Question

Drin uses labor time (L) and equipment time (K) to produce vintage alarm clocks. Drin’s daily production function is q = LK.

1. Does the production function displays increasing, constant, or decreasing returns to scale? Do you expect Drin’s long run average cost of production to be increasing (diseconomies of scale), constant, or decreasing (economies of scale) in quantity? Why? At Drin, the daily wage rate is $200 while the daily rental cost of capital is $400.
2. Leaving Drin’s level of output indicated as a parameter q, solve the firm’s cost minimization problem and find Drin’s conditional labor demand L^(q) and conditional equipment demand K^(q).
3. Substitute these functions into the expression for Total Cost = wL^ + rK^ and find Drin’s long run total cost function.
4. What share of total cost comes from the cost of labor? What share comes from the cost of equipment?
5. Find Drin’s Average Total Cost function.

Answer 1

Drin's production function suggests constant returns to scale, indicating that long run average cost will remain constant as output levels change. The firm's conditional labor and equipment demand functions can be calculated for cost minimization, which leads to the derivation of the long run total cost function and the shares of labor and capital in total costs. Lastly, Drin's Average Total Cost function is obtained by dividing total cost by output.

Step-by-step explanation:

Drin's production function q = LK suggests that the company experiences constant returns to scale, as doubling the input of labor (L) and equipment time (K) will also double the output (q). This would typically lead to the expectation that Drin's long run average cost (LRAC) will be constant over a range of output levels, providing neither economies nor diseconomies of scale.

To solve Drin's cost minimization problem for a given level of output q, we set up the minimization of Total Cost = wL + rK subject to the production function q = LK. Drin's daily wage rate is given as $200 (w) and the daily rental cost of capital is $400 (r). Taking the derivative of the cost function with respect to L and K gives the optimal ratio of labor to capital which is minimized cost. The conditional labor demand function L^(q) and conditional equipment demand K^(q) are derived from this optimization.

By substituting L^(q) and K^(q) into the expression for Total Cost = wL + rK, we arrive at Drin's long run total cost function which explains how total costs change with different levels of output, q, in the long run. The share of total cost coming from the cost of labor and the cost of equipment can be ascertained by looking at the respective coefficients of L and K in the total cost function. Drin's Average Total Cost function can be found by dividing the total cost function by q.