1 Introduction

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The public capital invested in roads, air navigation, canals, and water and sewage systems expands the productive capacity of an economy. Public capital enables greater geographic concentrations of economic resources and facilitates the movement of goods and people. It allows for broader market access and a greater range of choice for employers and employees. It affects input and output markets, helps determine spatial development patterns and provides a large network at low cost to individual users. Public capital is, in short, the foundation upon which the economy is built.

A body of literature based on these ideas suggests that public capital plays an important, often overlooked, role in private production (see, for example, Aschaeur 1989; Munnel 1990a, 1990b; Shah 1992; Berndt and Hanson 1992; Lynde and Richmond 1992; Nadiri and Mamuneas 1994; Conrad and Seitz 1994; Morrison and Schwartz 1996; Fernald 1999; Pereira 2000; and, Ramirez 2004).

These studies employ production and cost functions to estimate the elasticity of output with respect to public capital, or the marginal savings to the private sector of an extra unit of public capital, respectively. They argue that the impact of capital is noteworthy, that the impact is different over time and that changes in public investment in infrastructure have a differential impact across industries.

These arguments are examined using Canadian data in Harchaoui (1997), Harchaoui and Tarkhani (2003) and Brox and Fader (2005). All three papers employ cost functions for estimation. Their models assume that the real level of public capital enters the cost function as an unpaid factor of production. Firms minimize costs over private capital and labour, which constitute the variable cost function for the firm, but take public capital as given in the total cost function. Changes in public capital are assumed to change the height of the variable cost curve. If public capital is cost saving, then there will be a negative elasticity of total cost with respect to public capital. This means that an increase in the level of public capital reduces the total cost of private production.

Harchaoui (1997) uses a trans-log cost function and a panel of Canadian industry data for the 1961-to-1997 period. The author finds that the impact of public capital is significant, accounting for about 12% of overall business sector productivity growth. Harchaoui and Tarkhani (2003) re- examine the relationship using an expanded panel of Canadian industries from 1961 to 2000. They report that on average an increase in public capital reduces production costs in the private sector. Brox and Fader (2005) perform a similar exercise, arguing that public capital is an important input for firms. Taken together, the cost function studies imply that firms use public capital, and that its provision can affect cost structures.

Despite the general agreement that public capital enters a private production or cost function, there is little agreement on what a reasonable rate of return from public investment is. And there has been little discussion of how robust the estimates are to alternate formulations and methods used to deal with particular econometric problems.

The studies present a diverse set of rate of return estimates based on differing aggregations of economic time series, estimation techniques, sample spans and modelling approaches. The range of estimates makes it difficult to ascertain which rate of return is most plausible, or which method is relatively robust.

Investigating the robustness of econometric estimates, and their implied rates of return, is important because estimates of the latter are needed in order to guide those interested in evaluating the need for more infrastructure or those National Accountants who are trying to incorporate public infrastructure into the National Accounts. Both require estimates of the rate of return on public capital. But these estimates are not readily available—outside of econometric exercises.

National Accountants need an estimate of the value of public capital that is incorporated into public output. But markets for public sector products are rarely available and therefore the expenditure of final products and the sum of value added approaches for calculating public sector gross domestic product (GDP) in the National Accounts are not available to National Accountants. Instead, public sector GDP is estimated using factor payments

Using this approach, it is possible to calculate labour remuneration from payroll information; however, without a robust estimate for public capital's rate of return it is difficult to calculate the remuneration to publicly financed capital that should be added into public sector output. Because research to date has not reached a consensus about what rate of return is earned from public capital, the System of National Accounts (SNA) has assumed that the return from public capital is only equal to its depreciation rate. There is no economic return from public capital in current public sector GDP estimates. Robust public capital rate of return estimates are, therefore, necessary if an economic return from government assets is to be included in public sector GDP.

Estimating a robust rate of return is more difficult than it appears. Although the public sector owns buildings and machinery and equipment, the majority of public capital consists of roads, bridges, and water and sewer systems (Baldwin and Dixon 2008). These assets have no market price, and, in most cases, lack close parallels with privately owned capital goods in Canada. Because of this lack of information, it is not possible to directly calculate the gross return from public capital, nor is it possible to use the returns from privately owned assets as proxies in Canada. The lack of substitutes forces economists to rely on econometric techniques to infer the return from public capital, fuelling the debate about what a reasonable rate of return is by offering multiple methods for estimation and widely different answers.

This paper's primary purpose is to examine how robust the estimates of the impact of public capital are to alternative estimation methods and to ask what the range of rates of return to public capital is. It uses a variety of estimators and explicitly examines the time series properties of the data. Using a simple functional form, the relationship between real output and public capital, and the relationship between per unit costs and public capital, is estimated and used to infer public capital's rate of return. The paper then attempts to see whether information from the different methods can be combined in such a way as to 'triangulate' on a preferable estimate of public capital's rate of return.

The study is organized as follows. Section 2 examines how aggregate real GDP and public capital covary over time and shows the difficulty in disentangling the effects of public capital and growth in multifactor productivity. Section 3 describes the panel data sets employed to estimate the elasticity of public capital. Section 4 examines production function estimates of the elasticity of public capital, while section 5 examines cost function estimates for the elasticity of cost savings. Section 6 concludes the paper.