1. Consider a price-taking producer of a certain service. She requires two input
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Question
1. Consider a price-taking producer of a certain service. She requires two inputs to produce this service: input A and input B. She can adjust the quantity of input A immediately, but it takes some time for her to adjust the quantity of input B Denote the unit price of input A by wA, and that of input B by we. Denote the quantity of input A by gA, and that of input B by ga. She can sell her service at the price of p per unit. Suppose that her production technology can be represented by the following Cobb-Douglas production function fleA, ge) (BA)"(ge)b with 0Explanation / Answer
q = f(gA, gB) = (gA)a(gB)b
As per given data, input A is variable and input B is fixed.
Total cost, C = wA.gA + wB.gB
(a) Short-run objective is to minimize costs, subject to an output level of q. Expressed Mathematically,
Minimize C = wA.gA + wB.gB
Subject to
q = (gA)a(gB)b
(b) In short-run, total cost is minimized when Marginal product of variable input (MPA) equals its unit price, i.e.
MPA = wA
MPA = q / gA = a. (gB)b(gA)a - 1 = wA
(gA)a - 1 = wA / [a. (gB)b]
gA = {wA / [a. (gB)b]}[1 / (a - 1)]
(c)
Relationship between cost-minimizing quantity of input A (gA derived above) is inversely related to quantity of the fixed input used (gB), because as seen in above relationship, as gB increases (decreases), value of gA decreases (increases) ceteris paribus.
This relationship makes sense because amount of input B being fixed, as more units of input A are used, less units of input B are used.
(d)
We have
a. (gB)b(gA)a - 1 = wA
(gB)b = [wA / (a.(gA)a - 1)]
gB = [wA / (a.(gA)a - 1)](1/b)
Substituting above value of gB in production function,
q = (gA)a. [wA / (a.(gA)a - 1)](1/b)
q = (gA)(1/b). (wA / a)(1/b)
So, as gA increases (decreases), output (q) also increases (decreases). This makes correct sense because (at least in initial short-runstages), as more of the variable input is used, output increases
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