Aggregate Capital Accumulation in a Ramsey Model with Money

We next turn to the determinants of capital accumulation, output, and other real variables, such as the real interest rate and real wages.

7.2.1 The Production Function, the Real Interest Rate, and the Real Wage

As in the previous models, assume that output per efficiency unit of labor is determined by

where f is a neoclassical production function with all the usual properties.

We have already assumed an exogenous rate of growth of population n and an exogenous rate of technical progress g. The depreciation rate of the capital stock will be assumed equal to δ, where 1 > δ > 0.

In a competitive equilibrium, assuming that firms maximize profits, the real interest rate r(t) and the real wage per efficiency unit of labor w(t) will be determined by the usual marginal productivity conditions:

7.2.2 The Inflation Tax and the Accumulation of Capital

We shall assume that the assets of household j in equation (7.2) consist of capital, government bonds, and money,

where k_j, d_j, and m_j denote physical capital, real government bonds, and real money balances, respectively, held by an average member of household j.

Substituting (7.22) in (7.2), after multiplying by L(t), we get an accumulation equation for total household assets in the economy:

The rate of growth of real money balances is given by μ − π(t).

Substituting (7.24) into (7.23) and solving for the accumulation of physical capital and real government bonds, we get

The aggregate accumulation of capital and real government bonds by households depends on the difference between aggregate household disposable income and aggregate household consumption. The disposable income of households consists of their total asset and labor income minus taxes T(t) and the inflation tax imposed by the government through the growth rate of the money supply. Government revenue from the monopoly of issuing money, usually referred to as seigniorage, is equal to μ(M(t)/P(t)) and comprises a tax on real money balances held by households.

Equation (7.25) describes the budget constraint of households. The government budget constraint in an economy in which the government has the monopoly of money creation is described by

which suggests that the government accumulates debt to the extent that primary government expenditure C_g plus the real interest expenditure on existing debt rD exceeds total taxes (minus transfers) T plus the inflation tax μ(M/P).

Substituting the government budget constraint (7.26) into the household budget constraint (7.25), we end up with the well-known equation for aggregate capital accumulation:

Only primary government expenditure, and not the way it is financed, appears in the aggregate capital accumulation equation. Debt, taxes, and seigniorage revenue do not affect the accumulation of capital.

Expressing both sides of (7.27) per efficiency unit of labor (i.e., dividing by h(t)L(t)), we get

The economy accumulates capital per efficiency unit of labor when total savings per efficiency unit of labor exceed the investment required to maintain a constant capital stock per efficiency unit of labor. This is the same as the capital accumulation equation in the Ramsey model without money. The existence of money and money demand also does not affect the accumulation of real capital.

7.3