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Chapter 7: Problem 1

Initially a firm's wage is \(w=40\) and its rental cost of capital is \(r=40 .\) After its wage rate is halved, how do its isocost lines change?

Short Answer

Expert verified
The isocost lines become steeper and reflect a lower cost for labor relative to capital.

Step by step solution

01

Understanding Isocost Lines

Isocost lines represent combinations of inputs that result in the same total cost for a firm. The equation for an isocost line is given by: \[ C = wL + rK \]where \( C \) is total cost, \( w \) is the wage rate, \( L \) is the labor, \( r \) is the rental cost of capital, and \( K \) is the capital.
02

Initial Isocost Equation

Initially, the wage \( w = 40 \) and the rental cost of capital \( r = 40 \). Thus, the isocost line equation is:\[ C = 40L + 40K \]Simplifying, we can divide through by 40:\[ C/40 = L + K \]
03

New Isocost Equation with Halved Wage

The wage rate is halved, so the new wage \( w = 20 \). The rental cost of capital remains \( r=40 \). The new isocost equation is:\[ C = 20L + 40K \]Dividing through by 20 gives:\[ C/20 = L + 2K \]
04

Effect on Isocost Lines

Initially, the isocost equation was \( C/40 = L + K \). The new isocost equation is \( C/20 = L + 2K \), which means the slope of the isocost line has changed from -1 to -2. The isocost lines become steeper because labor is less expensive relative to capital.

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Key Concepts

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

Wage Rate
The wage rate is an important concept in microeconomic theory because it represents the cost to a firm for hiring labor. The wage rate is often denoted as \( w \). It directly affects a firm's overall cost structure and decision-making about hiring or using labor. In the context of isocost lines, the wage rate determines how much labor can be purchased for a certain cost. When the wage rate changes, it shifts the isocost lines and alters the firm's strategy for balancing labor and capital to maintain costs. For example, if a firm's wage rate is reduced by half, the firm can hire more labor for the same level of expenditure. This shift makes labor relatively cheaper compared to capital, and the slope of the isocost line becomes steeper. Thus, a decrease in the wage rate allows the firm to potentially employ more workers while staying within its budget.
Rental Cost of Capital
The rental cost of capital, represented as \( r \), refers to the cost a firm incurs for using capital equipment or resources. Like the wage rate, this cost impacts the firm's decisions on resource allocation. The rental cost of capital affects the firm's expenses for capital usage, much like renting a machine or equipment rather than purchasing outright. In the context of isocost lines, when this cost remains constant, such as in our example scenario where \( r = 40 \), the line's slope only changes with alterations in the wage rate. The cost of capital acts as a balancing act with wages, determining how much of each input (labor and capital) a firm can use while maintaining the same total cost. It helps firms decide between using more capital-intensive methods versus hiring more labor.
Microeconomic Theory
Microeconomic theory examines the behaviors and decisions of individual firms and consumers, focusing on how resources are allocated and distributed in an economy. Within this theoretical framework, concepts like isocost lines, wage rates, and rental costs of capital are prominent. Isocost lines are graphical representations of different combinations of labor and capital that a firm can employ without exceeding a certain cost. These lines are influenced by both wage rates and rental costs of capital. Microeconomic theory helps explain how changes in these costs affect a firm's production decisions and overall profitability. By understanding such interactions, businesses can better strategize and optimize their input mixes to improve efficiency and minimize costs, thus achieving greater productivity. These insights are vital for firms aiming to remain competitive in the market by optimizing their resource allocation strategies.

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Most popular questions from this chapter

A firm's learning curve, which shows the relationship between average cost and cumulative output (the sum of its output since the firm started producing \(),\) is \(A C=a+b N^{-r} ;\) where \(A C\) is its average cost; \(N\) is its cumulative output; \(a, b,\) and \(r\) are positive constants; and \(0

Rosenberg (2004) reported the invention of a new machine that serves as a mobile station for receiving and accumulating packed flats of strawberries close to where they are picked, reducing workers' time and the burden of carrying full flats of strawberries. A machine-assisted crew of 9 pickers produces as much output, \(q^{*},\) as that of an unaided crew of 15 workers. In a 6 -day, 50 -hour workweek, the machine replaces 300 worker hours. At an hourly wage cost of \(\$ 10,\) a machine saves \(\$ 3,000\) per week in labor costs, or \(\$ 78,000\) over a 26 -week harvesting season. The cost of machine operation and maintenance expressed as a daily rental is \(\$ 240,\) or \(\$ 1,440\) for a 6 -day week. Thus, the net savings equal \(\$ 1,560\) per week, or \(\$ 40,560\) for 26 weeks. a. Draw the \(q^{*}\) isoquant assuming that only two technologies are available (pure labor and labormachine). Label the isoquant and axes as thoroughly as possible. b. Add an isocost line to show which technology the firm chooses. (Be sure to measure wage and rental costs on a comparable time basis.) c. Draw the corresponding cost curves (with and without the machine), assuming constant returns to scale, and label the curves and the axes as thoroughly as possible.

A Chinese high technology manufacturing firm has a production function of \(q=7 L^{0.80} K^{0.20}\) (based on Zhang et al., 2012 ). It faces factor prices of \(w=8\) and \(r=2 .\) What are its short-run marginal cost and average variable cost curves?

In the twentieth century, department stores and supermarkets largely replaced smaller specialty stores, as consumers found it more efficient to go to one store rather than many stores. Consumers incur a transaction or search cost to shop, primarily the opportunity cost of their time. This transaction cost consists of a fixed cost of traveling to and from the store and a variable cost that rises with the number of different types of items the consumer tries to find on the shelves. By going to a supermarket that carries meat, fruits and vegetables, and other items, consumers can avoid some of the fixed transaction costs of traveling to a separate butcher shop, produce mart, and so forth. Use math or figures to explain why a shopper's average costs are lower when buying at a single supermarket than from many stores. (Hint: Define the goods as the items purchased and brought home.

A U.S. electronics firm is considering moving its production abroad. Its production function is \(q=L^{0.8} K^{0.2}\) (based on Hsieh, 1995 ), so its \(M P_{L}=0.8 K^{0.2} / L^{0.2}\) and its \(M P_{K}=0.2 L^{0.8} / K^{0.8}\) (as Appendix 6 C shows). The U.S. factor prices are \(w=8, r=2 .\) In Mexico, the wage is \(40 \%\) lower than in the United States but the firm faces the same cost of capital: \(w^{*}=4.80\) and \(r^{*}=2 .\) What \(\operatorname{are} L\) and \(K,\) and what is the cost of producing \(q=100\) units in both countries?
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