1. ECON 330 Lecture 14
Monday, November 9

2. Review questions for the midterm

3. Topics The competitive model The monopoly equilibrium, the DWL.
The equilibrium of the dominant firm
Oligopoly models
Price competition
The Cournot model

4. Monopoly question

5. A monopoly firm has TC(q) = 2q. This means AC = MC = 2
A monopoly firm has TC(q) = 2q. This means AC = MC = 2. Compute the monopoly equilibrium price, quantity, CS, and PS.
Demand is Q = 20 – 2P. The inverse demand is P = 10 – Q/2. Write the profit function (π = TR - TC) π = pxq – TC(q) π = [10 – q/2]xq – 2q Differentiate π with respect to q, set equal to zero: 10 – q – 2 = 0 (MR = 10 – q, MC = 2, so this condition is the MR = MC condition.) Solve 10 – q – 2 = 0 for q:  Q* = 8.

6. Use the inverse demand P = 10 – Q/2 to find the price P. : P. = 6
Use the inverse demand P = 10 – Q/2 to find the price P*: P* = 6. Profit is (p - AC)xq = (6 – 2)x8 = 32. CS is (10 – 6)x8/2 = 16. Social welfare is profit + CS = 48.

7. The competitive equilibrium question

8. Consider a competitive market with 9 firms
Consider a competitive market with 9 firms. All firms use the same production technology and have the same cost function: c(q) = F + 0.5q2; where F > 0. The market demand is Q(p) = 10 – p. Compute the competitive equilibrium (price, quantity, consumer surplus, producer surplus, etc).

9. Write the profit function for one of these firms π = pxq–0. 5q2
Write the profit function for one of these firms π = pxq–0.5q2. Note the p is independent (not a function of) q, the firm’s output. This is the price taking assumption: When choosing its output level, the firm assumes that it can sell any q at given fixed price p. Differentiate π with respect to q, set equal to zero: p – q = 0  (This is the P = MC condition, note that the firm’s MC is q.) Rewrite this as q(p) = …

10. The supply function of a single competitive firm is qs(p) = p
The supply function of a single competitive firm is qs(p) = p. Example: if p =0.5, the competitive firm will supply 0.5 units of output. if p =2, the competitive firm will supply 2 units of output. The market supply with 9 such firms is S(p) = 9xqs(p) = 9p.

11. Market equilibrium with market demand Q(p) = 10 – p.
We use the “demand = supply” condition to compute the competitive equilibrium price:
10 – p = 9p 
Solve this for p
p* = 1,
Put this back into the market demand Q(p) = 10 – p
Q* = 9.
At p = 1, each firm will produce 1 unit of output.

12. Profits CS etc
Producer’s surplus PS Either the blue area (pxq)/2 = (1x9)/2 = 4.5, or the total profits excluding the fixed costs (9xF) If we use the profit function. Per firm: π = 1x1 – 0.5(1)2 = 0.5. Producer’s surplus PS = 9x0.5 = 4.5. Consumer surplus is the pinkish/reddish area CS = [(10 – 1)x9]/2 = Total welfare = = 45.

13. Under the standard assumptions of competitive markets, can P = 1 be the long-run equilibrium price if F = 1? Please explain.

14. The cost function is c(q) = 0.5q2. The market demand is Q(p) = 10 – p.
Suppose these firms are not happy with competition and decide to join forces. They reorganize themselves as a single firm that becomes a monopoly. The monopoly firm’s cost function is MC(q) = q/9. Suppose the newly created monopolyfirm decides to produce Q units. It will allocate these Q units to each of the 9 units so that it is produced at lowest possible cost. Since the MC for all 9 units is increasing in that units q, and all units use the same technology, each unit will be allocated a production quota of Q/9 units. So the total minimum cost will be TC(Q) = 9x {0.5(Q/9)2} TC(Q) = 0.5Q2/9

15. Demand is Q = 10 – P. The inverse demand is P = 10 – Q
Demand is Q = 10 – P. The inverse demand is P = 10 – Q. We will use the MR = MC condition directly. For the monopoly q = Q! Total revenue: TR = (10 – q)xq. Marginal revenue MR = dTR/dQ = 10 – 2q. MC is q/9. 10 – 2q = q/9 Solve for q  Q* = 90/11 = 8.18

16. Q* = 8.18, inverse demand is P(Q) = 10 - Q
Use the inverse demand P = 10 – Q to find the price p*: p* = With 9 firms with identical cost functions, each firm produces 8.18/9 = units Profit is 8.18x1.82 – 9x(0.91)2/2 = = Per firm profit is 11.17/9 = 1.24 CS is (10 – 1.88)x8.18/2 = Social welfare, defined as PS + CS, is 44,38.

17. Next question
Dominant firm and computing DWL using Harberger’s method of approximation

18. The market demand is given by Q(P) = 120 – P
The market demand is given by Q(P) = 120 – P. The supply function of the small firms as a group is given by Sf(P) = 2P −100, if P > 50, Sf(P) = 0 if P ≤ 50. The cost function of the dominant firm is c(q) = 40q where q is the quantity produced by the dominant firm. Compute the profit maximizing price for the dominant firm and its market share in equilibrium. The dominant firm’s residual demand QR(P) is QR = market demand – Sf QR = [120 – P] – [2P −100] Collect terms: QR = 220 – 3P.

19. QR = 220 – 3P. The inverse residual demand is P = 220/3 – Q/3
Write the profit function for the dominant firm. π = P(q)xq – 40q. Use the inverse demand to substitute for P(q). π = [220/3 – q/3]xq – 40q. Differentiate π with respect to q, set equal to zero: 220/3 – 2q/3 – 40 = 0  Solve for q* = 50. Use this in P = 220/3 – Q/3 and compute p*: The dominant firm’s profit with p = is (56.66 – 40)x50 = 833 Small firms: Sf = 13.33, profits = (6.66x13.33)/2 = 44,40 Total (market) Q is 63.33

20. Can the dominant firm do better with p = 50 (and eliminate the small firms)? NO! With P = 50 q is 70 the dominant firm’s profit 700.

21. Now the DWL True DWL is complicated There are 2 sources of inefficiency here 1. Eq quantity < competitive quantity 2. The eq quantity is produced inefficiently Social welfare max: Only the dominant firm should produce, and the output level should be such that P = MC of the dominant firm.

22. Harberger’s short-cut for DWL
DWL = 0.5xr2xExTR r is (P-MC)/P, we use profits/TR by assuming AC = MC E is elasticity we assume it equals 1 TR is pxq total revenue Harberger uses industry figures for revenue and profit. Industry TR is 56.66x63.33 = 3589 Industry profits are = r = 877/3589 = 0,24 DWL = 0.5x(0.24)2x3589 = 107,3

23. Mueller and Cowlings short-cut for DWL
DWL = 0.5xprofits They use firm specific figures for revenue and profit. Dominant firm profits 833 DWL = 417 Small firms’ profits 44.4 DWL = 22 Total DWL = 439

24. Oligopoly models
The Cournot model

25. Suppose the small firms are organized as a single firm that has the cost function TC(q) = 50q + q2/4. The marginal cost function is MC(q) = 50 + q/2. Now that we have two firms, suppose they compete in the style of Cournot. Compute the Nash equilibrium quantities and the price of the Cournot model. Inverse demand: P = 120 – Q Call the ex-dominant firm firm 1, Write the profit function: π1 = {120 – (q1+q2)}xq1– 40q1. Differentiate w.r.t. q1; set equal to 0: dπ1/dq1 = 120 –2q1–q2 – 40 = 0  q1 = 60 – q2/2. Firm 1’s Best response function

26. Write the profit function for firm 2: π1 = {120 – (q1+q2)}xq2– (50q2 + q22/4) Differentiate w.r.t. q2; set equal to 0: dπ1/dq1 = 120 –q1–2q2 – 50 – q2/2= 0  q2 = (140 – 2q1)/5. Firm 2’s Best response function

27. q1 = 60 – q2/2. Firm 1’s Best response function q2 = (140 – 2q1)/5
q1 = 60 – q2/2. Firm 1’s Best response function q2 = (140 – 2q1)/5. Firm 2’s Best response function Solve these two for q1 and q2. q2 = (140 – 2{60 – q2/2})/5 q2 = 140/5 – 120/5+ q2/5 4q2/5 = 20/5 q2 = 5 q1 = 57.5 P = 57.5

28. After this…
You should to be able to do the following mathematical ….

29. A market with two identical firms. The cost function is c(q) = 0.1q2+q. The market demand is Q = 14 –2P.
A. Compute the competitive equilibrium price quantity profits etc. B. One of the firms behaves like a dominant firm, sets the price, the other behaves competitively. Compute equilibrium price quantity profits etc. C. The two firms collude and form a single firm that becomes a monopoly. The marginal cost function for the monopoly is MC(q) = 0.1q+0.5. Compute the monopoly equilibrium price quantity profits etc. D. The two firms compete in the style of Cournot.

30. A monopoly firm has AC = MC = c
A monopoly firm has AC = MC = c. The market demand is Q(P) = A/Pe, where A > 0, and e > 1. This is the so-called “constant elasticity” demand curve. The price elasticity of demand equals e for all price/quantity combinations.
Example with A = 100, e = 2
Q(P) = 100/P2

31. Compute the increase in the monopoly price if the marginal cost is increased by one dollar. How does the answer depend on the value of e?

32. Use the formula

33. (P – c)/P = 1/e. Solve this this for p: p = c x{e/(e-1)} If c goes up by 1, P must go up by e/(e-1). Example e = 2. So if c is up by 1 p goes up by 2! Let c = 5; then we have p = 10 If c′ = 6, p′ = 12!