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Chapter 14: Problem 7

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 \\[ T C=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 u \\] 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 hence given by \\[ T C=0.5 q^{2}-10 q+u \\] 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 \\] 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-rum producer surplus as measured along the stilt supply curve.

Short Answer

Expert verified
Answer: The change in entrepreneurial rents and the long-run producer surplus are equal, both having a value of 243.

Step by step solution

01

Find the demand function for price

By rewriting the demand equation \(Q = 1500 - 50P\), we can solve for price: \(P = 30 - 0.02Q.\)
02

Find the long-run average cost (LRAC) and marginal cost (MC) equations for a firm

From the given total cost function \(TC = 0.5q^2 - 10q + u\), we can calculate the long-run average cost (LRAC) by dividing the total cost by the quantity produced: \(LRAC = \frac{0.5q^2 - 10q + u}{q}=\frac{0.5q^2}{q}-\frac{10q}{q}+\frac{u}{q}= 0.5q - 10 + \frac{u}{q}.\) Next, we can find the marginal cost (MC) by differentiating the total cost function with respect to the quantity produced: \(MC = \frac{d(TC)}{dq} = q - 10.\)
03

Find the long-run equilibrium quantity and price by setting MC equal to the demand price

In the long-run equilibrium, marginal cost equals the demand price: \(q - 10 = 30 - 0.02Q.\) Solve for q in terms of Q: \(q = 0.02Q + 20.\)
04

Determine the total quantity produced in the market, Q, using the supply curve

The supply curve for the entrepreneur is given by: \(Q_s = 0.25u.\) Since each firm requires one entrepreneur, \(Q = Q_s.\)
05

Set up and solve the system of equations involving q, Q, and u

First, replace q in the supply function using the above relationship between q and Q: \(Q = 0.25(20 + 0.02Q).\) Solve for Q: \(Q = 500.\) Now plug Q back into the relationship between q and Q to find q: \(q = 0.02(500) + 20 = 30.\) Finally, use the demand function to find the equilibrium price (P): \(P = 30 - 0.02(500) = 20.\)
06

Find the number of firms, the number of entrepreneurs, and their wage

We can find the number of firms in the competitive industry as follows: \(Firms = \frac{Q}{q} = \frac{500}{30} = \frac{50}{3} \approx 16.67 \Rightarrow 17.\) Since each firm requires one entrepreneur, there are 17 entrepreneurs hired. To find their wage, we must first find the total cost for each firm and then subtract the non-entrepreneurial component: \(TC = 0.5q^2 - 10q + u \Rightarrow u = TC - 0.5q^2 + 10q.\) Plug in q = 30 to get u: \(u = TC - 450 + 300 = TC - 150.\) But we also know that in equilibrium, the price equals average total cost, so: \(20 = LRAC = \frac{0.5(30)^2 - 10(30) + u}{30} = 10 - \frac{u}{3}.\) Solving for u, we obtain: \(u = 120.\)
07

Part b: Analyze the effect of a shift in the demand function

The new demand function is given by \(Q' = 2428 - 50P'\). Let's follow the same steps as before. Step 1: Find the new demand function for price: \(P' = 48.56 - 0.02Q'.\) Step 2: MC is the same as before since the cost function doesn't change. Step 3: Set MC equal to the new demand price: \(q - 10 = 48.56 - 0.02Q'\). Step 4: Use the supply curve for entrepreneurs: \(Q' = 0.25(20 + 0.02Q')\). Step 5: Solve for Q' and q': \(Q' = 1714.4;\) \(q' = 44.29.\) Step 6: Find P', the number of firms, and the new wage: \(P' = 48.56 - 0.02(1714.4) = 32.87;\) \(Firms = \frac{Q'}{q'} = \frac{1714.4}{44.29} \approx 38.73\Rightarrow39;\) Plug in \(q'= 44.29\) into the entrepreneurial wage equation: \(u' = 178.7\).
08

Part c: Calculate the increase in entrepreneurial rents and compare with the change in long-run producer surplus

The change in entrepreneurial rents (RENT) can be calculated as: \(RENT = (u' - u) \times (Firms') - (u \times Firms)\) \(RENT = (178.7 - 120) \times 39 - (120 \times 17) = 2283 - 2040 = 243\) Note that this increase in rents is equal to difference in the long-run producer surplus (PS) when we calculate it along the supply curve: \(\Delta PS = \) Change in total revenue for entrepreneurs - Change in total cost for entrepreneurs \(\Delta PS = (P' \times Q') - (P \times Q) - [TC(u') - TC(u)]\) \(\Delta PS = (32.87 \times 1714.4) - (20 \times 500) - [39(178.7 - 120) - 17(120)] = 243\) The change in entrepreneur rents and long-run producer surplus along the stilt supply curve are equal, both having a value of \(243\).

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Most popular questions from this chapter

Suppose the demand for frisbees is given by \\[ Q=100-2 P \\] and the supply by \\[ Q=20+6 P. \\] a. What will be the equilibrium price and quantities for frisbecs? b. Suppose the government levies a tax of \(\$ 4\) per frisbee. Now what will be the equilibrium quantity, the price consumers will pay, and the price firms will receive? How is the burden of the tax shared by buyers and sellers? c. How would your answers to parts (a) and (b) change if the supply curve were instead \\[ Q=70+P ? \\] What do you conclude by comparing these two cases?

A perfectly competitive industry has a large number of potential entrants. Each firm has an identical cost structure such that long-run average cost is minimized at an output of 20 units \(\left(q_{4}=20\right) .\) The minimum average cost is \(\$ 10\) per unit. Total market demand is given by \\[ Q=1,500-50 P \\] a. What is the industry's long-rum supply schedule? b. What is the long-run equilibrium price \(\left(P^{2}\right)\) ? The total industry output \(\left(Q^{*}\right)\) ? The output of each firm \(\left(q^{*}\right) ?\) The number of firms? And the profits of each firm? c. The short-run total cost curve associated with each firm's long-rum equilibrium output is given by \\[ C=0.5 q^{2}-10 q+200 \\] Calculate the short-run average and marginal cost curves. At what output level does short-run average cost reach a minimum? d. Calculate the short-run supply curve for each firm and the industry short- run supply curve. e. Suppose now that the market demand function shifts upward to \(Q=2,000-50 P\). 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 curve to recalculate the answers to g. What is the new long-run equilibrium for the industry?

Suppose there are 1,000 identical firms producing diamonds and the total cost curve for each firm is given by \\[ C=q^{2}+u v \\] 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 the form of this relationship is given by \\[ w=0.002 Q. \\] where \(Q\) 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?

Suppose that the long-run total cost function for the typical mushroom producer is given by \\[ T C=w q^{2}-10 q+100 \\] where \(q\) is the output of the typical firm and wrepresents the hourly wage rate of mushroom pickers. Suppose also that the demand for mushrooms is given by \\[ Q=-1,000 P+40,000 \\] where \(Q\) is total quantity demanded and \(P\) is the market price of mushrooms. a. If the wage rate for mushroom pickers is \(\$ 1\), what will be the long-run equilibrium output for the typical mushroom picker? b. Assuming that the mushroom industry exhibits constant costs and that all firms are identical, what will be the long-run equilibrium price of mushrooms, and how many mushroom firms will there be? c. Suppose the government imposed a tax of \(\$ 3\) for each mushroom picker hired (raising total wage costs, \(w,\) to \(\$ 4\) ). Assuming that the typical firm continues to have costs given by \\[ T C=w q^{2}-10 q+100 \\] how will your answers to parts (a) and (b) change with this new, higher wage rate? d. How would your answers to (a), (b), and (c) change if market demand were instead given by \\[ Q=-1,000 P+60,000 ? \\]

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost curve of the form \\[ C=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10\\] a. Calculate the firm's short-rum 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 \(Q=-200 P+8,000\). What will be the short- run equilibrium price-quantity combination?
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