Problem set 4 solutionsEcon 100MWinter 2012Professor Justin Marion

Problem 1

Nicholson-Snyder 11.2Lump-sum tax: Tax on a firm that does not depend on its choices. Let the tax be T, and let thefirm’s revenue and cost functions be R(q) and C(q), respectively. The profit function is

Π = R(q)− C(q)− T

The first-order condition is R′(q) = C ′(q). The solution for q will not depend on T here.Proportional tax on profits: Here the firm pays a tax, t, that is a percentage of profits earned. Thefirm’s objective is to maximize after-tax profits:

Π = (1− t)(R(q)− C(q))

The first-order condition of this maximization problem is (1 − t)(R′(q) − C ′(q)) = 0. Divide bothsides by (1 − t), and we see that the FOC reduces to R′(q) = C ′(q), which indicates that theproportional profits tax does not distort the choice of quantity.Tax on each unit of output: Here the firm pays an amount, t, for each unit it sells. Profits:Π = R(q) − C(q) − tq. The FOC: R′(q) − C ′(q) − t = 0. This will affect the quantity produced.If t > 0, R′(q∗) > C ′(q∗) at the optimal solution. Let q0 solve R′(q0) = C ′(q0). Then q∗ < q0 ifR′′(q) ≤ 0 and C ′′(q) ≥ 0.Lastly consider a tax on labor: Here the tax shifts the firm’s cost curve by affecting the cost func-tion. Expand the cost function to include the tax inclusive wage rate, C(w + t, q). The firm’sfirst-order condition is R′(q) = C2(w + t, q). q will be lower for t > 0 than t = 0 since the taxincreases marginal cost.

Nicholson-Snyder 12.1

a. The marginal cost curve is MC = 1100q

2 + 0.4q + 4. We don’t have to worry about the firmshutting down in the short-run, since the average variable cost curve reaches a minimum at q = 0.So to find supply for the perfectly competitive firm, we just need to solve the first-order condi-tion that comes from its profit maximization problem, p = MC(q). So p = 1

100q2 + 0.4q + 4, or

0 = 1100q

2 + 0.4q + (4− p), which I solved using the quadratic formula to obtain q = −20 + 10√p.

b. Industry supply is the aggregation of the supply curves of the individual firms: Q =∑

i qi(p).Since all firms are identical, Q = 100qi(p), so Q = 100(−20 + 10

√p) = −2000 + 1000

√p.

c. To find market equilibrium, just set industry supply equal to demand: −2000 + 1000√p =

−200p+8000. After some rearranging, this reduces to 0.4p2− 5p+100 = 0. The quadratic formulayields two solutions for the equilibrium price, p = 25 and p = 100, however the latter would lead toa negative equilibrium quantity so is ruled out. Plugging 25 for price into either supply or demand,we get the equilibrium quantity of 3000.

Problem 2a. The equilibrium price is that which equates supply and demand.

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D(P ) = S(P ) → 100− 2P = 3P

Solving for P , we see that the equilibrium price is P = 20, and the equilibrium quantity isQ = 60.

Consumer surplus is the are under the demand curve and above the price.

CS = (1/2)(50− 20)(60) = 900

Producer surplus is the area above the supply curve and below the price.

PS = (1/2)(20)(60) = 600

b. With a subsidy of $10 per lesson, the supplier receives is P+10 per lesson – P from the consumer

and $10 from the government. Find the price that equates D(P ) = S(P + 10). Equating demandand supply:

100− 2P = 3(P + 10) → P = 14

The skiers pay $14 per lesson while the suppliers receive $24.To find quantity, plug either the consumer price or the supplier price into the demand curve or

supply curve, respectively. S(24) = 72, D(14) = 72.

CS = (1/2)(50− 14)(72) = 1296

PS = (1/2)(24)(72) = 864

The deadweight loss arises due to the overproduction of ski lessons. The deadweight loss isgiven by .5(24-14)(72-60)=60.

c. If the price ceiling is P = 15, then there is excess demand. Find quantity sold by the amountthe suppliers are willing to produce: S(15) = 3(15) = 45.

PS = (1/2)(15)45 = 337.5

CS = (1/2)(50− 27.5)(45) + (27.5− 15)45 = 1068.75

DWL = .5(27.5− 15)(60− 45) = 93.75

The consumer situation has been improved the most. Consumers gain XXX in surplus, whileproducers lose surplus.

Problem 3

Start this problem by finding average and marginal cost for each quantity.a. The long run price is the minimum of the average cost function. A higher price leads to a

profitable entry opportunity. Looking at the table, minAC = 15. Consider a higher price such asP = 20. At this price, a competitively behaving firm would produce 5 units, and earn a profit of$4 per unit.b. Each firm produces q = 4.

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Output Cost AC MC0 01 24 24 242 40 20 163 48 16 84 60 15 125 80 16 206 108 18 287 140 20 328 176 22 36

c. The industry level of production can be found by substituting P = 15 into the demand curve:Q(D) = 85− 3(15) = 40.d. Industry quantity will be 40, and each firm produces 4 units. Therefore, the number of firmswill be 10.e. This part just maps out the individual firm supply supply curve. Price would have to be at least$20 for a firm to be willing to supply the fifth unit. If price is less than $20, say $19, the fifth unitcosts 20 to make yet the firm only sells it for $19.

By similar logic, p6 = 28 and p7 = 32. In the short run, the number of firms is fixed at ten, soat p5, p6, and p7, you know that the industry supply is 50, 60, and 70, respectively.f. The market demand at p6, p7, and p8 is 84, 60, and 48, respectively. You can find this byplugging price into the market demand curve. The equilibrium price is the one that equates supplyand demand. We can see that this price is p6 = 28.g. Average cost when a firm outputs 6 is $18 compared to $15 in the long-run equilibrium.h. Each firm sells 6 units for $28 per unit, yielding revenues of 6*28=168. Total cost is 108 whenquantity is 6, so each firm earns $60 in profit.

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