Chapter 7. Production and Cost: Many Variable Inputs

1. Textbook problems # 6, 11, 12, 13, 14

2. True/ False

1. An LAC(y) curve is U-shaped because input prices are decreasing for low output levels and increasing for high output levels.

2. If a production technology exhibits IRTS, then a 10% increase in output will result in less than 10% increase in the total long-run costs of production.

3. If an equal percentage increase in the use of all inputs results in a smaller percentage increase in the quantity produced, a firm's production function is said to exhibit decreasing returns to scale.

4. To minimize the cost of producing a given amount of output, the marginal products of all inputs must be equal.

5. If a production process uses the two inputs of labor and capital, then an increase in the wage rate will cause a firm to increase its use of capital, with output held constant.

6. For strictly convex isoquants, an increase in the wage rate results in an increase in the use of capital, holding total costs constant.

7. The isoquants of a production technology characterized by fixed proportions are straight lines.

8. Any input bundle on an isocost curve yields a higher output than any point below the isocost curve.

9. The cost of producing a given level of output falls when the price of an inferior factor of production increases.

Answers: 1.F    2.T    3.T    4.F    5.T    6.F    7.F    8.F    9.F

3. Short Questions

1. Natural Farms Ltd. produces yogurt using milk and bacteria culture in fixed proportions. It takes one pint of milk and one ounce of bacteria culture to make a pint of yogurt. The price of milk is $0.50 per pint, and bacteria culture costs $0.10 an ounce. Draw one or two isoquants for Natural Farms Ltd. How much does it cost to produce one pint of yogurt? Two pints? Three pints? What is Natural Farms' cost function?

2. State whether the following production functions exhibit decreasing returns to scale, constant returns to scale, or increasing returns to scale:
a). f(x₁, x₂) = 5x₁^1/4x₂^1/4
b). F(K,L) = (K+L)²
c). f(x₁,x₂) = 2(x₁ + x₂)
Give an explanation for your answer.

3. For the production functions in the previous question calculate the marginal product of x₁ (or MP_K for part b) and the technical rate of substitution. Graph one or two isoquants for each production function (To check that you have the right answer remember: what is the relationship between the technical rate of substitution and the slope of the isoquant?)

Answers:  1. $0.60;$1.2;$1.8; LTC(y)=0.6y.
                2. a)DRTS; b)IRTS; c) CRTS
 

4. Long Questions

4. Consider the Cobb-Douglas production function: f(x₁, x₂) = A x₁^ax₂^b.
For the following sets of parameter values, does the function exhibit constant, increasing, or decreasing returns to scale?
a). A=1, a=1/3, b=2/3
b). A=2, a=1/3, b=2/3
c). A=1, a=1/4, b=1/4
d). A=a=b=1

5. Blooming Blossoms produces flowers using two inputs, seeds (measured in ounces - z₁) and fertilizer (measured in ounces - z₂). The production function for flowers is given by: y = 1000z₁^1/4z₂^1/4
a). Suppose initially that Blooming Blossoms only has only 16 ounces of Precious Petunia seeds. Calculate Blooming Blossoms' Total Product and Short Run Marginal Product functions.
b). Does the law of diminishing marginal product hold for Blooming Blossoms' total product function?
c). T/F/U. If the law of diminishing marginal product did not hold, Blooming Blossoms could grow the entire world's petunia supply from 16 ounces of seeds.
d). Now suppose that Blooming Blossom can buy as many Precious Petunia seeds as it needs. Graph one of Blooming Blossom's isoquants.
e). Calculate the solution to Blooming Blossom's cost minimization problem.

6. Dan Donaldson runs Draper Dan, a company that makes curtains. Curtain manufacturing uses cloth and labor in fixed proportions: it takes exactly five meters of cloth and 3 hours of labor to make 1 curtain. Derive Draper Dan's cost function (a) in terms of input prices and output and (b) when the price of cloth, w₁, is $3/metre and the wage rate w₂ is $10 per hour.

7. Herongate Horses produces saddles (y) using two inputs, leather (z₁) and labor (z₂). Herongate Horses's production function is given by: F(z₁, z₂) = (1/4)z₁z₂

a). Does Herongate Horses's production function exhibit constant, increasing, or decreasing returns to scale? Explain your answer fully.
b). Graph Herongate Horses's isoquant for y=4. Calculate the slope of the isoquant as a function of z₁ and z₂.
c). Calculate Herongate Horses's conditional input demands for z₁ and z₂ as a function of y, when w₁=5 and w₂=20.
d). Using your answer to (c), find Herongate Horses's total cost function. Graph the total cost function, TC(y). Find and graph the long run average cost function, LAC(y).
e). Suppose now that z₂ is fixed at 2. Find Herongate Horses's total product function, TP(z₁).
f). Continuing on from (e), find Herongate Horses's variable cost function, VC(y). Find and graph the average variable cost function, AVC(y).
g). Compare your answers from part (d) and part (f). Are the shape of the functions the same or different? Can you explain why?

8. A firm's production function is given by Q = (K+L)² where Q is output, and K and L are capital and labor inputs.

a). Does the production function exhibit constant, increasing, or decreasing returns to scale?
b). Draw isoquants for Q=4 and Q=9. What is the firm's marginal rate of technical substitution?
c). Suppose the price of capital, r, is $1 and the price of labor, w, is $2. How much capital and labor will the firm use to produce Q=4? Q=9?
d). What is the firm's cost function when r=$1 and w=$2?

9. Floppy Corp produces software using two inputs, large (5 1/4 inch) disks, L, and small (3 1/2 inch) disks, S. Its production function is given by: Q=(L/2) + S where Q is output.

a). Does the production function exhibit constant, increasing, or decreasing returns to scale?
b). Draw isoquants for Q=2 and Q=3. What is the firm's marginal rate of technical substitution?
c). Suppose small disks cost $3 each and large disks cost $1 each. How many of each type of disk will Floppy Corp use to produce Q=2? Q=3?
d). What is the firm's cost function when small disks cost $3 and large disks cost $1?

Answers:
4. a) CRTS; b) CRTS; c) DRTS; d) IRTS;
5. a) TP(z₂)=2000z₂^1/4; MP(z₂)=500z₂^(-3/4); b) yes; c) F; d)smooth and convex; e) In general, if y=Az₁^az₂^b then the conditional input demand functions are given by: z₁= (y/A) ^(1/(a+b))(aw₁/bw₂)^(b/(a+b)) and z₂=(y/A) ^(1/(a+b))(bw₁/aw₂)^(a/(a+b)) . In this case a=b=1/4 and A=1000.
6.a)  TC(y)=(5w₁+3w₂)y; b) TC(y)=45y where w₁ is the price of cloth and w₂ is the price of labor.
7. a) IRTS; b) slope=z₂/z₁; c) z₁=4y^1/2 and z₂=y^1/2 ; d) TC(y)=40y^1/2 LAC(y)=40/y^1/2 ; e) TP(z₁)=z₁/2 ; f) VC(y)=10y,  AVC(y)=10 ; g) different because LR allows for more flexibility (substitution in production).
8. a) IRTS; b) MRTS=1, isoquants are straight lines; c) K*=Q^1/2 ,L*=0, if Q=4 then K*=2 and L*=0, if Q=9 then K*=3 and L*=0; d) TC(Q) = rQ^1/2 .
9. a) CRTS; B)MRTS=1/2 (L on the horizontal axis); c) w_L/w_S=1/3 ; S*=0 and L*=2Q. If Q=2 then L*=4 and S*=0 and if Q=3 then L*=6 and S*=0; d) TC(Q)=2w_LQ.