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Chapter 12: Problem 1

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 \(Q=-200 P+8,000 .\) What will be the short-run equilibrium price-quantity combination?

Short Answer

Expert verified
In a perfectly competitive industry with 100 identical firms each having a total cost function of: \\[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \\] The marginal cost function (MC) of an individual firm is: \\[ MC(q) = \frac{3}{300} q^{2} + 0.4 q+ 4 \\] The individual firm's short-run supply curve is: \\[ q(P) = \frac{300(P - 4) - 120 q}{3} \\] The industry short-run supply curve is: \\[ Q(P) = 10000P - 40000 - 4000q \\] The short-run equilibrium price-quantity combination is P = $ \frac{25}{4} and Q = 6875.

Step by step solution

01

- Find the marginal cost function of an individual firm

The first step is to find the marginal cost function (MC) of an individual firm by taking the derivative of the total cost function (C(q)) with respect to the quantity (q). The given total cost function is: \\[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \\] To find the MC function, take the derivative of the total cost function with respect to q: \\[ MC(q) = \frac{dC(q)}{dq} = \frac{3}{300} q^{2} + 0.4 q+ 4 \\]
02

- Find the individual firm's short-run supply curve

In a perfectly competitive market, each firm's supply curve is determined by equating marginal cost (MC) with the market price (P). Setting MC(q) equal to P, we have: \\[ P = \frac{3}{300} q^{2} + 0.4 q+ 4 \\] Reorder the equation to solve for q: \\[ q(P) = \frac{300(P - 4) - 120 q}{3} \\] Now we have the short-run supply curve for an individual firm: \\[ q(P) = \frac{300(P - 4) - 120 q}{3} \\]
03

- Find the industry short-run supply curve

To find the short-run industry supply curve, multiply the individual firm's supply curve (q(P)) by the number of firms (100), since there are no interaction effects among the firms' costs. \\[ Q(P) = 100 \cdot q(P) = 100 \cdot \frac{300(P - 4) - 120 q}{3} \\] Expanding the equation, we get: \\[ Q(P) = 10000P - 40000 - 4000q \\]
04

- Find the short-run equilibrium price-quantity combination

To find the short-run equilibrium price-quantity combination, we will equate the industry supply curve (Q(P)) with the demand curve (Q) given by Q = -200P + 8000. So, set Q(P) equal to the given demand curve: \\[ 10000P - 40000 - 4000q = -200P + 8000 \\] Solve for P: \\[ P = \frac{7500}{10200} = \frac{25}{4} \\] Now that we have the equilibrium price, we can find the equilibrium quantity Q by substituting the value of P into the demand or supply curve (we will use the demand curve). \\[ Q = -200P + 8000 = -200\left(\frac{25}{4}\right) + 8000 = 6875 \\] So, the short-run equilibrium price-quantity combination is P = $ \frac{25}{4} and Q = 6875.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Short-run Supply Curve
In a perfectly competitive market, the short-run supply curve of an individual firm is crucial to understanding its behavior and responses to market prices. This curve depicts the quantity of goods a firm is willing to supply at different price levels in the short run.

To find a firm's supply curve from the cost function, we first derive the marginal cost (MC) by taking the derivative of the total cost function. For our exercise, the total cost function is given by:
  • \(C(q) = \frac{1}{300} q^{3} + 0.2 q^{2} + 4 q + 10\).
After deriving, the MC becomes:
  • \(MC(q) = \frac{3}{300} q^{2} + 0.4 q + 4\).
In perfect competition, firms equate their marginal cost to the market price (P), solving for output quantity (q) gives the firm's supply curve. Rearranging the equation \(P = MC(q)\), we get the supply function:
  • \(q(P) = q's \, expression \, in \, terms \, of \, P\).
This equation now represents the firm's responsive supply curve in the short run based on external market prices.
Industry Supply Curve
The industry supply curve is an aggregation of individual firms' supply curves and is essential for determining the overall market supply. It tells us the total quantity of goods supplied by all firms at varying price levels in the short run.

Given 100 identical firms in our problem, we multiply each firm's supply function by 100 to form the industry supply curve. Suppose each firm follows the supply curve:
  • \(q(P) = \text{individual \, supply \, equation \, in \, terms \, of \, P}\).
The short-run industry supply curve then is:
  • \(Q(P) = 100 \, \cdot \, q(P)\),
where \(Q(P)\) represents the total industry output. This mathematical relationship shows how sensitive the industry is to price changes and ensures a connection between individual supply action and industry-wide outcomes. The approach assumes no interaction effects among firms, meaning their actions don’t affect each other, which keeps calculations straightforward in perfect competition.
Equilibrium Price-Quantity
Understanding the equilibrium price and quantity in a competitive market is fundamental to predicting how supply and demand interactions balance out. Equilibrium occurs where the quantity demanded equals the quantity supplied, aligning market interest.

In this exercise, we are given the market demand function:
  • \(Q = -200P + 8000\).
To find equilibrium, we set the industry supply function \(Q(P)\) equal to the demand function. Solving this equation provides the equilibrium price:
  • \(P = \frac{7500}{10200} = \frac{25}{4}\).
Once price \(P\) is known, substitute back into either the supply or demand equation to find the equilibrium quantity \(Q\). For instance, using the demand equation yields:
  • \(Q = -200 \times \frac{25}{4} + 8000 = 6875\).
Thus, the equilibrium in this competitive market sits at a price of \(\frac{25}{4}\) dollars and a quantity of 6875 units. This point reflects an optimal balance where no excess supply or unmet demand exists at the given price.

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Most popular questions from this chapter

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.

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 $$\$ 10$$ 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 $$\$ 5$$ 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.

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.

One way to generate disequilibrium prices in a simple model of supply and demand is to incorporate a lag into producer's supply response. To examine this possibility, assume that quantity demanded in period \(t\) depends on price in that period \(\left(Q_{t}^{D}=a-b P_{t}\right)\) but that quantity supplied depends on the previous period's price-perhaps because farmers refer to that price in planting a crop \(\left(Q_{t}^{S}=c+d P_{t-1}\right)\) a. What is the equilibrium price in this model \(\left(P^{*}=P_{t}=P_{t-1}\right)\) for all periods, \(t\) b. If \(P_{0}\) represents an initial price for this good to which suppliers respond, what will the value of \(P_{1}\) be? c. By repeated substitution, develop a formula for any arbitrary \(P_{t}\) as a function of \(P_{0}\) and \(t\) d. Use your results from part (a) to restate the value of \(P_{t}\) as a function of \(P_{0}, P^{*},\) and \(t\) e. Under what conditions will \(P_{t}\) converge to \(P^{*}\) as \(t \rightarrow \infty ?\) f. Graph your results for the case \(a=4, b=2, c=1, d=1,\) and \(P_{0}=0 .\) Use your graph to discuss the origin of the term cobweb model.

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 \(\$ 3\) 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?
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