# 2 Aggregate gross domestic product and public capital

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The relationship between real gross domestic product (GDP) and public capital is complex because public capital is an enabling resource. Unlike most types of private capital, if public capital were to be removed from the economy, it would rapidly collapse.1 Moreover, public capital acts as a network connecting geographically separated economic agents. As a result, public capital's economic contribution is the full set of interactions that the network enables. It may, therefore, be difficult to accurately capture public capital's contribution to private sector value added.

Due to the complexity of the relationship, it is useful to start the examination of the relationship between public capital and real GDP using aggregate data before attempting to use sophisticated estimation techniques. Because public capital forms part of the foundation of the economy, one interpretation of how public capital affects real GDP is that public capital helps to define its trend.

This interpretation stems from how real GDP and the real stock of public capital co-vary over time (Figure 1). For most of the 1961-to-2005 period, real GDP and public infrastructure track each other closely. The only significant deviations occur during the recessions of the early 1980s and early 1990s.

In the absence of inputs, estimates of trend GDP have been interpreted as multifactor productivity (MFP), which is a proxy variable for intangible or difficult to measure production inputs. Since public capital is similar to trend GDP, Figure 1 implies that disentangling MFP and the marginal impact of public capital may be difficult.

A simple experiment helps to illustrate this. Suppose, for the moment, that the following models are specified:

Model 1: ln(GDP) = α + δt + γt2 + e;

Model 2: ln( GDP) = α + β ln(public capital) + e.

Model 1 supposes that the trend in log-level of GDP can be approximated by a quadratic trend while Model 2 assumes trend GDP can be modelled using public capital. If public capital and the commonly used time trend for MFP are capturing a similar feature of real GDP growth, then it is expected that the fitted values of the two models should be approximately the same. Moreover, the fitted values of the two models should mimic trend GDP growth.

When the fitted values are plotted against the log-level of real GDP, the hypothesized relationship emerges (Figure 2). Model 1 and Model 2 provide fitted values that closely resemble each other. Both models also track trend GDP well. The trend implied by public capital appears to better represent trend GDP during the first half of the period, while both models have difficulty during recessions and after the late 1990s when GDP overshoots the fitted values. Nevertheless, preliminary analysis supports the assertion that it will be difficult to separate trend changes in GDP from the contribution of public capital.

Of course, MFP is calculated as the difference between GDP growth and a weighted average of labour and capital input growth. Nevertheless, the same point applies. The growth in public capital is so closely related to the growth in overall GDP that it will be difficult to statistically separate its impact from other inputs that move in the same smooth fashion as does public capital.

1 The impact will be similar if any type of infrastructure is removed, including private sector infrastructure such as telecommunications or power networks.