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Chapter 11: Problem 3

The production function for a firm in the business of calculator assembly is given by \\[ q=2 \sqrt{l} \\] where \(q\) denotes finished calculator output and \(l\) denotes hours of labor input. The firm is a price-taker both for calculators (which sell for \(P\) ) and for workers (which can be hired at a wage rate of \(w\) per hour). a. What is the total cost function for this firm? b. What is the profit function for this firm? c. What is the supply function for assembled calculators \([q(P, w)] ?\) d. What is this firm's demand for labor function \([l(P, w)] ?\) e. Describe intuitively why these functions have the form they do.

Short Answer

Expert verified
#Answer# The supply function is given by q(P, w) = P/w. This indicates that the firm is willing to produce and supply more calculators as the product price (P) increases and/or labor cost (w) decreases.

Step by step solution

01

a. Total Cost Function

First, let's find the variable cost as a function of labor input l. Since the firm hires workers at a wage rate w per hour, the variable cost is given by: Variable Cost (VC) = w * l Now, we can find the total cost function by adding the fixed cost (assume it to be zero for simplicity) to the variable cost: Total Cost (TC) = VC = w * l
02

b. Profit Function

Profit is the difference between total revenue (TR) and total cost (TC). Total revenue is given by the product of price (P) and quantity (q) of calculators produced: TR = P * q = P * (2 * 鈭歭) Now, we can find the profit function by subtracting TC from TR: Profit = TR - TC = P * (2 * 鈭歭) - w * l
03

c. Supply Function

To find the supply function q(P, w), first, we should find the optimal labor input (l*) by maximizing the profit function. To maximize profit, differentiate with respect to l and set the result equal to zero: \\[ \frac{d(profit)}{dl} = \frac{d(P * (2 \sqrt{l}) - w * l)}{dl} = 0 \\] Solve for the optimal labor input (l*): \\[ l* = \frac{P^2}{4w^2} \\] Now, substitute l* into the production function (q = 2鈭歭) to get the supply function: q(P, w) = 2鈭歭* = 2鈭�(P虏/4w虏) = P/w
04

d. Demand for Labor Function

We already found the optimal labor input l* in the previous step. So, the firm's demand for labor function l(P,w) is given by: l(P, w) = l* = P虏/4w虏
05

e. Intuitive Explanation

a. The total cost function is directly proportional to the labor input and the wage rate, which reflects the fact that the firm's cost increases linearly as the labor input and wage rate increase. b. The profit function shows how the difference between the revenue (determined by the price of the product, and labor input) and labor costs impacts the overall profit that the firm can make. c. The supply function q(P, w) = P/w indicates that the firm is willing to produce and supply more calculators as the product price P increases and/or labor cost w decreases. d. The demand for labor function l(P, w) = P虏/4w虏 shows that firms will demand more labor as the price P increases relative to the wage rate w - it is a more cost-effective input when product price is high, or labor cost is low. e. These functions outline how different factors, such as product price (P), labor cost (w), and labor input (l), interact and shape a firm's decision-making process in the calculator assembly market. Understanding these relationships can help the firm optimize its operations to maximize profit.

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Key Concepts

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

Total Cost Function
The total cost function provides insight into how much a firm spends based on labor input and wage rates. In this scenario, since the firm incurs costs by hiring workers, the variable cost (VC) is calculated by multiplying the number of labor hours (\( l \)) by the wage rate (\( w \)). So, the equation is:
  • Variable Cost (VC): \( VC = w \times l \)
Assuming no fixed costs, the total cost (\( TC \)) becomes entirely the variable cost. So, we can say:
  • Total Cost (TC): \( TC = w \times l \)
This function helps businesses track their expenses, especially with rising labor needs or wage changes.
Supply Function
The supply function shows the relationship between the quantity of goods a firm is willing to make available to the market and factors like product price and cost of production. In this calculator assembly scenario, the equation for supply (\( q(P, w) \)) is found through optimizing labor input to maximize profits.
This is done by maximizing profit, differentiating with respect to labor, and finding the point where profit is highest. The optimal labor input is represented as:
  • Optimal labor input (\( l^* \)): \( l^* = \frac{P^2}{4w^2} \)
Substituting this back into the production function gives us the supply function:
  • Supply: \( q(P, w) = \frac{P}{w} \)
This reveals that as the product price rises, or labor cost falls, the firm is prepared to supply more calculators.
Profit Maximization
Profit maximization is the core motive of most firms. It involves determining the level of production and resource allocation where a firm's earnings outstrip costs by the greatest margin.
For our firm, profit is calculated as the difference between total revenue and total costs. Total revenue (\( TR \)) is simply the price per unit (\( P \)) multiplied by the number of units (\( q \)), thus:
  • Total Revenue: \( TR = P \times (2 \sqrt{l}) \)
Subtracting total costs, we derive the profit (\( ext{Profit} \)) function:
  • Profit: \( ext{Profit} = P \times (2 \sqrt{l}) - w \times l \)
Solving for labor where this function is maximized leads to greater profitability.
Labor Demand Function
The labor demand function delineates how much labor a firm requires to maximize profits based on external factors like wages and price of goods. For maximized profit, the firm calculates the optimal labor (\( l^* \)) necessary to produce the desired output efficiently.
This is obtained from the profit maximization process:
  • Demand for labor function: \( l(P, w) = \frac{P^2}{4w^2} \)
Essentially, as the price of the product increases relative to the wage, labor becomes a more attractive input. Consequently, firms will seek more labor resources to leverage the higher pricing power of their product.
This function underscores the sensitivity of labor demand to fluctuations in market conditions and production costs.

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Most popular questions from this chapter

With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.12 a. Use steps (b), (d), and (e) from Problem 11.12 to show that \\[ e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right) \quad \text { and } \quad e_{L, v}=s_{K}\left(\sigma+e_{Q, P}\right) \\] b. Describe intuitively why input shares appear somewhat differently in the demand elasticities in part (e) of Problem 11.12 than they do in part (a) of this problem. c. The expression computed in part (a) can be easily generalized to the many- input case as \(e_{x_{i}, w_{i}}=s_{j}\left(A_{i j}+e_{Q, P}\right),\) where \(A_{i j}\) is the Allen elasticity of substitution defined in Problem 10.12 . For reasons described in Problems 10.11 and 10.12 , this approach to input demand in the multi-input case is generally inferior to using Morishima elasticities. One oddity might be mentioned, however. For the case \(i=j\) this expression seems to say that \(e_{L, w}=s_{L}\left(A_{L L}+e_{Q . P}\right),\) and if we jumped to the conclusion that \(A_{L L}=\sigma\) in the two-input case, then this would contradict the result from Problem \(11.12 .\) You can resolve this paradox by using the definitions from Problem 10.12 to show that, with two inputs, \(A_{L L}=\left(-s_{K} / s_{L}\right) \cdot A_{K L}=\left(-s_{K} / s_{L}\right) \cdot \sigma\) and so there is no disagreement.

Because firms have greater flexibility in the long run, their reactions to price changes may be greater in the long run than in the short run. Paul Samuelson was perhaps the first economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Ch芒telier's Principle. The basic idea of the principle is that any disturbance to an equilibrium (such as that caused by a price change) will not only have a direct effect but may also set off feedback effects that enhance the response. In this problem we look at a few examples. Consider a price-taking firm that chooses its inputs to maximize a profit function of the form \(\Pi(P, v, w)=P f(k, 1)-w l-v k .\) This maximization process will yield optimal solutions of the general form \(q^{*}(P, v, w), I^{*}(P, v, w),\) and \(k^{*}(P, v, w) .\) If we constrain capital input to be fixed at \(\bar{k}\) in the short run, this firm's short-run responses can be represented by \(q^{s}(P, w, \bar{k})\) and \(I^{*}(P, w, \bar{k})\) a. Using the definitional relation \(q^{*}(P, v, w)=q^{s}\left(P, w, k^{*}(P, v, w)\right),\) show that $$\frac{\partial q^{*}}{\partial P}=\frac{\partial q^{s}}{\partial P}+\frac{-\left(\frac{\partial k^{*}}{\partial P}\right)^{2}}{\frac{\partial k^{*}}{\partial v}}$$ Do this in three steps. First, differentiate the definitional relation with respect to \(P\) using the chain rule. Next, differentiate the definitional relation with respect to \(v\) (again using the chain rule), and use the result to substitute for \(\partial q^{3} / \partial k\) in the initial derivative. Finally, substitute a result analogous to part (c) of Problem 11.10 to give the displayed equation. b. Use the result from part (a) to argue that \(\partial q^{*} / \partial P \geq \partial q^{s} / \partial P\). This establishes Le Ch芒telier's Principle for supply: Long-run supply responses are larger than (constrained) short-run supply responses. c. Using similar methods as in parts (a) and (b), prove that Le Ch芒telier's Principle applies to the effect of the wage on labor demand. That is, starting from the definitional relation \(l^{*}(P, v, w)=l^{s}\left(P, w, k^{*}(P, v, w)\right),\) show that \(\partial l^{*} / \partial w \leq \partial l^{s} / \partial w\) implying that long-run labor demand falls more when wage goes up than short-run labor demand (note that both of these derivatives are negative). d. Develop your own analysis of the difference between the short- and long-run responses of the firm's cost function \([C(v, w, q)]\) to a change in the wage \((w)\)

This problem has you work through some of the calculations associated with the numerical example in the Extensions. Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors (GM), who we imagine are deciding between remaining as separate firms or having GM acquire Fisher Body and thus become one (larger) firm. Let the total surplus that the units generate together be \(S\left(x_{F}, x_{G}\right)=x_{F}^{1 / 2}+a x_{G}^{1 / 2},\) where \(x_{F}\) and \(x_{G}\) are the investments undertaken by the managers of the two units before negotiating, and where a unit of investment costs \(\$ 1 .\) The parameter \(a\) measures the importance of GM's manager's investment. Show that, according to the property rights model worked out in the Extensions, it is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, in particular, if \(a>\sqrt{3}\)

Young's theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that \(\partial l(P, v, w) / \partial v=\partial k(P, v, w) / \partial w\). Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that \(\partial q / \partial w=-\partial l / \partial P\). Interpret this result. d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied.

The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for \(\$ 30\) per pound. In bad weather it sells for only \(\$ 20\) per pound. Caviar produced one weck will not keep until the next week. A small caviar producer has a cost function given by $$C=0.5 q^{2}+5 q+100$$ where \(q\) is weekly caviar production. Production decisions must be made before the weather (and the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5 a. How much caviar should this firm produce if it wishes to maximize the expected value of its profits? b. Suppose the owner of this firm has a utility function of the form \\[ \text { utility }=\sqrt{\pi} \\] where \(\pi\) is weekly profits. What is the expected utility associated with the output strategy defined in part (a)? c. Can this firm owner obtain a higher utility of profits by producing some output other than that specified in parts (a) and (b)? Explain. d. Suppose this firm could predict next week's price but could not influence that price. What strategy would maximize expected profits in this case? What would expected profits be?
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