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Chapter 12: Problem 2

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?

Short Answer

Expert verified
Based on the step-by-step solution, a short answer to the problem would be: To find the short-run supply curve for a single firm and the industry's supply curve when \(w=10\), we first calculated the firm's marginal cost (MC) and found it to be \(2q + w\). Then, the industry's supply curve is equal to 1,000 times the firm's MC. The total number of diamonds produced at prices 20 and 21 were 5,000 and 5,500, respectively. In part b, after being given the relationship between the wage rate, \(w\), and total industry output, \(Q\), we found the firm's MC in terms of Q. We then derived the industry supply curve and calculated the quantity produced at prices 20 and 21, which were 5,000 and 5,250, respectively. Lastly, we analyzed the shape of the short-run supply curve, which was upward-sloping with a linear slope of 4.

Step by step solution

01

Find the firm's marginal cost

In order to find the firm's (short-run) supply curve, we need to find the firm's marginal cost (MC). The marginal cost is the derivative of the total cost function with respect to the output level, \(q\). The total cost function is given by \(C(q,w)=q^{2}+wq\). Taking the derivative with respect to \(q\) gives us the marginal cost: MC = \(\frac{dC(q,w)}{dq} = 2q + w\)
02

Find the industry's supply curve

Given that there are 1,000 identical firms producing diamonds, the industry's supply curve is the sum of the individual firms' supply, which is equal to 1,000 times the firm's marginal cost. Industry Supply Curve = \(MC \times 1000 = (2q + w) 1000\)
03

Calculate the number of diamonds produced at a price of 20

When the price (\(P\)) is 20 and \(w=10\), we can equate the price with the firm's marginal cost: \(P = 2q + w = 20\) Now, we can solve for the output level \(q\): \(2q = 20 - 10\) \(2q = 10\) \(q = 5\) Since there are 1,000 identical firms, the total number of diamonds produced at the price of 20 is: Total Diamonds = \(Q = 1000 \times 5 = 5000\)
04

Calculate the number of diamonds produced at a price of 21

When the price is 21, we can equate the price with the firm's marginal cost: \(P = 2q + w = 21\) Solving for the output level \(q\): \(2q = 21 - 10\) \(2q = 11\) \(q = 5.5\) Since there are 1,000 identical firms, the total number of diamonds produced at the price of 21 is: Total Diamonds = \(Q = 1000 \times 5.5 = 5500\)
05

Calculate the firm's marginal cost with respect to Q

For part b, we are given the relationship between the wage rate, \(w\), and total industry output, \(Q\) as \(w=0.002Q\). Since \(Q = 1000 \times q\), we can write the wage rate as a function of \(q\): \(w = 0.002(1000q) = 2q\) Now, we can rewrite the firm's marginal cost function in terms of \(Q\): MC = \(2q + 2q\) MC = \(4q\)
06

Find the industry supply curve

The industry supply curve is the sum of the individual firms' supply, which is equal to 1,000 times the firm's marginal cost: Industry Supply Curve = \(MC \times 1000 = (4q) 1000\)
07

Calculate the quantity produced at prices 20 and 21

We can equate the price with the firm's marginal cost and find the quantities produced at prices 20 and 21: At \(P = 20\): \(20 = 4q \implies q = 5\) Total Diamonds = \(Q = 1000 \times 5 = 5000\) At \(P = 21\): \(21 = 4q \implies q = 5.25\) Total Diamonds = \(Q = 1000 \times 5.25 = 5250\)
08

Analyze the shape of the short-run supply curve

From the calculations, we can conclude that the short-run supply curve is upward-sloping. As the price increases, the quantity of diamonds supplied also increases. The shape of the short-run supply curve is linear, with a slope of 4.

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Key Concepts

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

Total Cost Function
Understanding the total cost function is fundamental in economics as it helps determine a firm's production expenses at different output levels. In the scenario given, the total cost function for a firm producing diamonds is expressed as \( C(q, w) = q^2 + wq \). Here, \( q \) is the quantity of diamonds produced by the firm, while \( w \) is the wage rate paid to diamond cutters.

The quadratic term \( q^2 \) represents the variable costs that increase with the square of the output level, capturing elements like the cost of raw materials which can increase exponentially with production. The linear term \( wq \) reflects the direct proportionality of wages to the quantity produced. Understanding this function allows a firm to compute the marginal cost, which is the increase in total cost when one additional unit of output is produced.
Short-Run Supply Curve
The short-run supply curve is crucial in representing how much of a good a firm is willing to offer at different price levels, keeping some inputs fixed due to short-term constraints. The firm’s supply curve in the exercise is derived from calculating the marginal cost (MC), which is the increase in total costs associated with a one-unit increase in production.

The relationship between marginal cost and the supply curve is direct because firms offer more goods for sale when the price is higher than the marginal cost. For the given cost function, the MC is computed as \( MC = \frac{dC(q, w)}{dq} = 2q + w \), which is a key determinant of a firm's short-run supply decision. When multiple identical firms are in the market, the industry supply curve can be constructed by aggregating individual supply curves, indicating how the entire market responds to price changes in the short run.
Industry Output
Industry output is a measure of the total production within a specific industry. It is the aggregate of all outputs from the individual firms. In our diamond industry example, industry output is symbolized as \( Q \), and since all firms are identical, \( Q \) is simply 1,000 times the output of a typical firm.

When analyzing industry output in relation to the wage rate or input prices, it's important to consider how increased production can affect these variables. For instance, if wage rates depend on output levels, this interdependence can impact firms' production decisions since it affects the total cost and, consequently, the supply curve of the industry as a whole. Therefore, the concept of industry output is intertwined with market supply and the equilibrium price levels.
Wage Rate
The wage rate is a critical factor in the production process as it constitutes the payment to labor, which is one of the key inputs into production. In our case, the wage rate \( w \) plays a direct role in the total cost function \( C(q, w) \) of diamond-cutting firms, illustrating how changes in wages impact production costs and supply decisions.

In many instances, the wage rate is considered fixed, but in our more complex scenario, it varies with the industry output. We're given that the wage rate for diamond cutters is determined by the formula \( w=0.002Q \), which means that as industry output increases, the wage rate increases linearly. This dynamic characteristic of the wage rate highlights the interrelatedness of labor costs and market supply, shaping the nature of the short-run supply curve and affecting how much firms are willing to produce at different price points.

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Most popular questions from this chapter

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.

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?

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 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?

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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