Role of money in the Solow model

A core proposition in neoclassical economics, especially textbook neoclassical economics, is that the income earned by each of “factors of production” (essentially, labour and “capital”) is equal to its marginal product.

Thus, the wage is alleged to be equal to the marginal product of labour, and the rate of profit equal to the marginal product of capital. A second core proposition is that changes in the price of a factor of production say, a fall in the rate of profit will lead to more of that factor being used in production.

A fall in this price means that more will be used since the law of diminishing returns implies that greater use of this input will imply a lower marginal product, all else equal.

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The traditional way to aggregate is to multiply the amount of each type of capital goods by its price and then to add up these multiples. Ideally, this sum would then be corrected for the effects of inflation.

A problem with this method arises from variations in the ratio of labour to the value of capital goods used in production across sectors. At different income distributions, prices would have to differ if the competitive market assumption of equal rates of profits in all sectors is to hold. For example, suppose a higher rate of profits and lower wage were to prevail than at the initial situation.

The prices of capital goods used in the less capital-intensive sectors would seem to need to rise with respect to the prices of capital goods used in more capital-intensive sectors, thereby ensuring the rate of profits remains identical across sectors.

But additional complications arise from the varying capital intensities in the sectors producing capital goods. At any rate, the price of a capital good, or of any arbitrary given set of capital goods, cannot be expected to remain constant across variations in the rate of profits.

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However, Sraffa then pointed out that this accurate measuring technique still involved the rate of profit: the amount of capital depended on the rate of profit. This reversed the direction of causality that neoclassical economics assumed between the rate of profit and the amount of capital.

According to neoclassical production theory, an increase in the amount of capital employed should cause a fall in the rate of profit. Sraffa instead showed that a change in the rate of profit would change the measured amount of capital, and in highly nonlinear ways: an increase in the rate of profit might initially increase the perceived value of the truck more than the laser, but then reverse the effect at still higher rates of profit.

This aggregation problem was thus a serious challenge to the neoclassical theories of income distribution and of production, which is why the debate was so important.

In neoclassical economics, a production function is often assumed, for example,

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Where q is the sum of all output, K is the sum of all capital goods, and L is the sum of all labour input. Both of the inputs have a positive impact on output, with diminishing marginal returns. In neoclassical growth theory, this function is assumed to apply to the entire economy.

Then, the neoclassical theory of the distribution of income sketched above is assumed to apply: under perfect competition, the rate of return on capital goods (r) equals the marginal product of capital goods, while the wage rate equals the marginal product of labour.

The problem here can be understood by thinking about an increase in the r (corresponding to a fall in w). This causes a change in the distribution of income, thus a change in the prices of different capital goods, and finally a change in the value of K.

So the rate of return on K (i.e., r) is not independent of the measure of K as assumed in the neoclassical model of growth and distribution. Causation goes both ways, from K to r and from r to K.

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This is a problem that is also indicated by general equilibrium theory as with the Sonnenshein Mantel-Debreu theorem, which rejects all representative agent models and other inappropriate aggregations, except under very restrictive conditions. Note that this says that it’s not simply K that is subject to aggregation problems: so is L.