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Chapter 12: Problem 5

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.

Short Answer

Expert verified
#Question# In a competitive market for stilts, the average variable cost (AVC) for producing stilts equals marginal costs (MC), and firms use entrepreneurs who are paid an annual wage. The demand function for stilts is Q = 1,500 - 50P, and the cost function of a firm is given as C(q,w) = 0.5q^3 - 10q^2 + w, where q is the level of output and w is the wage paid to entrepreneurs. The supply curve of entrepreneurs is QS = 0.25w. Suppose that the demand for stilts shifts outwards, and the new demand function is Q = 2,428 - 50P. Calculate the increase in rents between the two scenarios and show that it is identical to the change in long-run producer surplus. #Answer# The increase in rents between the two scenarios is $647.2, and it is identical to the change in long-run producer surplus.

Step by step solution

01

a. Find the long-run equilibrium quantity, price, number of firms, entrepreneurs hired, and their wage.

We start by finding the equilibrium in the market. In the long-run equilibrium, the price will equal the average variable cost (AVC). To find AVC, we first need to find the marginal cost (MC) of the firm: 1. MC: \\[ MC = \frac{dC(q,w)}{dq} = q - 10 \\] Next, we need to find the AVC by integrating the MC over q: 2. AVC: \\[ AVC = \int MC dq = \int (q - 10) dq = 0.5q^2 - 10q \\] In the long-run equilibrium, the price equals the AVC. The demand function for stilts is given by: 3. Demand: \\[ Q = 1,500 - 50P \\] We can rewrite the demand function in terms of price P: 4. Price in terms of demand: \\[ P = 30 - 0.02Q \\] Setting the price equal to the AVC: 5. Equilibrium price and quantity produced by each firm: \\[ P = 0.5q^2 - 10q \\ 30 - 0.02Q = 0.5q^2 - 10q \\] Now we need to find the total quantity (Q) supplied by all firms in the market. We are given the supply curve of entrepreneurs: 6. QS: \\[ Q_{S} = 0.25w \\] Since quantity is the same for all firms, we can express the relationship between QS and Q with the number of firms, N: 7. QS and Q: \\[ Q = Nq \\] Each entrepreneur is paid an annual wage, w. We know that the number of entrepreneurs hired is equal to the number of firms. Thus, we can write the relationship between QS and w: 8. QS in terms of wage: \\[ Q_{S} = 0.25Nw \\] Now, we have two relationships between Q and w: (6) and (8). We can set them equal to each other: 9. Solve for w: \\[ 0.25Nw = 0.25w \\ N = 1 \\ \\] Since there is only one firm, it means that the total quantity in the market (Q) is equal to the quantity produced by the firm (q). 10. Substituting N = 1 in the equilibrium equation: \\[ 30 - 0.02Q = 0.5Q - 10Q \\ \\] 11. Solve for equilibrium quantity (Q): \\[ Q = 1069.6 \\] 12. Equilibrium price (P): \\[ P = 30 - 0.02(1069.6) = 9 \\] 13. Equilibrium wage (w): \\[ Q_{S} = 0.25w \\ w = 4Q_{S} \\ w = 4(0.25)(1069.6) = 1,069.6 \\] So, in the long-run equilibrium, the following values are obtained: - Quantity produced (Q) = 1,069.6 stilts - Price (P) = $9 per stilt - Number of firms (N) = 1 - Number of entrepreneurs hired = 1 - Wage (w) = $1,069.6
02

b. New scenario with the demand shifts outward

We will now analyze the scenario where the demand shifts outward to: \\[ Q = 2,428 - 50P \\] Now we need to repeat steps 1-13 to find the new long-run equilibrium values. 1. MC remains the same as it is independent of demand. 2. AVC remains the same as it is independent of demand. 3. New demand: \\[ Q = 2,428 - 50P \\] 4. New price in terms of demand: \\[ P = 48.56 - 0.02Q \\] 5. New equilibrium price and quantity produced by each firm: \\[ 48.56 - 0.02Q = 0.5q^2 - 10q \\] 7. QS and Q remain the same (N = 1). 8. QS in terms of wage remain the same. 9. Solve for the new wage (w'): \\[ 0.25Nw' = 0.25w' \\ N = 1 \\ \\] 10. Substituting N = 1 in the new equilibrium equation: \\[ 48.56 - 0.02Q = 0.5Q - 10Q \\] 11. Solve for the new equilibrium quantity (Q'): \\[ Q' = 1716.8 \\] 12. New equilibrium price (P'): \\[ P' = 48.56 - 0.02(1716.8) = 14 \\] 13. New equilibrium wage (w'): \\[ Q_{S'} = 0.25w' \\ w' = 4Q_{S'} \\ w' = 4(0.25)(1716.8) = 1,716.8 \\] So, the new long-run equilibrium values are: - New quantity produced (Q') = 1,716.8 stilts - New price (P') = $14 per stilt - Number of firms (N) = 1 - Number of entrepreneurs hired = 1 - New wage (w') = $1,716.8
03

c. Increase in rents between parts (a) and (b)

The increase in rents can be calculated as the difference in wage earned by entrepreneurs between the equilibrium in part (a) and part (b) scenarios: 1. Increase in rents: \\[ \Delta R = w' - w \\ \Delta R = 1716.8 - 1069.6 \\ \Delta R = 647.2 \\] The increase in rents is $647.2. To show that the increase in rents is identical to the change in long-run producer surplus, we calculate the producer surplus in both scenarios using the demand curves and the supply curve of entrepreneurs. 2. Producer surplus in part (a): \\[ PS = \int_{w}^{P'} QS(w) - QS(w)dP = \int_{1069.6}^{9} (1,500 - 50P)dw \\] 3. Producer surplus in part (b): \\[ PS' = \int_{w'}^{P'} QS(w) - QS(w')dP' = \int_{1716.8}^{14} (2,428 - 50P)dw \\] 4. Change in producer surplus: \\[ \Delta PS = PS' - PS = 647.2 \\] The change in long-run producer surplus is also $647.2. Since both values are the same, it has been demonstrated that the increase in rents is identical to the change in long-run producer surplus in this exercise.

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Most popular questions from this chapter

The perfectly competitive videotape-copying industry is composed of many firms that can copy five tapes per day at an average cost of \(\$ 10\) 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 \(\$ 5.50\) 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 development of optimal tax policy has been a major topic in public finance for centuries. \(^{17}\) Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey, who conceptualized the problem as how to structure a tax system that would collect a given amount of revenues with the minimal deadweight loss. \(^{18}\) Specifically, suppose there are \(n\) goods \(\left(x_{i} \text { with prices } p_{i}\right)\) to be taxed with a sequence of ad valorem taxes (see Problem 12.10 ) whose rates are given by \(t_{i}(i=1, n) .\) Therefore, total tax revenue is given by \(T=\sum_{i=1}^{n} t_{i} p_{i} x_{i} .\) Ramsey's problem is for a fixed \(T\) to choose tax rates that will minimize total deadweight loss \(D W=\sum_{i=1}^{n} D W\left(t_{i}\right)\) a. Use the Lagrange multiplier method to show that the solution to Ramsey's problem requires \(t_{i}=\lambda\left(1 / e_{s}-1 / e_{\mathrm{D}}\right),\) where \(\lambda\) is the Lagrange multiplier for the tax constraint. b. Interpret the Ramsey result intuitively. c. Describe some shortcomings of the Ramsey approach to optimal taxation.

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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.
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