Question

There are 20 firms in a perfectly competitive industry. Ten firms have production function q = 2L^0.5 while the other ten firms have production function q = 4L^0.5. We label the first type of firms, “low productivity” and the second type of firms, “high productivity.” All firms sell their output at a unit price of P = $1. All firms are price takers in the labor market.

Find the profit maximizing demand for labor of the two types of firms.

Find the overall demand for labor.

Suppose there are 200 workers, find the equilibrium wage and the number of workers employed by the two types of firms.

In two separate diagrams, draw the production function of low productivity and high productivity firms and illustrate their profit maximizing number of workers.

Workers are efficiently allocated across firms when overall output is maximized.

How many workers would you allocate to the low productivity firms and how many to the high productivity firms to make sure the allocation is efficient? Show your work.

Suppose government imposes a 50% tax on the wages “high productivity” firms pay to their employees but not on the wages “low productivity” firms pay to theirs. The tax lowers the profit of “high productivity” firms at each number of workers employed to Profit(L) = q(L) – (1 +50%)wL.

How does this tax affect the net wage w paid to workers?

How does it affect the allocation of workers between types of firms?

What are the total output consequences of this tax?

Suppose government imposed the 50% tax on wages on firms with “low productivity” as well. Would efficiency increase or decrease? Justify your answer.

Answer 1

The production functions are and . All firms are price taker in the product and input market, and price is $1. The cost function will be , where w is wage.

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The profit function for the low productivity firms will be , ie . The profit maximization problem will be hence . The FOC will be , ie or or or or . Which will be the demand for labor by the low productivity firm.

The profit function for the high productivity firms will be , ie . The profit maximization problem will be hence . The FOC will be , ie or or or or . Which will be the demand for labor by the high productivity firm.

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The overall demand for labor will be the aggregate/sum of the demand for labor of both firms, which is , ie or .

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The labor supply is given as a constant of amount 200, ie . The labor market will be in equilibrium when labor demand is equal to the labor supply, ie or or or , which is the equilibrium wage. At this wage, the labor demand in both firms are and .

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The low productivity scenario:

The high productivity scenario: