Question 1 (8 points)
Consider the following relationships that describe cost and pricing behaviour of a monopolistically competitive firm producing SUVs
AC = a + (F0/S0) N Firm average cost relationship, a> 0 P = a + (b/N) Firm pricing function, b > 0
P = AC Equilibrium condition
The first equation describes the relationship between the firm’s average cost (AC) and the number of competing firms (N) in the SUV industry, with exogenously given market size (S0) and fixed cost (F0). The constant a is the firm’s average variable cost. Since fixed costs and market size are positive, this equation implies that as the number of firms N increases, the average cost of each firm rises since each firm will be producing a smaller output. The second equation describes pricing behaviour. Price can be seen as a mark-up over average variable cost, the mark-up being (b/N). Since b is a positive constant, this means that as the number of firms in the industry increases, competition leads to lower prices.
The equilibrium condition simply means that the industry will be in equilibrium when price equals average cost. This implies that firms are making only normal profits, and there is no incentive to leave or enter the industry.
Use the equilibrium condition to solve for the equilibrium number of firms in the industry. (3 points)
Then use calculus to determine the impact on the equilibrium number of firms of a change market size, assuming that each firm has fixed cost of $750 million). (3 points)
Using the price equation to determine the impact of market size on equilibrium price. (2 points)
Question 2 (10 points) Parts A, B and C are unrelated
Suppose consumption C is related to income X as follows: C = a + bX where a > 0 and 0 < b < 1.
What is the relationship between the average and marginal functions, assuming that X > 0? Draw a rough graph to illustrate your answer (3 points)
Show that the elasticity of C with respect to X is (C-a)/C. (2 points)
The supply function of a commodity is: Y = aXb, where Y is quantity supplied and X is the price received by the producer, and a and b are positive. If X = hP, where P is the market price, and h is a positive constant, representing the fraction of the market price actually received by suppliers (after
paying transportation costs e.t.c), find the impact on Y of a change in the market price P. What does this derivative tell you about the nature of the relationship between Y and P ? That is, is it a positive or inverse relationship? (2 points)
If a firm’s revenue R =100Q – 2Q2, where Q is output and a and b are positive constants, and Q = 3L0.5, find
labour’s marginal revenue product (that is, dR/dL). (3 points)
Question 3 (7 points)
Consider the production function: Q = aXb, where Q is output, and X is the stock of knowledge, and a and
b are constants such that a > 0, and 0 < b < 1.
Use derivatives to determine whether the law of diminishing returns to knowledge holds. (2 points)
If the production function of knowledge is: X = g + ht, where t stands for time, and g and h are positive constants, obtain the derivative dQ/dt. (2 points)
Another way of interpreting the stock of knowledge X is to think of it as the stock of human capital – that is, the number of workers (N) adjusted for quality or efficiency (v). For instance, we can define X = vN, where X is now labour input measured in efficiency units. According to efficiency wage theory, efficiency is positively related to the real wage; that is, higher wages tend to promote worker efficiency. Suppose then that v = wh, where h is a positive constant. Use calculus to show that the impact of the real wage on output if N=10. (3 points)