimated excess 13 We are implicitly assuming here that (a) the available time series for each country is sufficiently long that government-private differences in initial capital stocks are sufficiently depreciated that they do not matter, and (b) that flows of government and private investment result in increases in corresponding capital stocks. As emphasized by Pritchett [2000] the latter is a non-trivial assumption.

23
**Table 3:  Calibration results**

|Empirical est|imates|Calibratedpa|rameters|Excess return to<br>|public investment<br>|||
|---|---|---|---|---|---|---|---|
|||||Average(λ=3)|10thpercentile(λ=1)|threshold λ|% above threshold|
|1|0.5|-0.35|0.4|0.25|-0.04|1.35|82.05|
|1|1|||||||
|1|1.5|||||||
|1|2|||||||
|1.5|0.5|-1.19|0.17|0.17|-0.10|2.06|53.85|
|1.5|1|-0.67|0.49|0.66|-0.01|1.02|89.74|
|1.5|1.5|-0.54|0.65|1.17|0.11|0.67|100.00|
|1.5|2|-0.48|0.73|1.7|0.22|0.51|100.00|
|2|0.5|-3.18|0.02|0.11|-0.13|2.54|48.72|
|2|1|-1.72|0.22|0.6|-0.09|1.59|71.79|
|2|1.5|-1.43|0.4|1.11|-0.04|1.18|84.62|
|2|2|-1.32|0.5|1.5|0|1.00|89.74|
|2.5|0.5|-10.48|0.00|0.05|-0.13|2.92|38.46|
|2.5|1|-4.96|0.007|0.54|-0.13|2.30|51.28|
|2.5|1.5|-4.13|0.03|1.05|-0.13|1.97|53.85|
|2.5|2|-3.8|0.06|1.56|-0.12|1.77|53.85|

Notes: The calibrated values of the parameters of the CES production function _σ_ and _γ_ (equation 1), as a function of the empirical estimates of the response of private investment to government investment ( _β_ ) and the response of output to government investment ( _βy_ ). The excess returns to government investment for _λ_ = 3 and _λ_ = 1 implied by this calibration are calculated using equation 11, with _r_<sup>_∗_</sup> + _δ_ = 0 _._ 13. The last two columns present the threshold level of _λ_ =<sup>_kp_above which there are positive excess returns to government</sup> _kg_<sup>,</sup> investment, and the percent of countries in our sample for which _λ_ is above that threshold. The highlighted row presents the calibrated values for our point estimates of _β_ and _βy_ .

24
returns may be overstated, for several reasons. First, as previously noted, our estimation of the costs of financing abstract from distortions associated with taxation. Second, the excess returns are scaled by _r_<sup>_∗_</sup> + _δ_ - a smaller value of _r_<sup>_∗_</sup> or _δ_ would imply lower excess returns (though the sign would not change). Third, our estimates of _βy_ rely on the assumption that shocks to government investment affect output only through the accumulation of public and private capital. If there are other indirect “stimulus” effects, our estimates of _βy_ may be overstated; as illustrated in Table 3, for a given _β_ , higher values of _βy_ imply higher excess returns.

Despite high calibrated excess returns to government investment in the “average” country, the excess returns for countries with _λ_ = 1 tend to be negative. The seventh column presents the threshold values of _λ_ , above which the calibrated marginal return to government investment is higher than the world interest rate. This threshold ranges from 0.5 to about 2.5. The implication is that for countries with _λ >_ 2 _._ 5, we can say with some confidence that there are positive excess returns to government investment; for countries with _λ <_ 0 _._ 5, we can say with some confidence that there are negative excess returns to government investment. The last column presents the percent of countries in our sample for which the calibrated excess returns are positive. The number ranges between 38%-100%. For most parameter values, there are at least some countries for which the calibrated excess returns are negative. Table 4 presents calibrated excess returns by country, for the point estimates _β_ = 2 and _βy_ = 1 _._ 5.

These findings suggest that, for most developing countries, there is underinvestment in government capital. However, for some countries such as Ethiopia, Central African Republic, Comoros, Guyana, Malawi and Mozambique, there may be excess government capital.

25
|||λ (ratio of private and|Excess<br>returns|
|---|---|---|---|
|Country|code|public investment)|(%)|
|Armenia|ARM|5.39|505.51|
|Burundi|BDI|4.09|253.29|
|Benin|BEN|1.78|22.32|
|Burkina Faso|BFA|1.77|21.83|
|Bangladesh|BGD|2.33|54.61|
|Bolivia|BOL|1.73|19.79|
|Central African Republic|CAF|0.99|-4.64|
|Côte d'Ivoire|CIV|1.38|5.83|
|Cameroon|CMR|7.31|1074.88|
|Congo, Rep.|COG|3.32|147.06|
|Comoros|COM|0.88|-6.59|
|Djibouti|DJI|1.39|6.38|
|Ethiopia|ETH|0.92|-5.93|
|Ghana|GHA|2.70|83.42|
|Guinea|GIN|3.00|112.08|
|Guyana|GUY|0.78|-8.28|
|Honduras|HND|3.43|160.52|
|Kenya|KEN|3.85|216.11|
|Cambodia|KHM|1.96|31.51|
|Sri Lanka|LKA|5.39|506.36|
|Moldova|MDA|4.91|401.68|
|Madagascar|MDG|1.19|0.26|
|Mali|MLI|2.22|47.16|
|Mozambique|MOZ|1.06|-3.11|
|Mauritania|MRT|3.32|146.60|
|Malawi|MWI|1.17|-0.36|
|Niger|NER|1.68|17.61|
|Nicaragua|NIC|4.24|277.37|
|Nepal|NPL|2.90|102.38|
|Pakistan|PAK|2.85|97.66|
|Rwanda|RWA|1.35|5.00|
|Sudan|SDN|5.13|448.69|
|Senegal|SEN|1.87|26.78|
|Sierra Leone|SLE|1.43|7.71|
|Togo|TGO|1.75|20.96|
|Tanzania|TZA|3.70|194.92|
|Uganda|UGA|2.95|107.07|
|Uzbekistan|UZB|3.75|202.69|
|Yemen,Rep.|YEM|2.61|76.37|

Table 4: The estimated excess returns by country, for the point estimates _β_ = 2 and _βy_ = 1 _._ 5.

26
# **5 Conclusion**

This paper studies the returns to government investment in developing countries. We estimate the extent to which government investment crowds in private investment, and the effect that it has on output. We use predicted disbursement of loans from official creditors as an instrument, and find that government investment has positive and significant effects on both private investment and output. Our point estimates suggest that one dollar of government investment is associated with 2 additional dollars of private investment, and 1.5 additional dollars of output in the subsequent period.

While, at first glance, “crowding in” tends to suggest favorable implications regarding the desirability of government investment, a structural interpretation of these findings suggests that the implications are rather mixed. We use our empirical estimates to calibrate a CES production function, and use the estimated functional form to derive implications for the marginal return to government investment. For the “average” country in our sample, the calibration suggests underprovision of government capital, and high rates of return associated with additional government investment. However, for countries that already have high rates of government investment (such as Ethiopia, Central African Republic, Comoros, Guyana, Malawi and Mozambique), even these favorable estimates suggest that the rate of return on government investment is below the world interest rate. Since our estimates for the “average” country imply rather high rates of return, the negative excess return to government investment in these countries is a rather robust conclusion.

# **References**

Pol Antras. Is the us aggregate production function cobb-douglas? new estimates of the elasticity of substitution. _Contributions to Macroeconomics_ , 4(1):Article 4, 2004.

27
- David A. Aschauer. Does public capital crowd out private capital? _Journal of Monetary Economics_ , 24(2):171–188, 1989.

- Marianne Baxter and Robert G. King. Fiscal policy in general equilibrium. _The American Economic Review_ , 83(3):315–334, June 1993.

- Cesar Calderon and Luis Serven. The effects of infrastructure development on growth and income distribution. Vol. 3400, Washington, DC: World Bank., 2004.

- Cesar Calderon, Enrique Moral-Benito, and Luis Serven. Is infrastructure capital productive? a dynamic heterogeneous approach. World Bank Policy Research Working Paper 5682, 2011.

- Eduardo Cavallo and Christian Daude. Public investment in developing countries: A blessing or a curse? _Journal of Comparative Economics_ , 39(1): 65–81, March 2011.

- Walter H. Fisher and Stephen J. Turnovsky. Public investment, congestion, and private capital accumulation. _The Economic Journal_ , 108:399–413, March 1998.

- Edward M. Gramlich. Infrastructure investment: A review essay. _Journal of Economic Literature_ , 32(3):1176–1196, September 1994.

- A.S. Hadi. Identifying multiple outliers in multivariate data. _Journal of the Royal Statistical Society_ , 54:761–771, 1992.

- Aart Kraay. How large is the government spending multiplier? evidence from world bank lending. _Quarterly Journal of Economics_ , 127(2):829– 887, 2012.

- Aart Kraay. Government spending multipliers in developing countries. evidence from lending by official creditors. _American Economic Journal: Macroeconomics_ , 2013. Forthcoming.

28
- Sylvain Leduc and Daniel Wilson. Roads to prosperity or bridges to nowhere? theory and evidence on the impact of public infrastructure investment. _NBER Macroeconomics Annual_ , 27(1):89–142, 2013.

- Lant Pritchett. The tyranny of concepts: Cudie (cumulated depreciated investment effort) is not capital. _Journal of Economic Growth_ , 5(4):361– 384, 2000.

- Luis Serven. Does public capital crowd out private capital? evidence from india. World Bank Policy Research Working Paper, 1613, 1996.

# **A Proofs**

## **A.1 Proof of Lemma 1**

The first derivative of _Y_ with respect to _kp_ is given by: In the ratio of the second derivatives, the constants _αA_ (1 _− γ_ ) will cancel out. We thus ignore them and take the derivatives: 29
Taking the ratio of the two yields: _<u>α</u>_ Canceling out the terms ( _γkg_<sup>_σ_+ (1</sup><sup>_−γ_)</sup><sup>_k_</sup> _p_<sup>_σ_)</sup> _σ_<sup>_−_2</sup> and _kp_<sup>_σ−_1</sup> : Multiplying the numerator and the denominator by _kp_ yields: Denote _λ_ = _k_<sup>_k_</sup> _g_<sup>_<u>p</u>_.Then,</sup><sup>_kp_=</sup><sup>_λkg_,andtheaboveexpressioncanberewritten</sup> as: Canceling out _kg_<sup>_σ_yields:</sup> Multiplying the above expression by _−_ 1 yields the expression in equation 12. To derive equation 13, note that equating the expression in equation 16 30
with _r_<sup>_∗_</sup> + _δ_ yields: It follows that: Thus:

# **A.2 Proof of Lemma 2**

First, consider the process of backing out _γ_ given _σ_ and _λ_ , using the relationship in equation 12: It follows that:
And: An estimate of _γ ∈_ (0 _,_ 1) requires that the denominator is positive; this is true for _σ_ sufficiently low (provided that _λ > β_ ). The estimated excess return to public investment is then: This expression is increasing in _σ_ ; thus, underestimating the degree of complementarity between public and private capital is overestimating the rate of return on public investment.

32