ECON 330 Lecture 14 Monday, November 9

Review questions for the midterm

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

Monopoly question

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.

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.

The competitive equilibrium question

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).

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) = …

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.

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.

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 = 40.5. Total welfare = 40.5 + 4.5 = 45.

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

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

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

Q* = 8.18, inverse demand is P(Q) = 10 - Q Use the inverse demand P = 10 – Q to find the price p*: p* = 1.82. With 9 firms with identical cost functions, each firm produces 8.18/9 = 0.908 units Profit is 8.18x1.82 – 9x(0.91)2/2 = 14.89 - 3.72 = 11.17 Per firm profit is 11.17/9 = 1.24 CS is (10 – 1.88)x8.18/2 = 33.21. Social welfare, defined as PS + CS, is 44,38.

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

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.

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*: 56.66 The dominant firm’s profit with p = 56.66 is (56.66 – 40)x50 = 833 Small firms: Sf = 13.33, profits = (6.66x13.33)/2 = 44,40 Total (market) Q is 63.33

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.

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.

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 833 + 44.4 = 877.4 r = 877/3589 = 0,24 DWL = 0.5x(0.24)2x3589 = 107,3

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

Oligopoly models The Cournot model

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

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

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

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

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.

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

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?

Use the formula

(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!