Now let us introduce the income tax rate t in order to investigate the effect of fiscal policy on economic growth. Suppose the government imposes a linear income tax t to finance public consumption. Equation (5.2) may be rewritten as

For simplicity, we assume that saving is proportional to disposable income as in the conventional Keynesian model in Chap. 2. Thus, substituting Eq. (5.4) into Eq. (5.2), economic growth is determined as

which decreases with the tax rate.

An increase in the tax rate reduces private savings and hence the economic growth rate. Namely, when the government raises the tax burden on the private sector, private saving declines. This reduces capital accumulation and the long-run growth rate. It seems plausible to find a negative relationship between tax and growth rate.

The Incorporation of Public Investment

So far, we have assumed that government spending financed by taxes is used only for ordinary spending and does not affect production capacity; we can then confirm the negative effect of tax on growth. However, in reality, a part of tax revenue (and public spending) is used for public investment and contributes to the expansion of macroeconomic production capacity by improving the social infrastructure. If we incorporate the supply-side benefit of public investment into the model, how is the result altered?

Considering this possibility, we now assume that a part of public spending is used for public investment, which accumulates public capital and raises production capacity by the size of X (<1) in terms of private investment. Here X is the share of public investment in public spending. We also assume that public capital and private capital are perfect substitutes. Then, the capital accumulation equation AK = I = S is rewritten as

where g = G/Y is government spending per GDP and G is government spending, S denotes savings for private capital accumulation, and XgY means public investment for public capital accumulation. Based on this equation, the growth rate ю is given

as

where t = T/Y is tax per GDP and T is tax revenue. Namely, the growth rate is higher by the amount of Xg. A higher X means a higher growth rate ю.

In the long run, the government budget must be balanced. Government budget constraint is given as

Considering this equation, Eq. (5.6) is now rewritten as

If the ratio of public investment from tax revenue X is greater than the private saving rate s, an increase in the size of government spending enhances economic growth (and vice versa). Thus, the relationship between the tax rate and economic growth is generally ambiguous; it depends upon the sign of X — s in this simple model.

In reality, public capital and private capital are not necessarily perfect substitutes. Let us denote 0 as the relative productivity of public capital in terms of private capital. 0XgY means the contribution to capital accumulation by public investment. Then, the saving-investment equation in the economy is

Hence, Eq. (5.6') is rewritten as

If 0 is low, the sign of (0X — s) is likely to become negative, and an increase in t reduces ю. In other words, if the productivity of public capital is very low compared with the amount of public capital, an increase in the tax rate is likely to reduce the growth rate. Such a negative relationship occurs even if X is high.