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Consider a firm that produces using capital (K) and labor (N) given a production technology represented with the following production function:

zF (K, N) = zK^αN^(1−α)

The firm has a fixed amount of capital K and can hire labor at a market (given) wage (w) (per unit of labor hired). Then the profits of the firm are:

π = max N zF (K, N) − wN

= max zK^αN^(1−α) − wN

(a) Find the FOC of the firm and interpret it.

(b) Solve for the optimal labor demand of the firm.

(c) Is the labor demand increasing in K? Why?

(d) The elasticity of the labor demand to wages is defined as: ξw =∂ ln N/∂ ln w

Find it and interpret it.

(e) The elasticity with respect to capital and productivity measures how much the demand for labor varies when the amount capital or the productivity vary. We can use our labor demand function to find these elasticities:

ξK =∂ ln N/∂ ln K, ξz =∂ ln N/∂ ln z

Find them and interpret them.

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Jean Keeling
Jean KeelingLv2
4 Aug 2019

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Related questions

Consider a representative firm that produces using capital (K) and labor (N), but that owns no capital. The firm can hire workers in the labor market at a wage w and can rent capital market at a rental rate r (that is, the firm can rent K units of capital for which it would have to pay rK). The production function of the firm is given by zF (K, N) where: F (K, N) = KaN B and a + B < 1.

a) (2 points) Does the technology exhibit constant returns to scale? If not does it exhibit increasing returns, decreasing returns, or neither?

b) (2 points) Pose the profit maximization problem of the firm.

c) (2 points) Find the firm's optimal capital and labor demand. Hint: You can combine the firm's FOC by taking their ratio (diving one by the other) to get the labor demand as a function of the capital. Then you can use this to replace the FOC to get the capital demand as a function of prices and parameters.

d) (2 points) Find the firm's optimal profits. Hint: Use the firm's FOC to replace for the prices (r and w), then replace for labor as a function of capital (that you found above) and finally replace for the optimal capital demand.

e) (2 points) What happens to the optimal profits when B = 1 - a?

Consider an economy with a representative consumer like the one described above, and a representative firm like The production function of the firm is given by zF (K, N) where: F (K, N) = KαN β and α + β < 1.The firm can hire workers in the labor market at a wage w and can rent capital market at a rental rate r (that is, the firm can rent K units of capital for which it would have to pay rK). For this problem assume that there is one unit of capital owned by a foreign investor. The investor’s capital supply is perfectly inelastic (that is she will supply one unit of capital at any rental rate).

a) (2.5 points) The equilibrium rental rate must be such that the demand and supply of capital are equalized. Find the equilibrium rental rate. Hint: You can express the equilibrium rental as a function of the wage. For this is is useful to use the FOC of the firm with respect to capital.

b) (2.5 points) The equilibrium wage must be such that the demand and supply of labor are equalized. Find the equilibrium wage. Hint: You can get an easier expression for the labor demand when you use the labor demand as a function of capital (obtained by combining the firm’s FOC) and set capital to its market clearing value. Then you can solve for the wage. Remember to replace by the equilibrium interest rate from the previous point.

c) (2.5 points) With the equilibrium prices solve explicitly for the equilibrium labor. Hint: You can get a better expression for the equilibrium labor using the labor supply and replacing by the wage you obtained above. d) (2.5 points) In equilibrium, what happens to the wage, interest rate, labor and consumption when the government increases the labor taxes? Hint: Find out first what happens to the wage, and then use your result for the equilibrium interest rate to find out what happens to that variable. The effect on labor can be obtained directly from your answer to (c) or from the FOC of the firm with respect to labor. Finally, get the effect on profits before finding out the effect that extra taxes have on consumption.

Short-run factor demand

A price-taking firm (p = 100) chooses an amount of labor to employ that maximizes its profits. It has a fixed amount of capital (K= 100), and constant returns to scale CobbDouglass production function: q(L, K) = L ^(1/2) K ^(1/2). The wage rate and rental rate of capital are 10 (w = r = 10).

(a) Set up the profit function.

(b) Solve for the optimal level of labor IN THE SHORT RUN.

(c) With additional capital, would the optimal level of labor increase or decrease?

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