# 6 Public capital's rate of return

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A salient question that arises, once the marginal impact of public capital is estimated, pertains to what public capital's rate of return is. Identifying the rate of return is important because it informs researchers and policy makers about what the flow of public capital services may be.

The rate of return on public capital can be estimated by solving the equation describing the user cost of capital for the implied rate of return. Often, the user cost formula includes a term to account for capital appreciation over time. For public capital, this is not included because there is no market for public capital in Canada. Moreover, even if there were markets, the transaction costs would preclude these gains from being realized. When it is not possible to realize capital gains, they do not constitute an opportunity cost which should be accounted for when examining an asset's rate of return.

The relationship between the cost of capital and marginal revenue is described by the user cost of capital formula:

where *P _{y}* is the price of output,

*P*is the price of a new unit of public capital,

_{g}*r*is the rate of return on public capital and

_{g}*δ*is public capital's depreciation rate.

_{g}To relate

from the production function estimates using Equation(2), note that the logarithmic derivative can re-written asAs a result, Equation (7) can be re-arranged as:

where *β _{g}* is the elasticity of public
capital,

*P*is nominal business sector GDP,

_{y}Y*P*is the nominal value of the stock of public capital.

_{g}GFor the public sector, it is unclear that raising funds is costless. It is argued that the real cost of government funds is higher than the government bond rate—though for national accounts purposes, it is not obvious that the cost of raising funds is a relevant consideration in valuing the output that should be attributed to public output. The cost of raising capital can be accounted for by scaling the marginal revenue from an additional unit of public capital by the marginal cost of

funds (*MC _{g}*) . Measuring

*MC*is beyond the scope of this paper; however, Usher (1986) provides an estimate of 1.6, which is employed here. Including the marginal cost of funds adjusts Equation (8) so that it takes the following form:

_{g}Because the argument about government efficiency is contentious, as is the size of the scaling factor, rates of return based on both Equation (8) and Equation (9) are reported. As well, it is important to note that there are 42 yearly rates of return calculated and that only the average rate of return is initially discussed.

Before examining the estimates of public capital's rate of return, note that when the implied rate of return to private capital is calculated by solving the user cost of capital, the rate of return is close to average yearly TSE (Toronto Stock Exchange) returns over the 1962-to-2003 period (Figure 20). The implied return is calculated, including and excluding price appreciation for capital goods in the user cost of capital formula. The production function point estimates provide a reasonable estimate of the return to private capital, suggesting that the production function specification is adequate for inferring a private rate of return. The issue that remains is whether, and to what degree, a reasonable return to public capital can be estimated given the intertwined effects of public capital and multifactor productivity (MFP).

Previous studies that have estimated public capital's rate of return find that the return is large. This type of result is criticised as being improbable (Aaron 1990) and is one of the main reasons attention shifts from production function to the cost function estimates.

The implied rate of return for public capital generated in this
paper is also large when the elasticity of public capital point
estimates from the production function excluding MFP are used
(Figure 20 Point Estimates). The FGLS_{1} and
FGLS_{2} estimators for the production function imply that
public capital's rate of return is over 50% if the marginal
cost of raising funds is one. When Usher's marginal cost of
funds estimate is employed the rate of return decreases to around
35%, but remains about four times larger than the average long-run
government bond yield.

Although these large rates of return are criticized as implausibly large, they do warrant comment. First, if the elasticity of public capital estimate is capturing the enabling nature of public capital, which allows for agglomeration effects, or the effect of increasing the network of public roads that may generate significant externalities, then a large historical rate of return may be plausible. However, it would not be replicable because, as Fernald (1999) points out: "A . . . plausible interpretation . . . is that building one network may have a very high rate of return; but building a second network may have a very low marginal return" (p. 630).

**
Figure 20
Nominal rate of return to private capital
**

**
Figure 21
Public capital's implied nominal rate of return across
estimates
**

Second, and more plausibly, the estimates in this paper are generated from a model that excludes MFP growth because it is difficult to disentangle the effect of MFP and public capital over time. Therefore, it is probable that if both effects could be separated, the marginal impact of public capital would be lower than estimated. Since a lower estimate will necessarily decrease the return from public capital, if MFP growth is reflected in the elasticity of public capital estimate, the rate or return is biased upward.

It is important to note that estimating the rate of return from public capital demands too much of a data series that increases at a fairly constant rate over time. In this setting, it may not be reasonable to assume the point estimate accurately represents the elasticity of public capital. The cost function estimates provide an external of source information that, as argued above, can be used to condition expectations about the magnitude of a reasonable elasticity estimate. The cost function includes the marginal impact of public capital and MFP, overcoming lack of variation in the public capital stock in the production function.

Estimating the rate of return using a range of elasticity
estimates of 0.2, 0.15 and 0.1, which are centered on the
unweighted average S-estimate of the marginal cost savings from
public capital, generates rates of return that are lower. And the
weighted average of around 0.10 provides a rate of return of 17%.
These rates of return are slightly above the long-run private rate
of return on capital and the long-term government bond rate (Figure
20 Hypothesized Values).^{7}

It should still be noted that there is uncertainty about the
point estimates for the rates of return associated with the
hypothesized elasticity of public capital coefficients taken from
the cost function approach. For the rates of return of around 17%,
excluding the marginal cost adjustment, a 95% confidence interval
spans roughly 12 percentage points above and below that of the
average estimates while the rates of return including the marginal
cost of funds adjustment spans 8 percentage points above and below
the average estimates.^{8} In the prior case, the confidence
interval surrounding the rate of return centered on the cost
function average and weighted cost savings from public
infrastructure include the long-term government bond rate.

The analysis of the bounds within which the true estimate of the rate of return falls considers only the statistical error that is associated with the estimation of the elasticity of business-sector cost with respect to public capital. There is, however, a non-systematic error that needs to be acknowledged—especially when comparing estimates of rates of return from different authors. Formula (9) for the rate of return involves three terms—the ratio of GDP to public capital ρ, the infrastructure costs elasticity β, and the depreciation rate δ. The use of different rates of depreciation will affect the estimate for the rate of return, both directly through δ and indirectly through the estimate of ρ. Lower depreciation rates applied to the perpetual inventory technique that is used to aggregate capital leads to a lower ρ.

The estimation of δ has important consequences for how large the rate of return estimate is. In addition to improving the accuracy of the elasticity estimate, improvements in estimating δ provide a second avenue along which more certain estimates for the rate of return can be generated. In the future, improved databases and estimation techniques may offer more accurate estimates for public capital's rate of return.

^{
7
} Harchaoui and Tarkhani (2003) report a 12% rate
of return from their elasticity estimate of 0.06 but use different
depreciation rates that were adopted here, based on the most recent
estimates that the productivity program has recently
introduced.

^{
8
}For this estimate, we have used the average value
of the ratio of gross domestic product to public capital in
Equation (8). Additionally, we ignore the variance of the
depreciation estimate.

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