Question

You own a small business that repairs furnaces and air conditioners. The market is competitive so you must charge the same price as all other firms - $50 per house visit. You can take on as many customers as you would like when you charge this price.

Your daily production function is given below, where K = the number of vans you have and L = the number of employees you have each day: = (KK, ) = 10KK21/2 (round down if you find a fractional unit of Q. For example, if you pick K and L such that Q=3.12 then round down to Q=3 in this case)

You currently lease 2 vans and cannot change this in the short run. The daily wage for one of your workers is w=$250, and since there is a competitive market for labor you cannot change this either. The daily rental rate on one van is r=$500.

If everything is the same in this problem except r = $500, how many workers do you want to hire? Show your calculations / explain.
Suppose again r = $500 but instead K = 3 (and everything else remains unchanged). How many workers do you want to hire? Show your calculations / explain. [Use the expression for MPL that I give above, or solve in Excel].
Using your answers to parts #1 and 2 above, what is the marginal benefit from the 3rd unit of K? What is the marginal cost of the 3rd unit of K?
If you could choose to buy a 3rd unit of K or not, would you? You can't choose any other value: it's either 2 or 3. Explain.

Answer 1

A profit-maximizing small business that repairs furnaces and air conditioners would hire 4 workers when they have 2 vans (K=2) due to the market conditions described. Calculations are based on equating the marginal revenue product to the market wage. Analysis for 3 vans (K=3) follows the same procedure to find profit-maximizing employment level and deciding whether to hire an additional van involves comparing extra revenue to its cost.

Step-by-step explanation:

To determine the number of workers to hire for a profit-maximizing firm, we must equate the marginal revenue product (MRP) to the market wage. Since this is a competitive market, the price of service per house visit is fixed at $50, therefore the revenue generated per house visit is constant at this price.

Assuming the production function Q = 10K√L and K=2, the marginal product of labor (MPL) would be dQ/dL = 10K/(2√L) = 10(2)/(2√L) = 10/√L. The MRP is the MPL multiplied by the price of output, and since each house visit costs $50, the MRP is 50(10/√L).

Setting the MRP equal to the wage rate, we get 50(10/√L) = 250, which simplifies to 10/√L = 5. Solving for L gives us L = 4. So, when K=2 and the van rental rate r is $500, the profit-maximizing level of employment is 4 workers.

For K=3, the marginal product of labor would be MPL = 10(3)/(2√L), and following a similar setup, you would find the number of workers that equalizes the market wage with the marginal revenue product. Lastly, the marginal benefit from the 3rd unit of K is the increase in Q, while the marginal cost is the increase in costs from hiring the additional van, which is $500. Comparing the additional revenue generated from the 3rd van to its cost can help decide whether to hire the extra unit of K or not.