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Chapter 14: Problem 1

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?

Short Answer

Expert verified
The following short-answer question can be formulated based on the step-by-step solution above: Q: Given a short-run total cost function for a single firm in a perfectly competitive industry, and information about the number of identical firms in the industry and the market demand function, explain the process of finding the firm's short-run supply curve, the short-run industry supply curve, and the short-run equilibrium price and quantity. A: To find these values, one must follow these three steps: 1. Calculate the firm's short-run supply curve by finding the marginal cost (MC) by differentiating the total cost function with respect to quantity, and then equating the marginal cost to the price. 2. Calculate the short-run industry supply curve by multiplying the individual firm's supply curve (in terms of quantity) by the number of firms in the industry. 3. Find the short-run equilibrium price and quantity by setting the market demand equal to the short-run industry supply curve and solving for the price, then plugging the equilibrium price back into the industry supply curve equation to find the equilibrium quantity.

Step by step solution

01

Calculate the firm's short-run supply curve

In order to find the firm's short-run supply curve, we first need to compute the marginal cost (MC) for the firm. Marginal cost is the additional cost of producing one more unit of output. We can find the marginal cost by taking the derivative of the total cost function with respect to quantity: \\[MC=\frac{dC}{dq}\\] Differentiating the given total cost function with respect to \(q\), we get: \\[MC = \frac{d (\frac{1}{300} q^3 + 0.2 q^2 + 4q + 10)}{dq} = \frac{1}{100} q^2 + 0.4 q + 4\\] In the short run, a firm will produce at the level where the price is equal to its marginal cost (i.e., \(P = MC\)). Therefore, we can write the firm's short-run supply curve as: \\[P = \frac{1}{100} q^2 + 0.4 q + 4\\] Now, we need to solve for the quantity \(q\) as a function of price \(P\). Rearranging the equation above, we get: \\[q^2 + 40q + 400 -100P = 0\\] This is a quadratic equation in terms of \(q\), and we can find its real roots using the quadratic formula: \\[q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]
02

Calculate the short-run industry supply curve

Since there are 100 identical firms in the industry, the short-run industry supply curve will be the total quantity produced by all the firms combined at a given market price. Assuming there are no interaction effects among firms' costs, we can simply multiply the individual firm's supply curve (in terms of quantity) by the number of firms (100) to find the industry supply curve: \\[Q_s = 100q = 100\left(\frac{-40 \pm \sqrt{40^2 - 4(1)(400 -100P)}}{2}\right)\\] Now we have the short-run industry supply curve in terms of the market price \(P\).
03

Find the short-run equilibrium price-quantity combination

To find the short-run equilibrium price and quantity, we need to set the market demand equal to the short-run industry supply and solve for \(P\): \\[-200P+8000 = 100\left(\frac{-40 \pm \sqrt{40^2 - 4(1)(400 -100P)}}{2}\right)\\] This equation might seem challenging to solve directly. However, since demand has a negative slope and the supply has a positive slope, we know that the equilibrium is reached when the positive solution (+) of the quadratic expression is used: \\[-200P+8000 = 100\left(\frac{-40 + \sqrt{40^2 - 4(1)(400 -100P)}}{2}\right)\\] Solving this equation for \(P\), we find the equilibrium price: \\[P = 19.6 \\] Plugging the equilibrium price back into the industry supply curve equation, we can find the equilibrium quantity: \\[Q_s = 100\left(\frac{-40 + \sqrt{40^2 - 4(1)(400 -100(19.6))}}{2}\right) = 4104\\] So the short-run equilibrium price-quantity combination is \((P, Q_s) = (19.6, 4104)\).

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Most popular questions from this chapter

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 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.

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 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 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?
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