Capital We are now introducing a capital, an alternative asset besides fiat money, which enables individual to acquire consumption when old. Consider the following production technology: o If k t units of the consumption good are converted into capital goods at time t, we will receive xk t at time t+1. o x is the gross real rate of return on capital, and x is some positive constant. o The depreciation rate is 100 percent. Under the same basic overlapping generations model, each initial old holds a stock of capital that produces xk 0 goods in the first period.
A Model of Capital Suppose that individual does not hold fiat money for now. The capital technology enables the young to use some of their endowment to produce goods at a later date. Capital produces goods and thus affects an economy s output. The budget constraint in the first period : c + k 1, t t y The budget constraint in the second period: c xk. The lifetime budget constraint : c 2, t+ 1 t c 2, t+ 1 1, t + y x..
x determines the slope of the budget line. Since we assume that x is constant, the rate of return on capital is also constant. However, there is a case when capital exhibits a diminishing marginal product. The marginal product of capital represents the added output resulting from the addition of one extra unit of capital. The diminishing marginal product represents the fact that, as capital is increased, the added output from an extra unit of capital gets smaller. We will write output from capital as a function, f (k). The marginal product can be written as f (k).
Rate-of-Return Equality Indeed, capital is not the only alternative asset people can hold besides money. For example, they can store value overtime through land or loans. What are the implications of the presence of alternative assets? Consider an economy with two alternative assets: capital and private debt (loans). The rate of return on capital is x, while loans offer the rate of return r. If r < x, people would make no loans.
If r > x, people would not invest in capital. People are willing to hold both assets when r = x. We refer it as rate-of return equality. This equality is held no matter how many assets exist in the economy. Suppose now there is money in the economy together with capital and loans. For lenders to hold three assets simultaneously, their rate of return must be identical. Recall that the rate of return on fiat money is n/z, this implies n/z = r = x.
Example 6.1 Consider an overlapping generations economy with two assets; capital and money. Suppose the number of young people born in period t is determined by N t = 1.5N t-1. Capital pays the gross rate of return x = 1.25. For what values of z will fiat money be valued?
The Tobin Effect Suppose now capital exhibits a diminishing marginal product. There are two assets in the economy; fiat money and capital. People hold both assets as a form of saving when their rates of return are equal: n f '( k) =. z At the equality, the stock of capital, k*, is determined. Suppose there is a permanent increase in the anticipated rate of fiat money creation from z to z. The inflation lowers the rate of return on fiat money, inducing individuals to hold more capital instead of fiat money. The switch to capital increases the capital stock but lowers the marginal product of capital.
The substitution of capital for fiat money in reaction to an increase in anticipated inflation is called the Tobin effect. Since more capital today generates more output in the following period, the larger capital stock implies a subsequent increase in output. Consider the total real output (gross domestic product, GDP) at time t: GDP t = Nt y + Nt 1 f ( k 1). Should we use inflation as a tool to increase output? We should not because: o The goal is to maximize the welfare (utility) not the output. The increase in stock of capital lowers the rate of return on capital. o In the real world, the stock of money is very small relative to the capital stock. The effect will be insignificant.