Competitive Markets, the Real Interest Rate, and Real Wages

As we have studied it so far, the Solow model assumes a single domestic firm and one national household, which owns this firm. However, because of the constant returns to scale hypothesis, all the properties of this model go through when one assumes competitive markets with many identical firms and households.

Suppose we have many households owning capital and supplying one unit of labor per member. Firms rent capital and labor in competitive capital and labor markets. The interest rate is r(t), and the real wage (per efficiency unit of labor) is w(t). Each firm uses capital and labor and produces according to a production function that, in intensive form, is given by (3.2). Each firm pays the return on capital to households holding its shares and real wages to its workers.

The conditions for profit maximization on the part of firms are that the marginal product of capital equals the user cost of capital (the real interest rate plus the depreciation rate) and that the marginal product of labor equals the real wage. Therefore it holds that

It is easy to see that when (3.17) and (3.18) are satisfied, firms have zero profits, and factor payments exhaust real output. This is a consequence of constant returns to scale.

The total household income per efficiency unit of labor is equal to gross output and is given by

The condition equating savings and investment per efficiency unit of labor is given by

Substituting (3.17) and (3.18) in (3.19), we have the basic accumulation equation of the Solow model:

Consequently, the Solow model is compatible with the existence of competitive markets for goods, labor, and capital.

In the process of adjustment toward the balanced growth path from the left (i.e., when the initial capital per efficiency unit of labor is less than its steady state value), real wages are rising and real interest rates are falling, reflecting the evolution of the falling marginal product of capital and the rising marginal product of labor.

On the balanced growth path, the real wage (per efficiency unit of labor) remains constant, and the same happens with the real interest rate. However, the real wage per employee, along with all other per capita variables, is growing at a rate g, the exogenous rate of technical progress.

Exercise 3.3 Assuming a Cobb-Douglas production function, as in exercise 3.1, and competitive product and factor markets, derive and discuss the conditions determining the real interest rate and the real wage in the Solow model. Also derive and discuss the steady state real interest rate and the steady state real wage per efficiency unit of labor, as functions of the savings rate and the other parameters of the model. What is the growth rate of real wages per worker in the steady state?

3.3