1. The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost o
ID: 1192365 • Letter: 1
Question
1. The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost of production is
C(Q) = Q^3-20Q^2+125Q. The firm’s problem is to choose the value of Q> or = 0 that maximizes its profit. You
may occasionally find an irrational number and in those cases simplify your answer as much as possible.
(a) Calculate the firm’s inverse demand function.
(b) Calculate the firm’s marginal and average cost functions.
(c) Over what range of Q does the firm have economies to scale? Over what range of Q does it have
diseconomies to scale? What is the firm’s lowest possible average cost of production?
(d) Does the firm’s profit-maximization problem satisfy the global SOC?
(e) Find all values of Q (if any) that satisfy the first-order condition for the firm’s problem.
(f) Calculate the firm’s profit-maximizing price and quantity. Justify your answer carefully.
(g) Calculate the firm’s maximized profit, and the revenue and cost that produce that profit.
(h) Calculate the elasticity of demand at the profit-maximizing point.
(i) What is the firm’s markup at the profit-maximizing point? Confirm that this markup has the
expected relationship to the elasticity of demand calculated in part (h).
(j) Calculate the price(s) that would cause the firm to break even, meaning: earn exactly zero profit.
(k) For this part only, change the demand function by assuming that demand (at any given price) is
half of what it was before. In this new situation, calculate the firm’s inverse demand function,
profit-maximizing point, and maximized profit.
(l) For this part only, suppose that the problem is to maximize revenue instead of profit. Does this
problem satisfy the global SOC? Find all points (if any) that satisfy the FOC. Calculate the
revenue-maximizing price and quantity. (Justify your answer carefully.) Calculate the firm’s
maximized revenue. How much profit does the firm sacrifice by choosing to maximize revenue
instead of profit?
[The remaining parts of the original assignment are postponed to HW #10.]
2. The demand function for a firm’s product is Q = P-3. The firm’s marginal cost of production is
constant at MC(Q) = 12.
(a) Calculate the elasticity of demand, as a function of Q.
(b) Does the firm’s profit maximization problem satisfy the global SOC?
(c) Using your answers to (a) and (b), what is the firm’s profit-maximizing markup? (Justify your
answer carefully. Do not forget about the possibility of a boundary solution.)
(d) Based on your answer to (c), what is the firm’s profit-maximizing price?
(e) Based on your answer to (d), what is the firm’s profit-maximizing quantity
Explanation / Answer
a) Demand function : Q(P) = 50-P/10
Inverse demand function: P = 50*10 - Q*10
P = 500 - 10Q
b) Total cost function : C(Q) = Q3-20Q2+125Q
Marginal Cost function: dC(Q)/dQ = d(Q3-20Q2+125Q)/dQ = 3Q2 - 40Q + 125
Average Cost function : C(Q)/Q = (Q3-20Q2+125Q) / Q = Q2 - 20Q + 125
c) Economies of scale occur when MC < AC and diseconomies of scale occur when MC > AC.
putting MC<AC to get range of economies of scale.
3Q2 - 40Q + 125 < Q2 - 20Q + 125
Q < 10 units
Hence less than 10 units firm will be having economies of scale.
Now putting MC>AC to get diseconomies of scale
3Q2 - 40Q + 125 > Q2 - 20Q + 125
Q > 10
Hence more than 10 units will give diseconomies of scale.
At firms lowest average cost, average cost equals marginal cost.
Equating Average cost function and marginal cost function.
Q2 - 20Q + 125 = 3Q2 - 40Q + 125
3Q2 - Q2 = 40Q - 20Q
2Q2 = 20Q
Q = 20/2 = 10 units
at 10 units of Q, average cost will be minimum.
d) second order condition of profit maximization is that second order derivative of profit function should be negative.
Profit = Q*P - Cost
Profit = (500 - 10Q)Q - (Q3-20Q2+125Q) = 500Q - 10Q2 - Q3 + 20Q2 - 125Q
First order derivative = dProfit/dQ = 500 - 20Q - 3Q2 + 40Q - 125
Second order derivative = -20 - 6Q + 40 = - 6Q + 20
as second order derivate is negative it satisfies global SOC.
f) profit maximizing output occurs at point where MC equals MR
TR = P*Q = 500Q - 10Q2
MR = dTR/dQ = 500 - 20Q
Equating MR and MC
500 - 20 Q = Q2 - 20Q + 125
375 = Q2
Q = 19.36 = approx 20 units.
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