### An Efficiency Wage Approach to the Krugman Puzzle

**Matthew Martin**1/04/2013 09:17:00 PM

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Just a thought to add on the whole "capital-biased" technological change discussion that has been going on in the econosphere (see Krugman here, here, here, here, and here; Nick Rowe here, here, and here; and me here, among others.) The puzzle is to explain why

So, lets re-examine the Krugman puzzle assuming that that is exactly what happens in the aggregate economy. That's a big assumption, because not all markets are alike, and it is not profitable in every market for firms to pay higher wages inexchange for more productive workers. But lets roll with it anyway.

I'm unfamiliar with modeling efficiency wages in modern rational expectations macro models. . So, take this with a grain of salt. We start with households--I'm ignoring for the moment the microfoundations of the efficiency wage phenomenon (it will simply be assumed in the production function), so the household seeks to seeks only to choose consumption and capital to maximize expected lifetime utility:

Which gives us the standard Euler consumption equation:

Now we turn to the firm problem. The firm has a production function that looks like this:

Note that as before, capital and consumption are perfect substitutes, and that alpha is the parameter of interest controlling the degree of capital-biasedness. However, there is an additional input e, representing worker efficiency, which is a continuously differentiable function that is increasing in the real wage rate (there are some other conditions needed to guarantee an efficiency wage solution, so let me summarize: etc.) Here is the firm's profit maximization problem:

and the first order conditions (FOCs)

combining the first two gives us the Solow condition:

Now, there are many different possible formulations for the efficiency function e, but supposing in this case that it depends only on the wage and not any other factors like unemployment, the Solow condition fully characterizes the equilibrium wage rate, denoted w* here on out. Note that e with the subscript w means the first derivative of e with respect to the wage rate. Now, plugging the second FOC (L) into the last FOC (K) gives us the rental rate on capital:

Recalling from the Euler equation, this implies that in the steady state we must have

which is not in general true (except accidentally with probability zero). That means the steady state does not exist at an interior solution--firms either want to accumulate more capital or sell it. Now, I hate Karush-Kuhn-Tucker conditions almost as much as I hate tedious algebra, so let me divine some answers here, and check the math at a later date. There are two cases. The first is

which we can ignore because it is not characterized by an efficiency wage equilibrium (that is, competitive firms will bid up wages to market clearing, and we have a whole different ball game. The second case is

which implies that firms want to accumulate capital. From the second FOC (L) we know that labor demand is increasing in the amount of capital, so labor demand also rises. Hence, firms will accumulate capital until we hit the boundary condition where L=1, the total amount of labor available. Thus, the steady state capital is where

Note that as alpha approaches one, steady state capital increases, the rental rate on capital decreases, while labor and the wage rate do not change.

This seems to me to solve the Krugman puzzle. Then again, I'm doubtful both that I did this math right and that this is a realistic model to begin with. That's why I'm publishing it on a blog.

- an increasing share of income is going to capital instead of workers
- workers incomes are stagnating
- interest rates are falling

So, lets re-examine the Krugman puzzle assuming that that is exactly what happens in the aggregate economy. That's a big assumption, because not all markets are alike, and it is not profitable in every market for firms to pay higher wages inexchange for more productive workers. But lets roll with it anyway.

I'm unfamiliar with modeling efficiency wages in modern rational expectations macro models. . So, take this with a grain of salt. We start with households--I'm ignoring for the moment the microfoundations of the efficiency wage phenomenon (it will simply be assumed in the production function), so the household seeks to seeks only to choose consumption and capital to maximize expected lifetime utility:

Which gives us the standard Euler consumption equation:

Now we turn to the firm problem. The firm has a production function that looks like this:

Note that as before, capital and consumption are perfect substitutes, and that alpha is the parameter of interest controlling the degree of capital-biasedness. However, there is an additional input e, representing worker efficiency, which is a continuously differentiable function that is increasing in the real wage rate (there are some other conditions needed to guarantee an efficiency wage solution, so let me summarize: etc.) Here is the firm's profit maximization problem:

and the first order conditions (FOCs)

combining the first two gives us the Solow condition:

Now, there are many different possible formulations for the efficiency function e, but supposing in this case that it depends only on the wage and not any other factors like unemployment, the Solow condition fully characterizes the equilibrium wage rate, denoted w* here on out. Note that e with the subscript w means the first derivative of e with respect to the wage rate. Now, plugging the second FOC (L) into the last FOC (K) gives us the rental rate on capital:

Recalling from the Euler equation, this implies that in the steady state we must have

which is not in general true (except accidentally with probability zero). That means the steady state does not exist at an interior solution--firms either want to accumulate more capital or sell it. Now, I hate Karush-Kuhn-Tucker conditions almost as much as I hate tedious algebra, so let me divine some answers here, and check the math at a later date. There are two cases. The first is

which we can ignore because it is not characterized by an efficiency wage equilibrium (that is, competitive firms will bid up wages to market clearing, and we have a whole different ball game. The second case is

which implies that firms want to accumulate capital. From the second FOC (L) we know that labor demand is increasing in the amount of capital, so labor demand also rises. Hence, firms will accumulate capital until we hit the boundary condition where L=1, the total amount of labor available. Thus, the steady state capital is where

Note that as alpha approaches one, steady state capital increases, the rental rate on capital decreases, while labor and the wage rate do not change.

This seems to me to solve the Krugman puzzle. Then again, I'm doubtful both that I did this math right and that this is a realistic model to begin with. That's why I'm publishing it on a blog.

Nick Rowe1/05/2013 01:17:00 AM