Question

For each of the following production functions:

(i) Calculate the marginal rate of technical substitution (if it is defined).
(ii) Sketch two isoquants, one for q = 81 and another for q = 243. Add in some isocost lines, assuming w = 2 and r = 4.
(iii) Solve for the firm’s cost function C(q) as a function of q. (a) f(L, K) = LK³ (b) f(L, K) = L + 3K (c) f(L, K) = [min{L, K}]³

Answer 1

The student's economics question involves understanding and applying production functions and cost analysis by finding the marginal rate of technical substitution, sketching isoquants with isocost lines, and deriving the cost function for different production scenarios in a firm.

Step-by-step explanation:

The student's question pertains to calculating the marginal rate of technical substitution (MRTS), sketching isoquants, and deriving the firm’s cost function C(q) for three different production functions. The MRTS can be found using calculus by taking the ratio of the marginal products of labor (L) and capital (K). However, detailed calculations for each function are not shown here. Sketching isoquants involves plotting level sets of the production function to show combinations of L and K that yield the same output. Isocost lines, representing all combinations of L and K that cost the same, can be drawn given the wage rate (w) and rental rate of capital (r). Finally, the firm's cost function is derived by finding the lowest cost combination of inputs required to produce a given level of output q.

To assist with the understanding of production functions and their related cost functions, it is useful to consider a production function as an equation that shows the relationship between inputs (like labor and capital) and output. The cost function then follows from this by determining the least cost way to produce a given level of output, incorporating input costs such as wages and rent.