Math Problem Solver

Let 𝑝1 and 𝑝2 be the prices of the input factors 𝑧1 and 𝑧2 respectively, and let 𝑞 be a given

output target. Find the cost function of the following firms with different CES production

functions. Does either technology exhibit increasing returns to scale? Decreasing returns?

Constant returns? How can you tell by looking at the cost functions you derived above?

(a) 𝑓(𝑧1,𝑧2) = √𝑧1 +√

𝑧2

(b) 𝑓(𝑧1,𝑧2) = (√𝑧1 +√

𝑧2)2.

[2 ×10 = 20 points]

2. Suppose that a firm owns two plants, each producing the same good. Every plant 𝑗′𝑠

average cost is given by

𝐴𝐶𝑗(𝑞𝑗) = 𝛼+𝛽𝑗𝑞𝑗

for 𝑞𝑗 ≥ 0 and both 𝑗 = 1,2. The coefficient 𝛽𝑗 may differ from plant to plant, i.e., if 𝛽1 > 𝛽2

plant 2 is more efficient than plant 1, since its average costs increase less rapidly in output.

Assume that you are asked to determine the cost-minimizing distribution of aggregate output

𝑞 = 𝑞1 +𝑞2, among the two plants (i.e., for a given aggregate output in output 𝑞, how much

𝑞1 to produce in plant 1 and how much 𝑞2 to produce in plant 2). Here 𝑞 is a given output

target.

For simplicity, assume that the output target 𝑞 is such that

𝑞 <

𝛼

max{|𝛽1|,|𝛽2|} .

(a) If 𝛽𝑗 > 0 for every plant 𝑗, how should output be located among the two plants?

(1)

(b) If 𝛽𝑗 < 0 for every plant 𝑗, how should output be located among the two plants? Why

is the restriction on 𝑞 in (1) necessary?

(c) Suppose that 𝛽1 > 0 and 𝛽2 < 0. How should output be located among the two plants?

[3 ×10 = 30 points