Solutions to the End-of-Term exam
questions
1. You manage a plant that produces cars by teams of workers using
assembly machines. The technology is summarized by the production
function:
q = 5 KL
where q is the number of engines per week, K is the number of assembly
machines, and L is the number of labor teams.
Each assembly machine rents for r = 12,000 KZT per week, and each team
costs w = 7000 KZT per week.
Engine costs are given by the cost of labor teams and machines, plus 4000
KZT per engine for raw materials.
Your plant has a fixed installation of 7 assembly machines as part of its
design.
What is the cost function for your plant (average cost function) — namely,
how much would it cost to produce q cars? What are average and marginal
costs for producing q cars (for example, when q1=0, and q2=1)?
The short-run production function is q = 5(7)L = 35L, because K is fixed at 7.
This implies that for any level of output q, the number of labor teams hired
will be L = q/35
The total cost function is thus given by the sum of the costs of capital, labor,
and raw materials:
TC(q) = rK +wL +4000q = (12,000)*(7) + (7,000)*(q/35) + 4,000q
TC(q) = 84,000 + 4200q
The average cost function is then given by:
AC (q) = TC (q)/q = (84,000 +4200q)/q
Marginal cost function is given by:
MC (q) = dTC/dq = ((84,000+(4200*1))- (84,000+(4200*0)))/(1-0)= 4 200 KZT
2. Suppose you are the manager of a wallet making firm operating in a
competitive market. Your cost of production is given by C = 400 + 2q^2,
where q is the level of output and C is total cost. (The marginal cost of
production is 4q; the fixed cost is 400 KZT).
If the price of wallets is 400 KZT, how many wallets should you produce to
maximize profit? What will the profit level be?
Profits are maximized where price equals marginal cost. Therefore,
400 = 4q
or q = 100
Profit is equal to total revenue minus total cost: π = Pq – (400 + 2q^2).
Thus,
π = (400)*(100) – (400 + 2(100)^2) = 19 600 KZT
3. A firm produces a product in a competitive industry and has a total cost
function
C = 70 + 4q + 2q^2
and a marginal cost function MC = 4 + 4q.
At the given market price of $40, the firm is producing 5 units of output.
Find the current and profit maximizing levels of the profit of this firm, and the
profit maximizing quantity?
If the firm is maximizing profit, then price will be equal to marginal cost.
P = MC => results in 40 = 4 + 4q
or q = 9
The firm is not maximizing profit, it is producing not enough output.
The current level of profit is
π = Pq – C = 40(5) – (70 + 4(5) + 2(5)^2) = $60
and the profit maximizing level is
π = Pq – C = 40(9) – (70 + 4(9) + 2(9)^2) = $92
4. Suppose the firm’s cost function is C(q) = 6q^2 + 20.
Marginal cost is given by MC = 12q.
Find average cost, average variable cost, and average fixed cost.
Find the output that minimizes average cost.
Variable cost is that part of total cost that depends on q (so VC = 6q^2) and fixed cost
is that part of total cost that does not depend on q (FC = 20).
VC = 6q^2 and FC =20
AC = C(q)/q = (6q^2 + 20)/q = 6q + 20/q
AVC = VC/q = 6q^2 /q = 6q
AFC = FC/q = 20/q
Minimum average cost occurs at the quantity where MC is equal to AC:
AC = 6q + 20/q = 12q = MC
20/q = 6q
20 = 6q^2
3,33 = q^2
q = sqrt of 3,33 = 1,83
5. A firm faces the following average revenue (demand) curve:
P = 140 – 0.02Q
where Q is weekly production and P is price, measured in cents per unit.
The firm’s cost function is given by C = 80Q + 25,000.
Assume that the firm maximizes profits.
What is the level of production, price, and total profit per week?
The profit-maximizing output is found by setting Marginal Revenue equal to Marginal
Cost.
Given a linear demand curve in inverse form, P = 140 – 0.02Q, we know that the
marginal revenue curve has the same intercept and twice the slope of the demand
curve.
Thus, the marginal revenue curve for the firm is MR = 140 – 0.04Q.
Marginal cost is the slope of the total cost curve. The slope of TC = 80Q + 25,000 is 80,
so MC equals 80. Setting MR = MC to determine the profit-maximizing quantity:
140 – 0.04Q = 80
Q = 1500
Substituting the profit-maximizing quantity into the inverse demand function to
determine the price:
P = 140 – (0.02)*(1500) = 110 (cents).
Profit equals total revenue minus total cost:
π = (110)*(1500) – (25,000 + (80)*(1500))
π = 20,000 cents per week, or $200 per week
6. Suppose that Lada (AvtoVAZ) can produce any quantity of cars at a constant
Marginal Cost equal to $0,4 and a fixed cost of $12. You are asked to advise
the CEO as to what prices and quantities Lada company should set for sales in
Kazakhstan and in Russia. The demand for Lada Kalina in each market is given
by
Q(Kz) = 60 - 10P(Kz) and
Q(Ru) = 120 - 20P(Ru)
If Lada Kalina were forced to charge the same price in each market, what
would be the quantity sold in each market, the equilibrium price, and the
company’s profit?
If Lada Kalina must charge the same price in both markets, they must find total demand, Q = Q(Kz) + Q(Ru),
where each price is replaced by the common price
Q(Kz) + Q(Ru) = 60 - 10P + 120 - 20P
Q = 180 – 30P, or in inverse form,
P = (180/30) − (Q/30) = 6 - (Q/30).
Marginal revenue has the same intercept as the inverse demand curve and twice the slope:
MR= 6 – (Q/15)
To find the profit-maximizing quantity, set marginal revenue equal to marginal cost:
MC=0,4 = MR= 6 – (Q/15)
Q = 84 cars
Substituting Q* into the inverse demand equation to determine price:
P = 6 – (84/30) = $3,2
Substitute into the demand equations for the European and American markets to find the quantity sold in
each market:
Q(Kz) = 60 – 10* (3,2), or Q(Kz) = 28 cars in Kazakhstan, and
Q(Ru) = 120 – (20)*(3,2), or Q(Ru) = 56 cars in the Russia
Profit is
π = 3,2* (84) – [12 + 0,4* (84)], or
π = $223,2