Chapter 12

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form \[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \] a. Calculate the firm's short-run supply curve with $q$ as a function of market price (P). b. On the assumption that there are no interaction effects among costs of the firms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by $Q=-200P+8,000.$ What will be the short-run equilibrium price-quantity combination?

Suppose there are 1,000 identical firms producing diamonds. Let the total cost function for each firm be given by \[ C(q, w)=q^{2}+w q \] where $q$ is the firm's output level and $w$ is the wage rate of diamond cutters. a. If $w=10$, what will be the firm's (short-run) supply curve? What is the industry's supply curve? How many diamonds will be produced at a price of 20 each? How many more diamonds would be produced at a price of $21?$ b. Suppose the wages of diamond cutters depend on the total quantity of diamonds produced, and suppose the form of this relationship is given by \[ w=0.002 Q \] here $Q$ represents total industry output, which is 1,000 times the output of the typical firm. In this situation, show that the firm's marginal cost (and short-run supply) curve depends on $Q$. What is the industry supply curve? How much will be produced at a price of $20?$ How much more will be produced at a price of $21?$ What do you conclude about the shape of the short-run supply curve?

A perfectly competitive market has 1,000 firms. In the very short run, each of the firms has a fixed supply of 100 units. The market demand is given by \[ Q=160,000-10,000 P \] a. Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one firm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that, in the short run, each firm has a supply curve that shows the quantity the firm will supply $\left(q_i\right)$ as a function of market price. The specific form of this supply curve is given by \[ q_{i}=-200+50 P \] Using this short-run supply response, supply revised answers to (a)-(d).

demand is given by \[ Q=1,500-50 P \] a. What is the industry's long-run supply schedule? b. What is the long-run equilibrium price $\left(P^{\ast }\right)?$ The total industry output $\left(Q^{\ast }\right)?$ The output of each firm $\left(q^{\ast }\right)?$ The number of firms? The profits of each firm? c. The short-run total cost function associated with each firm's long-run equilibrium output is given by \[ C(q)=0.5 q^{2}-10 q+200 \] Calculate the short-run average and marginal cost function. At what output level does short-run average cost reach a minimum? d. Calculate the short-run supply function for each firm and the industry short-run supply function. e. Suppose now that the market demand function shifts upward to $Q=2,000-50P.$ Using this new demand curve, answer part (b) for the very short run when firms cannot change their outputs. f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry?

Suppose that the demand for stilts is given by \[ Q=1,500-50 P \] and that the long-run total operating costs of each stilt-making firm in a competitive industry are given by \[ C(q)=0.5 q^{2}-10 q \] Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by \[ Q_{S}=0.25 w \] where $w$ is the annual wage paid. Suppose also that each stilt-making firm requires one (and only one) entrepreneur (hence the quantity of entrepreneurs hired is equal to the number of firms). Long-run total costs for each firm are then given by \[ C(q, w)=0.5 q^{2}-10 q+w \] a. What is the long-run equilibrium quantity of stilts produced? How many stilts are produced by each firm? What is the long-run equilibrium price of stilts? How many firms will there be? How many entrepreneurs will be hired, and what is their wage? b. Suppose that the demand for stilts shifts outward to \[ Q=2,428-50 P \] How would you now answer the questions posed in part (a)? c. Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the increase in rents between parts (a) and (b). Show that this value is identical to the change in long-run producer surplus as measured along the stilt supply curve.

The handmade snuffbox industry is composed of 100 identical firms, each having short-run total costs given by \[ S T C=0.5 q^{2}+10 q+5 \] and short-run marginal costs given by \[ S M C=q+10 \] where $q$ is the output of snuffboxes per day. a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole? b. Suppose the demand for total snuffbox production is given by \[ Q=1,100-50 P \] What will be the equilibrium in this marketplace? What will each firm's total short-run profits be? c. Graph the market equilibrium and compute total short-run producer surplus in this case. d. Show that the total producer surplus you calculated in part (c) is equal to total industry profits plus industry short-run fixed costs. e. Suppose the government imposed a $\mathrm{\$}3$ tax on snuffboxes. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between snuffbox buyers and sellers? g. Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short-run profits in the snuffbox industry. Why do fixed costs not enter into this computation of the change in short-run producer surplus?

The perfectly competitive videotape-copying industry is composed of many firms that can copy five tapes per day at an average cost of $\mathrm{\$}10$ per tape. Each firm must also pay a royalty to film studios, and the per-film royalty rate $(r)$ is an increasing function of total industry output (Q): \[ r=0.002 Q \] Demand is given by \[ Q=1,050-50 P \] a. Assuming the industry is in long-run equilibrium, what will be the equilibrium price and quantity of copied tapes? How many tape firms will there be? What will the per-film royalty rate be? b. Suppose that demand for copied tapes increases to \[ Q=1,600-50 P \] In this case, what is the long-run equilibrium price and quantity for copied tapes? How many tape firms are there? What is the per-film royalty rate? c. Graph these long-run equilibria in the tape market, and calculate the increase in producer surplus between the situations described in parts (a) and (b). d. Show that the increase in producer surplus is precisely equal to the increase in royalties paid as $Q$ expands incrementally from its level in part (b) to its level in part (c). e. Suppose that the government institutes a $\mathrm{\$}5.50$ per-film tax on the film-copying industry. Assuming that the demand for copied films is that given in part (a), how will this tax affect the market equilibrium? f. How will the burden of this tax be allocated between consumers and producers? What will be the loss of consumer and producer surplus? g. Show that the loss of producer surplus as a result of this tax is borne completely by the film studios. Explain your result intuitively.

The domestic demand for portable radios is given by $$Q=5,000-100\mathrm{P}$$ where price $(\mathrm{P})$ is measured in dollars and quantity $(\mathrm{Q})$ is measured in thousands of radios per year. The domestic supply curve for radios is given by $$Q=150\mathrm{P}$$ a. What is the domestic equilibrium in the portable radio market? b. Suppose portable radios can be imported at a world price of $$\mathrm{\$}10$$ per radio. If trade were unencumbered, what would the new market equilibrium be? How many portable radios would be imported? c. If domestic portable radio producers succeeded in having a $$\mathrm{\$}5$$ tariff implemented, how would this change the market equilibrium? How much would be collected in tariff revenues? How much consumer surplus would be transferred to domestic producers? What would the deadweight loss from the tariff be? d. How would your results from part (c) be changed if the government reached an agreement with foreign suppliers to "voluntarily" limit the portable radios they export to $1,250,000$ per year? Explain how this differs from the case of a tariff.

Suppose that the market demand for a product is given by $Q_D=A-BP.$ Suppose also that the typical firm's cost function is given by $C(q)=k+aq+b{q}^2$ a. Compute the long-run equilibrium output and price for the typical firm in this market. b. Calculate the equilibrium number of firms in this market as a function of all the parameters in this problem. c. Describe how changes in the demand parameters $A$ and $B$ affect the equilibrium number of firms in this market. Explain your results intuitively. d. Describe how the parameters of the typical firm's cost function affect the long-run equilibrium number of firms in this example. Explain your results intuitively.

Throughout this chapter's analysis of taxes we have used per-unit taxes-that is, a tax of a fixed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes-that is, taxes on the value of the transaction (or, what amounts to the same thing, proportional taxes on price). Given an ad valorem tax rate of $t(t=0.05\text{ for a }5$ percent tax), the gap between the price demanders pay and what suppliers receive is given by $P_D=(1+t)P_S$ a. Show that for an ad valorem tax \[ \frac{d \ln P_{D}}{d t}=\frac{e_{S}}{e_{S}-e_{D}} \quad \text { and } \quad \frac{d \ln P_{S}}{d t}=\frac{e_{D}}{e_{S}-e_{D}} \] b. Show that the excess burden of a small tax is \[ D W=-0.5 \frac{e_{D} e_{S}}{e_{S}-e_{D}} t^{2} P_{0} Q_{0} \] c. Compare these results with those derived in this chapter for a unit tax.