Problem Set #3: Production & Costs
Name__________________________ ECO 302—Spring 2011 Prof. Laura Giuliano
Problem Set #3: Production & Costs Due: Tuesday, April 5th
1. A bottling company uses two inputs to assemble bottles of the soft drink Squish: bottling machines and workers. Running the bottling machine is an imperfect substitute for the labor performed by the workers. The production function is: ¼
q = 100 K
½
L
The implicit rental rate plus hourly operating cost of the machine is r = $15 per hour; and the workers each earn w = $20 per hour. The current level of input use for the production of q=1000 bottles is: K=16 machine hours and L=25 labor hours. Question: Is this firm producing at minimum cost? If it is minimizing costs, explain why. If it is not minimizing costs, explain how the firm should change its input use to lower its costs. For full credit, you must discuss the two efficiency conditions for minimizing cost. That is: Is the bundle technologically efficient? (yes/no/why or why not) Is it economically efficient? (yes/no/why or why not) (You are not required to solve for the optimal bundle of inputs, but you must justify your answers.)
Name__________________________
2.
Sid and Nancy both need to buy pens for school. The bookstore has only two types of pens: Basic blue pens (B), which cost $.60 each. Long-lasting green pens (G), which have twice as much ink as the blue pens, and cost $1.00 each.
Nancy needs the pens to draw multi-colored graphs in her Economics class. For Nancy, these two types of pens are perfect complements. The graphs Nancy must draw require the ink of two basic blue pens for every one longlasting green pen, and her production function is: q = min(25B,50G). Nancy will need to draw 100 graphs. Sid needs the pens to write essays in his English class. One essay requires the ink of one basic blue pen or ½ of a long-lasting pen, and ink color is irrelevant. So, for Sid, these two types of pens are perfect substitutes, and his production function is: q = B + 2G. Sid will need to write 10 essays. A. Sketch Nancy’s q=100 isoquant here:
B. Sketch Sid’s q=10 isoquant here:
B
B
G
G
C. At the current prices, what is Nancy’s cost-minimizing bundle (for drawing 100 graphs)? Basic blue: _____ Long-lasting green: _____ How much will she spend on pens in order to draw 100 graphs? ________ D. At the current prices, what is Sid’s cost-minimizing bundle of pens (for writing 10 essays)? Basic blue: _____ Long-lasting green:______ How much will he spend on pens in order to write 10 essays? ________
Suppose that when Sid and Nancy go to the bookstore, basic blue pens are on sale for half price ($.30 ea.). E. At the sale prices, what is Nancy’s cost-minimizing bundle? Basic blue: _____ Long-lasting green: _____ How much will she spend on pens in order to draw 100 graphs? ________ F. At the sale prices, what is Sid’s cost-minimizing bundle of pens? Basic blue: ______ Long-lasting green: _____ How much will he spend on pens in order to write 10 essays? ________
Name__________________________
3.
Bill has two sources of income:
(1) He works at a local coffee shop for wage of $7.50 per hour. The job is very flexible; he can work there as many hours as he wants. (2) When he’s not working at the coffee shop, Bill assembles and sells widgets with the help of his friend Ted, whom he pays $10 per hour. Bill’s labor (LB) and Ted’s labor (LT) are imperfect substitutes (Bill and Ted have somewhat different skills and their labor is somewhat complementary in the production process). The isoquant for assembling q=10 widgets per day is shown in the graph below. Also shown in the isocost line representing Bill’s current economic cost (C1) of assembling q=10 widgets, given the wage he currently pays Ted and the implicit cost of his own labor, along with Bill’s current cost-minimizing bundle of labor inputs (LB1 and LT1). Suppose that Bill gets promoted to manager at the coffee shop and his wage goes up from $7.50 to $15.00 per hour. Show graphically and describe verbally the effect that this has on Bill’s choice of labor inputs in his widgetassembly business. Assume that Bill continues to produce 10 widgets per day, that he chooses an input bundle that minimizes the economic costs of production, and that both Bill’s and Ted’s hours are flexible (Bill can still divide his time between the coffee shop and widget assembly however he wants; Ted has nothing else to do and will work as many or as few hours as Bill wants him to at the wage of $10 per hour.) In particular: A. Explain in words why Bill’s raise at the coffee shop is relevant to his choice of inputs in his widget assembly business.
B. Sketch the isocost line representing the new minimum cost (C2) of assembling 10 widgets (it does not need to be drawn exactly to scale). Label the endpoints. C. Label the new optimal bundle of inputs on the graph. D. What happens to the number of hours that Bill and Ted each work at widget assembly per day?
LB (Hours of Bill’s labor)
C1/7.5
LB1
q=10 assembled widgets
LT1
C1 10
.
LT (Hours of Ted’s labor)
Name__________________________
4. On July 24, 2009, the federal minimum wage increased from $6.55 to $7.25 per hour. Demonstrate the possible effect of this minimum wage increase on employment, using a model of a fastfood restaurant’s choice of two types of labor inputs (M=managers and F=frontline workers) in the production of fast-food meals. In particular, compare the effect of an increase in the minimum wage on the optimal choice of labor inputs, given two different estimates of the production function (obtained from two different economists). Isoquants based on these two different estimates are shown in the graphs on the next page. A. Which estimate suggests greater substitutability of managers (M) and frontline workers (F)? (A or B) In each graph, I have indicated the effect of the minimum wage increase on the isocost line, holding the total cost of production (C1) constant. The pre-legislation wages of managers and frontline workers are denoted wM and wF, respectively. B. Explain why an increase in the minimum wage is likely to rotate the isocost line in the direction indicated. (What does the graph imply about the effect of the minimum wage increase on the wages of each type of labor?)
C. On both graphs, sketch the isocost line containing the new cost-minimizing bundle of managers and frontline workers, after the minimum wage increase, and label the new optimal bundle, assuming that restaurants continue to sell 100 meals per day. D. Assuming that fast-food restaurants continue to sell 100 meals per day, under which scenario does the minimum wage increase cause a larger reduction in the employment of frontline workers? (A or B)
E. Which economist would predict a larger increase in the total cost of producing 100 meals? (A or B)
F. If demand for fast food is fairly elastic, explain why managers at fast food restaurants might worry about losing their jobs as a result of the minimum wage increase. [Hint: Fast food restaurants might not continue to sell 100 meals per day.] Which economist’s estimate should cause them more concern? Explain briefly.
Name__________________________
Hours of Manager Labor per day (M)
C1/wM
M1
q=100 meals per day C1 F 1 7.25
C1 wF
Hours of Frontline Labor per day (F)
From Economist A’s Production Function Estimate
Hours of Manager Labor per day (M)
C1/wM
M1
q=100 meals per day C1 F 1
7.25
C1 . wF
Hours of Frontline Labor per day (F)
From Economist B’s Production Function Estimate
Name__________________________ 5. Suppose a firm employs labor as its only variable input in the short run. In the short run, it has a fixed cost of capital equal to $50. Labor costs $20 per day. Each unit of output can be sold at a competitive market price of $10. Output per day (q) for various amounts of labor (L=days of labor) is shown in the table below. (The firm cannot produce anything without at least one day of labor.) A. First, complete the table by calculating each level of labor usage (L)the (approximate) marginal product of labor (MPL), average product of labor (APL), total variable cost (VC), marginal cost (MC), average variable cost (AVC), average fixed cost (AFC), average total cost (AC), total cost (TC), total revenue, and profit. L
q
MPL
APL
VC
MC
AVC
AFC
AC
TC
Revenue
Profit = Revenue – TC
0.0 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0 12.0 14.0
0.0 4.0 10.0 20.0 30.0 36.0 40.0 46.0 50.0 52.0 52.0
-4.00 6.00 10.00 10.00 6.00 4.00
-4.00
-$20.00 $40.00
-$5.00 $3.33 $2.00 $2.00 $3.33
-$5.00
-$12.50 $5.00 $2.50 $1.67 $1.39 $1.25 $1.09
-$17.50 $9.00 $5.50 $4.33 $4.17 $4.25 $4.57
-$70 $90 $110 $130 $150 $170 $210
-$40 $100 $200
--$30 $10 $90
$0.96 $0.96
$5.58 $6.35
$290 $330
$360 $400 $460 $500 $520 $520
$210 $230 $250 $250 $230 $190
2.00 1.00 0.00
6.67 7.50 7.20 6.67 5.75 5.00 4.33 3.71
$80.00 $100.00 $120.00 $160.00 $200.00 $240.00 $280.00
mc
$6.67 $10.00 $20.00 ∞
$3.00 $2.67 $2.78 $3.00 $3.48 $4.00 $4.62 $5.38 avc
afc in the graph below. (Use the boxesac B. Label the curves shown provided.)
$
20
15
p* = 10
5
0 0
20
40
60
q
Answer the following as precisely as possible, using the table and graph. C. Where (at approximately what quantity) does the MC curve cross the AVC curve? q = between____and_____ D. Where (at approximately what quantity) does the MC curve cross the AC curve? q = between____and______ E. At what quantity is MC equal to the market price? q = _______ What is AC at this level of q? AC = _______. Calculate profit at this level of q as profit = (p-AC)×q = _______ F. If the firm produces the quantity that maximizes profit, approximately how much will it produce, and at what profit? q = ___________ ; Profit = __________
Name__________________________ 6. Ned has a small lawn-mowing business, in which the two inputs, a lawn-mower and labor, are perfect complements. Each employee can mow a lawn in half an hour, using exactly ½ hour of labor, and using the mower for exactly ½ hour. Thus, the production function for mowing lawns is: q=min(2K,2L)
where q=lawns mowed, K=hours of lawn-mower use and L=hours of labor.
Ned has one lawn mower. Lawn-mowers cannot feasibly be rented by the hour, and due to safety regulations, they can only be ridden during the 18 hours of daylight per day. Thus, Ned’s lawn-mower capital is fixed in the short run at 18 hours per day (K=18 hours). The implicit rental rate of the lawn-mower is r=$4/day. Labor is variable and can be hired by the half hour. The wage rate is w=$10 per hour. A. What is the marginal product (MPL) of the 1st hour of labor per day? = the 10th hour ?= the 19th hour ?= B. What is the average product (APL) of the 1st hour of labor per day? = the 10th hour ?= the 19th hour ?= C. What is the marginal cost of the 1st lawn mowed per day? = the 10th lawn mowed ?= the 19th lawn mowed ?= the 37th lawn mowed ?= D. What is the average variable cost of the 1st lawn mowed per day? = the 10th lawn mowed ?= the 19th lawn mowed ?= the 37th lawn mowed ?=
E. Sketch the MC and AVC functions here. Label both axes appropriately.
F. Write an equation showing the total cost per day (C) as a function of lawns mowed (q):
Problem Set #3: Production & Costs Due: Tuesday, April 5th
1. A bottling company uses two inputs to assemble bottles of the soft drink Squish: bottling machines and workers. Running the bottling machine is an imperfect substitute for the labor performed by the workers. The production function is: ¼
q = 100 K
½
L
The implicit rental rate plus hourly operating cost of the machine is r = $15 per hour; and the workers each earn w = $20 per hour. The current level of input use for the production of q=1000 bottles is: K=16 machine hours and L=25 labor hours. Question: Is this firm producing at minimum cost? If it is minimizing costs, explain why. If it is not minimizing costs, explain how the firm should change its input use to lower its costs. For full credit, you must discuss the two efficiency conditions for minimizing cost. That is: Is the bundle technologically efficient? (yes/no/why or why not) Is it economically efficient? (yes/no/why or why not) (You are not required to solve for the optimal bundle of inputs, but you must justify your answers.)
Name__________________________
2.
Sid and Nancy both need to buy pens for school. The bookstore has only two types of pens: Basic blue pens (B), which cost $.60 each. Long-lasting green pens (G), which have twice as much ink as the blue pens, and cost $1.00 each.
Nancy needs the pens to draw multi-colored graphs in her Economics class. For Nancy, these two types of pens are perfect complements. The graphs Nancy must draw require the ink of two basic blue pens for every one longlasting green pen, and her production function is: q = min(25B,50G). Nancy will need to draw 100 graphs. Sid needs the pens to write essays in his English class. One essay requires the ink of one basic blue pen or ½ of a long-lasting pen, and ink color is irrelevant. So, for Sid, these two types of pens are perfect substitutes, and his production function is: q = B + 2G. Sid will need to write 10 essays. A. Sketch Nancy’s q=100 isoquant here:
B. Sketch Sid’s q=10 isoquant here:
B
B
G
G
C. At the current prices, what is Nancy’s cost-minimizing bundle (for drawing 100 graphs)? Basic blue: _____ Long-lasting green: _____ How much will she spend on pens in order to draw 100 graphs? ________ D. At the current prices, what is Sid’s cost-minimizing bundle of pens (for writing 10 essays)? Basic blue: _____ Long-lasting green:______ How much will he spend on pens in order to write 10 essays? ________
Suppose that when Sid and Nancy go to the bookstore, basic blue pens are on sale for half price ($.30 ea.). E. At the sale prices, what is Nancy’s cost-minimizing bundle? Basic blue: _____ Long-lasting green: _____ How much will she spend on pens in order to draw 100 graphs? ________ F. At the sale prices, what is Sid’s cost-minimizing bundle of pens? Basic blue: ______ Long-lasting green: _____ How much will he spend on pens in order to write 10 essays? ________
Name__________________________
3.
Bill has two sources of income:
(1) He works at a local coffee shop for wage of $7.50 per hour. The job is very flexible; he can work there as many hours as he wants. (2) When he’s not working at the coffee shop, Bill assembles and sells widgets with the help of his friend Ted, whom he pays $10 per hour. Bill’s labor (LB) and Ted’s labor (LT) are imperfect substitutes (Bill and Ted have somewhat different skills and their labor is somewhat complementary in the production process). The isoquant for assembling q=10 widgets per day is shown in the graph below. Also shown in the isocost line representing Bill’s current economic cost (C1) of assembling q=10 widgets, given the wage he currently pays Ted and the implicit cost of his own labor, along with Bill’s current cost-minimizing bundle of labor inputs (LB1 and LT1). Suppose that Bill gets promoted to manager at the coffee shop and his wage goes up from $7.50 to $15.00 per hour. Show graphically and describe verbally the effect that this has on Bill’s choice of labor inputs in his widgetassembly business. Assume that Bill continues to produce 10 widgets per day, that he chooses an input bundle that minimizes the economic costs of production, and that both Bill’s and Ted’s hours are flexible (Bill can still divide his time between the coffee shop and widget assembly however he wants; Ted has nothing else to do and will work as many or as few hours as Bill wants him to at the wage of $10 per hour.) In particular: A. Explain in words why Bill’s raise at the coffee shop is relevant to his choice of inputs in his widget assembly business.
B. Sketch the isocost line representing the new minimum cost (C2) of assembling 10 widgets (it does not need to be drawn exactly to scale). Label the endpoints. C. Label the new optimal bundle of inputs on the graph. D. What happens to the number of hours that Bill and Ted each work at widget assembly per day?
LB (Hours of Bill’s labor)
C1/7.5
LB1
q=10 assembled widgets
LT1
C1 10
.
LT (Hours of Ted’s labor)
Name__________________________
4. On July 24, 2009, the federal minimum wage increased from $6.55 to $7.25 per hour. Demonstrate the possible effect of this minimum wage increase on employment, using a model of a fastfood restaurant’s choice of two types of labor inputs (M=managers and F=frontline workers) in the production of fast-food meals. In particular, compare the effect of an increase in the minimum wage on the optimal choice of labor inputs, given two different estimates of the production function (obtained from two different economists). Isoquants based on these two different estimates are shown in the graphs on the next page. A. Which estimate suggests greater substitutability of managers (M) and frontline workers (F)? (A or B) In each graph, I have indicated the effect of the minimum wage increase on the isocost line, holding the total cost of production (C1) constant. The pre-legislation wages of managers and frontline workers are denoted wM and wF, respectively. B. Explain why an increase in the minimum wage is likely to rotate the isocost line in the direction indicated. (What does the graph imply about the effect of the minimum wage increase on the wages of each type of labor?)
C. On both graphs, sketch the isocost line containing the new cost-minimizing bundle of managers and frontline workers, after the minimum wage increase, and label the new optimal bundle, assuming that restaurants continue to sell 100 meals per day. D. Assuming that fast-food restaurants continue to sell 100 meals per day, under which scenario does the minimum wage increase cause a larger reduction in the employment of frontline workers? (A or B)
E. Which economist would predict a larger increase in the total cost of producing 100 meals? (A or B)
F. If demand for fast food is fairly elastic, explain why managers at fast food restaurants might worry about losing their jobs as a result of the minimum wage increase. [Hint: Fast food restaurants might not continue to sell 100 meals per day.] Which economist’s estimate should cause them more concern? Explain briefly.
Name__________________________
Hours of Manager Labor per day (M)
C1/wM
M1
q=100 meals per day C1 F 1 7.25
C1 wF
Hours of Frontline Labor per day (F)
From Economist A’s Production Function Estimate
Hours of Manager Labor per day (M)
C1/wM
M1
q=100 meals per day C1 F 1
7.25
C1 . wF
Hours of Frontline Labor per day (F)
From Economist B’s Production Function Estimate
Name__________________________ 5. Suppose a firm employs labor as its only variable input in the short run. In the short run, it has a fixed cost of capital equal to $50. Labor costs $20 per day. Each unit of output can be sold at a competitive market price of $10. Output per day (q) for various amounts of labor (L=days of labor) is shown in the table below. (The firm cannot produce anything without at least one day of labor.) A. First, complete the table by calculating each level of labor usage (L)the (approximate) marginal product of labor (MPL), average product of labor (APL), total variable cost (VC), marginal cost (MC), average variable cost (AVC), average fixed cost (AFC), average total cost (AC), total cost (TC), total revenue, and profit. L
q
MPL
APL
VC
MC
AVC
AFC
AC
TC
Revenue
Profit = Revenue – TC
0.0 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0 12.0 14.0
0.0 4.0 10.0 20.0 30.0 36.0 40.0 46.0 50.0 52.0 52.0
-4.00 6.00 10.00 10.00 6.00 4.00
-4.00
-$20.00 $40.00
-$5.00 $3.33 $2.00 $2.00 $3.33
-$5.00
-$12.50 $5.00 $2.50 $1.67 $1.39 $1.25 $1.09
-$17.50 $9.00 $5.50 $4.33 $4.17 $4.25 $4.57
-$70 $90 $110 $130 $150 $170 $210
-$40 $100 $200
--$30 $10 $90
$0.96 $0.96
$5.58 $6.35
$290 $330
$360 $400 $460 $500 $520 $520
$210 $230 $250 $250 $230 $190
2.00 1.00 0.00
6.67 7.50 7.20 6.67 5.75 5.00 4.33 3.71
$80.00 $100.00 $120.00 $160.00 $200.00 $240.00 $280.00
mc
$6.67 $10.00 $20.00 ∞
$3.00 $2.67 $2.78 $3.00 $3.48 $4.00 $4.62 $5.38 avc
afc in the graph below. (Use the boxesac B. Label the curves shown provided.)
$
20
15
p* = 10
5
0 0
20
40
60
q
Answer the following as precisely as possible, using the table and graph. C. Where (at approximately what quantity) does the MC curve cross the AVC curve? q = between____and_____ D. Where (at approximately what quantity) does the MC curve cross the AC curve? q = between____and______ E. At what quantity is MC equal to the market price? q = _______ What is AC at this level of q? AC = _______. Calculate profit at this level of q as profit = (p-AC)×q = _______ F. If the firm produces the quantity that maximizes profit, approximately how much will it produce, and at what profit? q = ___________ ; Profit = __________
Name__________________________ 6. Ned has a small lawn-mowing business, in which the two inputs, a lawn-mower and labor, are perfect complements. Each employee can mow a lawn in half an hour, using exactly ½ hour of labor, and using the mower for exactly ½ hour. Thus, the production function for mowing lawns is: q=min(2K,2L)
where q=lawns mowed, K=hours of lawn-mower use and L=hours of labor.
Ned has one lawn mower. Lawn-mowers cannot feasibly be rented by the hour, and due to safety regulations, they can only be ridden during the 18 hours of daylight per day. Thus, Ned’s lawn-mower capital is fixed in the short run at 18 hours per day (K=18 hours). The implicit rental rate of the lawn-mower is r=$4/day. Labor is variable and can be hired by the half hour. The wage rate is w=$10 per hour. A. What is the marginal product (MPL) of the 1st hour of labor per day? = the 10th hour ?= the 19th hour ?= B. What is the average product (APL) of the 1st hour of labor per day? = the 10th hour ?= the 19th hour ?= C. What is the marginal cost of the 1st lawn mowed per day? = the 10th lawn mowed ?= the 19th lawn mowed ?= the 37th lawn mowed ?= D. What is the average variable cost of the 1st lawn mowed per day? = the 10th lawn mowed ?= the 19th lawn mowed ?= the 37th lawn mowed ?=
E. Sketch the MC and AVC functions here. Label both axes appropriately.
F. Write an equation showing the total cost per day (C) as a function of lawns mowed (q):
