# Optimal Size of Public Investment

So far, the total amount of public investment is fixed. Finally, then, let us investigate the amount of public investment that should be conducted.

In a simple two-period model, we have the following constraints:

where Y_{1} is the output available in period 1, which is exogenously given. Equation (5.11) is the budget constraint in period 1. Equation (5.12) is the budget constraint in period 2, which means that public capital and the output from public capital are used for consumption in period 2. C_{1} and C_{2} are the outputs used for consumption in both periods. K_{G} is public capital.

The government intends to maximize total consumption over time. In other words, the government maximizes the total consumable output given by the following function by choosing KG:

where p is the discount rate of future consumption.

Then, the optimal rule for the size of public investment is given as follows:

The marginal product of public investment = the rate of time preference

Alternatively, this condition means

The rate of time preference p compares utility from current consumption with utility from future consumption. People normally evaluate current utility more than future utility. The rate of time preference means the relative evaluation of current utility compared with future utility. If this rate is high, people pay significant attention to evaluating current utility.

For example, if the rate of time preference is 10 %, amounts of 100 for current utility and 110 for future utility are equivalent for the agent. If the rate becomes 20 %, amounts of 100 for current utility and 120 for future utility are equivalent. The higher the rate of time preference, the larger the evaluated current consumption. Thus, public investment becomes desirable only if it produces a large benefit in the future. Note that public investment requires funds and thus a sacrifice in current consumption. Figure 5.2 illustrates optimal allocation over time.

In Fig. 5.2, the vertical axis denotes the marginal product of public investment and the horizontal axis denotes the level of public investment. E is the initial

Fig. 5.2 **Optimal allocation over time**

optimal point. As shown in Fig. 5.2, if the marginal product curve of public investment somehow moves upward (E ! E^{0}) or the rate of time preference exogenously declines (E ! E^{00}), the optimal level of public investment rises. However, if the marginal product curve of public investment moves downward or the rate of time preference rises, the optimal level of public investment declines.

In Japan, significant public investment has been conducted following World War II. The reason is that in Japan the initial level of public capital was too little in the 1950s; hence, the marginal product of public investment has been significantly large. In addition, the government has been politically stable and the rate of time preference has been significantly low.