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microeconomics
Microeconomics 4th Edition David Besanko, Ronald Braeutigam - Solutions
7.1. A computer-products retailer purchases laser printers from a manufacturer at a price of $500 per printer. During the year the retailer will try to sell the printers at a price higher than $500 but may not be able to sell all of the printers. At the end of the year, the manufacturer will pay
9. For a given quantity of output, under what conditions would the short-run quantity demanded for a variable input (such as labor) equal the quantity demanded in the long run?
8. In Chapter 5 you learned that, under certain conditions, a good could be a Giffen good: An increase in the price of the good could lead to an increase, rather than a decrease, in the quantity demanded. In the theory of cost minimization, however, we learned that, an increase in the price of an
7. What is the difference between the expansion path and the input demand curve?
6. Explain why, at an interior optimal solution to the firm’s cost-minimization problem, the additional output that the firm gets from a dollar spent on labor equals the additional output from a dollar spent on capital. Why would this condition not necessarily hold at a corner point optimal
5. Could the solution to the firm’s cost-minimization problem ever occur off the isoquant representing the required level of output?
4. How does an increase in the price of an input affect the slope of an isocost line?
3. Why does the “sunkness” or “nonsunkness” of a cost depend on the decision being made?
2. You decide to start a business that provides computer consulting advice for students in your residence hall.What would be an example of an explicit cost you would incur in operating this business? What would be an example of an implicit cost you would incur in operating this business?
1. A biotechnology firm purchased an inventory of test tubes at a price of $0.50 per tube at some point in the past. It plans to use these tubes to clone snake cells.Explain why the opportunity cost of using these test tubes might not equal the price at which they were acquired.
The short-run cost-minimization problem involves a choice of inputs when at least one input quantity is held fixed.
All variable costs are nonsunk. Fixed costs can be sunk (unavoidable) or nonsunk (avoidable) if the firm produces no output.
In the short run, at least one input is fixed. Variable costs are output sensitive—they vary as output varies.Fixed costs are output insensitive—they remain the same for all positive levels of output.
When the elasticity of substitution between inputs is small, the price elasticity of demand for each input is also small. When the elasticity of substitution is large, so is the price elasticity of demand.
The price elasticity of demand for an input is the percentage change in the cost-minimizing quantity of that input with respect to a 1 percent change in its price.
An input demand curve shows how the costminimizing quantity of the input varies with its input price.
The expansion path shows how the cost-minimizing quantity of inputs varies as quantity of output changes.
An increase in the quantity of output will cause the cost-minimizing quantity of an input to go up if the input is a normal input and will cause the cost-minimizing quantity of the input to go down if the input is an inferior input.
An increase in the price of an input causes the costminimizing quantity of that input to go down or stay the same. It can never cause the cost-minimizing quantity to go up.
At corner point solutions to the cost-minimization problem, the ratios of marginal products to input prices may not be equal.
At an interior solution to the long-run cost-minimization problem, the firm adjusts input quantities so that the marginal rate of technical substitution equals the ratio of the input prices. Equivalently, the ratio of the marginal product of one input to its price equals the corresponding ratio for
An isocost line shows all combinations of inputs that entail the same total cost. When graphed with quantity of labor on the horizontal axis and quantity of capital on the vertical axis, the slope of an isocost line is minus the ratio of the price of labor to the price of capital.
The long run is the period of time that is long enough for the firm to vary the quantities of all its inputs. The short run is the period of time in which at least one of the firm’s input quantities cannot be changed.
Sunk costs are costs that have already been incurred and cannot be recovered. Nonsunk cost are costs that can be avoided if certain choices are made.
Accounting costs include explicit costs only.Economic costs include explicit and implicit costs.
Explicit costs involve a direct monetary outlay.Implicit costs do not involve an outlay of cash.
From a firm’s perspective, the opportunity cost of using the productive services of an input is the current market price of the input.
Opportunity costs are forward looking. When evaluating the opportunity cost of a particular decision, you need to identify the value of the alternatives that the decision forecloses in the future.
The opportunity cost of a decision is the payoff associated with the best of the alternatives that are not chosen.
Describe the firm’s cost-minimization problem in the short run and analyze the firm’s choice of inputs when the firm has at least one fixed factor of production and one or more variable factors.
Employ comparative statics analysis to explain how changes in the prices of inputs and the level of output affect a firm’s choices of inputs and its costs of production.
Describe a firm’s cost-minimization problem in the long run, using the concept of isocost lines (the combinations of inputs such as labor and capital that have the same total cost).
Identify and apply different concepts of costs that figure in a firm’s decision making, including explicit versus implicit costs, opportunity cost, economic versus accounting costs, and sunk versus nonsunk costs.
6.30. Suppose that in the 21st century the production of semiconductors requires two inputs: capital (denoted by K) and labor (denoted by L). The production function takes the form However, in the 23rd century, suppose the production function for semiconductors will take the form In other words, in
6.28. A firm’s production function is initially with and Over time, the production function changes to with and . (Assume, as in LearningBy-Doing Exercise 6.5, that for this production process, L and K must each be greater than or equal to 1.)a) Verify that this change represents technological
6.27. Suppose a firm’s production function initially took the form Q ! 500(L # 3K ). However, as a result of a manufacturing innovation, its production function is now Q ! 1,000(0.5L # 10K ).
6.26. The following table presents information on how many cookies can be produced from eggs and a mixture of other ingredients (measured in ounces):
6.25. Consider the following production functions and their associated marginal products. For each production function, indicate whether (a) the marginal product of each input is diminishing, constant, or increasing in the quantity of that input; (b) the production function exhibits decreasing,
6.24. Consider a CES production function given by Q !(K0.5 " L0.5)2.a) What is the elasticity of substitution for this production function?b) Does this production function exhibit increasing, decreasing, or constant returns to scale?c) Suppose that the production function took the form Q ! (100 "
6.23. A firm’s production function is Q ! 5L2/3 K1/3 with MPK ! (5/3)L2/3K#2/3 and MPL ! (10/3)L#1/3K1/3a) Does this production function exhibit constant, increasing, or decreasing returns to scale?b) What is the marginal rate of technical substitution of L for K for this production function?c)
6.22. Consider a production function whose equation is given by the formula Q ! LK2, which has corresponding marginal products, MPL ! K2 and MPK ! 2LK.Show that the elasticity of substitution for this production function is exactly equal to 1, no matter what the values of K and L are.
6.21. A firm produces a quantity Q of breakfast cereal using labor L and material M with the production function The marginal product functions for this production function area) Are the returns to scale increasing, constant, or decreasing for this production function?MPM ! 25B LM" 1 MPL ! 25B ML"
6.20. What can you say about the returns to scale of the Leontief production function Q ! min(aK, bL), where a and b are positive constants?
6.19. What can you say about the returns to scale of the linear production function Q ! aK " bL, where a and b are positive constants?
6.18. To produce cake, you need eggs E and premixed ingredients I. Every cake needs exactly one egg and one package of ingredients. When you add two eggs to one package of ingredients, you produce only one cake. Similarly, when you have only one egg, you can’t produce two cakes even though you
6.17. Let B be the number of bicycles produced from F bicycle frames and T tires. Every bicycle needs exactly two tires and one frame.a) Draw the isoquants for bicycle production.b) Write a mathematical expression for the production function for bicycles.
6.16. Two points, A and B, are on an isoquant drawn with labor on the horizontal axis and capital on the vertical axis. The capital–labor ratio at B is twice that at A, and the elasticity of substitution as we move from A to B is 2. What is the ratio of the MRTSL,K at A versus that at B?
6.15. Suppose that a firm’s production function is given by Q ! KL " K, with MPK ! L " 1 and MPL ! K. At point A, the firm uses K ! 3 units of capital and L ! 5 units of labor. At point B, along the same isoquant, the firm would only use 1 unit of capital.a) Calculate how much labor is required
6.14. Consider the following production functions and their associated marginal products. For each production function, determine the marginal rate of technical substitution of labor for capital, and indicate whether the isoquants for this production function exhibit diminishing marginal rate of
6.13. You might think that when a production function has a diminishing marginal rate of technical substitution of labor for capital, it cannot have increasing marginal products of capital and labor. Show that this is not true, using the production function Q ! K2 L2, with the corresponding
6.12. Suppose the production function is given by the following equation (where a and b are positive constants):Q ! aL " bK. What is the marginal rate of technical substitution of labor for capital (MRTSL,K) at any point along an isoquant?
6.11. Consider again the production function for DVDs: Q ! KL2 # L3.
6.10. Suppose the production function is given by the equation Graph the isoquants corresponding to Q ! 10, Q ! 20, and Q ! 50. Do these isoquants exhibit diminishing marginal rate of technical substitution?
6.9. Suppose the production function for automobiles is where Q is the quantity of automobiles produced per year, L is the quantity of labor (man-hours), and K is the quantity of capital (machine-hours).a) Sketch the isoquant corresponding to a quantity of Q ! 100.b) What is the general equation
6.8. Widgets are produced using two inputs, labor, L, and capital, K. The following table provides information on how many widgets can be produced from those inputs:
6.7. The following table shows selected input quantities, total products, average products, and marginal products. Fill in as much of the table as you can:Labor, L Total Product, Q APL MPL 0 0 0—1 19 19 2 36 34 256 64 103 5 375 6 129 7 637 91 133 8 96 9 891 10 100 11 1089 89 12 96 13 14 #7 15 75
6.6. Economists sometimes “prove” the law of diminishing marginal returns with the following exercise:Suppose that production of steel requires two inputs, labor and capital, and suppose that the production function is characterized by constant returns to scale. Then, LK Q 0 100 0 1 100 1 2 100
6.5. Are the following statements correct or incorrect?a) If average product is increasing, marginal product must be less than average product.b) If marginal product is negative, average product must be negative.c) If average product is positive, total product must be rising.d) If total product is
6.4. Suppose that the production function for DVDs is given by Q ! KL2 # L3, where Q is the number of disks produced per year, K is machine-hours of capital, and L is man-hours of labor.a) Suppose K ! 600. Find the total product function and graph it over the range L ! 0 to L ! 500. Then sketch the
6.2. A firm is required to produce 100 units of output using quantities of labor and capital (L, K) ! (7, 6). For each of the following production functions, state whether it is possible to produce the required output with the given input combination. If it is possible, state whether the input
6.1. A firm uses the inputs of fertilizer, labor, and hothouses to produce roses. Suppose that when the quantity of labor and hothouses is fixed, the relationship between the quantity of fertilizer and the number of roses produced is given by the following table:Tons of Number of Tons of Number of
9. Suppose the production of electricity requires just two inputs, capital and labor, and that the production function is Cobb–Douglas. Now consider the isoquants corresponding to three different levels of output: Q !100,000 kilowatt-hours, Q ! 200,000 kilowatt-hours, and Q ! 400,000
8. What is the elasticity of substitution? What does it tell us?
7. Why would a firm that seeks to minimize its expenditures on inputs not want to operate on the uneconomic portion of an isoquant?
6. Could the isoquants corresponding to two different levels of output ever cross?
5. Why must an isoquant be downward sloping when both labor and capital have positive marginal products?
4. What is the difference between diminishing total returns to an input and diminishing marginal returns to an input? Can a total product function exhibit diminishing marginal returns but not diminishing total returns?
3. What is the difference between average product and marginal product? Can you sketch a total product function such that the average and marginal product functions coincide with each other?
2. Suppose a total product function has the “traditional shape” shown in Figure 6.2. Sketch the shape of the corresponding labor requirements function (with quantity of output on the horizontal axis and quantity of labor on the vertical axis).
1. We said that the production function tells us the maximum output that a firm can produce with its quantities of inputs. Why do we include the word maximum in this definition?
If the elasticity of substitution is large, there is substantial opportunity to substitute between inputs. In equation (6.6), this corresponds to the fact that ! will be large if the percentage change in MRTSL,K is small, as illustrated in Figure 6.11(b).
If the elasticity of substitution is close to 0, there is little opportunity to substitute between inputs. We can see this from equation (6.6), where ! will be close to 0 when the percentage change in MRTSL,K is large, as in Figure 6.11(a).
When the production function offers abundant input substitution opportunities, the MRTSL,K changes gradually as we move along an isoquant. In this case, the isoquants are nearly straight lines, as in Figure 6.11(b).
When the production function offers limited input substitution opportunities, the MRTSL,K changes substantially as we move along an isoquant. In this case, the isoquants are nearly L-shaped, as in Figure 6.11(a).
When average product neither increases nor decreases in labor because we are at a point at which APL is at a maximum (point A in Figure 6.3), then marginal product is equal to average product.
When average product is decreasing in labor, marginal product is less than average product. That is, if APL decreases in L, then MPL " APL.
When average product is increasing in labor, marginal product is greater than average product. That is, if APL increases in L, then MPL ! APL.
Verify whether a change in a production function represents technological progress, and if it does, determine whether the technological progress is labor-saving, neutral, or capital-saving.
Determine whether a production function exhibits increasing, constant, or decreasing returns to scale.
Compare and contrast a number of special production functions that are frequently used in microeconomic analysis: the linear production function, the Leontief production function, the Cobb–Douglas production function, and the CES production function.
Describe how the concept of elasticity of substitution measures the firm’s input substitution opportunities.
Show graphically how a firm’s input substitution opportunities determine the shape of the firm’s isoquants.
Explain how the concept of marginal rate of the technical substitution is related to the concept of marginal product.
Derive the equation of an isoquant from the equation of the production function.
Demonstrate how a production function with two variable inputs can be represented by isoquants.
Illustrate graphically how the graphs of the marginal product and average product functions relate to the graph of the total product function.
Describe the concept of diminishing marginal returns.
Distinguish between the concepts of total product, marginal product, and average product for a production function with a single input.
Illustrate the difference between technologically efficient combinations of inputs and outputs and technologically inefficient combinations of inputs and outputs.
Explain how a production function represents the various technological recipes the firm can choose.
5.33. Gina lives in Chicago and very much enjoys traveling by air to see her mother in Italy. On the accompanying graph, x denotes her number of round trips to Italy each year. The composite good y measures her annual consumption of other goods; the price of the composite good is py, which is
5.32. Julie buys food and other goods. She has an income of $400 per month. The price of food is initially$1.00 per unit. It then rises to $1.20 per unit. The prices of other goods do not change. To help Julie out, her mother offers to send her a check each month to supplement her income. Julie
5.31. Raymond consumes leisure (L hours per day) and other goods (Y units per day), with preferences described by The associated marginal utilities are MUY ! 1 and The price of other goods is 1 euro per unit. The wage rate is w euros per hour.a) Show how the number of units of leisure Raymond
5.30. Consider Noah’s preferences for leisure (L) and other goods The associated marginal utilities are and MUY !Suppose that Is Noah’s supply of labor backward bending?
5.29. Terry’s utility function over leisure (L) and other goods (Y ) is U(L, Y ) ! Y " LY. The associated marginal utilities are MUY ! 1 " L and MUL ! Y. He purchases other goods at a price of $1, out of the income he earns from working. Show that, no matter what Terry’s wage rate, the optimal
5.28. Consider the optimal choice of labor and leisure discussed in the text. Suppose a consumer works the first
5.27. Joe’s income consumption curve for tea is a vertical line on an optimal choice diagram, with tea on the horizontal axis and other goods on the vertical axis.a) Show that Joe’s demand curve for tea must be downward sloping.b) When the price of tea drops from $9 to $8 per pound, the change
5.26. Suppose that Bart and Homer are the only people in Springfield who drink 7-UP. Moreover their inverse demand curves for 7-UP are, respectively, P ! 10 " 4QB and P ! 25 " 2QH, and, of course, neither one can consume a negative amount. Write down the market demand curve for 7-UP in Springfield,
5.25. One million consumers like to rent movie videos in Pulmonia. Each has an identical demand curve for movies. The price of a rental is $P. At a given price, will the market demand be more elastic or less elastic than the demand curve for any individual? (Assume there are no network
5.24. There are two consumers on the market: Jim and Donna. Jim’s utility function is U(x, y) ! xy, with associated marginal utility functions MUx ! y and MUy ! x. Donna’s utility function is U(x, y) ! x2y, with associated marginal utility functions MUx ! 2xy and MUy ! x2. Income of Jim is IJ !
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