Chapter 6 Problems

1.  Tickle Me Billy Inc., a manufacturer of presidential "look-a-like" dolls, has the following production function:

Q = 20L - L2 + 12K - .5K2

Additionally, the company’s input costs are PL = $2 and PK = $4.
 

a.  What is the company’s marginal product of labor (MPL) and marginal product of capital (MPK)?

MPL = Q/ L = 20 - 2L

MPK = Q/ K = 12 - K
 

b.  Does the company’s production function exhibit diminishing returns to each input?

Yes. Both marginal products decline, and as a result, returns are diminished.
 

c.  In the range 0 to 10 units, if both inputs can be purchased at the same price per unit, will production be relatively labor intensive or capital intensive?

Labor intensive. For L = K , the marginal product of labor is greater than the marginal product of capital. Thus, at the same input prices, the company will use more labor than capital.
 

d.  Using the company’s given input costs (PL = $2 and PK = $4), what is the optimal combination of capital and labor?

MPL/PL = MPK/PK

(20 - 2L)/2 = (12 - K)/4

10- L = 3 - .25K

10 + .25K = 3 + L

L = .25K + 7
 

e.  What input mix satifies the optimal combination of capital and labor?

L = .25K + 7

Set L = 8, thus:

8 = .25K + 7

.25K = 1

K = 4

 

f.  What is the resulting output using this input mix?

When L = 8 and K = 4,

Q = 20(8) - (8)2 + 12(4) - .5(4)2

Q = 160 - 64 + 48 - 8

Q = 136

 

g.  What is the company’s total input cost using this mix?

When L = 8 and K = 4,

TC = ($2)(8) + ($4)(4)

TC = $16 + $16

TC = $32

Conclusion:  The minimum cost of producing 136 dolls is $32 using 8 units of labor and 4 units of capital.

 

 

2. Manager Bob at Bob’s Better Beanies and More figures he wants to buy between 10 and 25 new patterns for his Bobby Beanie line of products. PL=$10, PK=$30

a. Using the production function Q=20L0.8 K0.2, if Bob’s goal is produce 1000 Beanies, how many workers can Bob hire if he buys 10, 15, or 25 patterns?

1000=20L0.8 K0.2

L=[(50)/(K.2)]1.25

Using this equation, plug in the values of K to find L: K L

(K,L): 10,75; 15,68; 25,60
 

b.Which combination of labor and capital is the most cost efficient using the values of K and L from question 1?

 

TC=PL + PKK

at K=10, TC=10(75)+30(10)= $1050 **most cost efficient

at K=15, TC=10(68)+30(15)= $1130

at K=25, TC=10(60)+30(25)= $1350

 

 c. Graph the production isocost and isoquant lines if Bob’s budget =$2000 and Q=1000.

TC=PLL + PKK

K=(TC/PK)-( PL/PK)L

K=(2000/30)-(10/30)L

K=66.67-.33L

 

d.Find MRTSLK

 

MRTSLK=( Q/ L)/( Q/ K)= [.8(20)L-0.2K0.2]/[.2(20)L0.8K-.8]= 4K/L

 

e.What is the lowest total cost Bob can have if he makes 1000 Bobby Beanies?

 

Set the slope of the isoquant (MRTSLK) to the slope of the isocost. (PL/PK).

MRTSLK =PL /PK

4K/L=.33

L=12.12K

Q=20L0.8 K0.2

1000=20(12.12K).8 K.2

K=6.79

L= 12.12K= 82.35

PL =10(82.35)= $823.50

PK =30(6.79)= $203.70

TC=203.70+823.50= $1027.20

 f.What is the maximum number of Bobby Beanies Bob can produce with a budget of $2000?

TC=PLL + PKK

2000=10(4K)+30K

K=28.57

L=4K= 114.29

Q=20(114.29)0.8(28.57)0.2

Q=1732 Bobby Beanies

 

g.Because of the unusually hot market for this fad item, will Bob follow a cost minimization or an output maximization method to produce his Bobby Beanies?

 

Because the demand for fad items is typically short-lived, Bob will most likely follow the output maximization method in an attempt to meet the current market demand for Bobby Beanies.

 

3. Carpenter’s Knock-Off Golf manufacturers exact replications of the most popular name brand golf clubs. The following equation is their production function:

 
10L.4K.6

Where K represents inputs of capital (including robotic machines), and L represents inputs of labor. Using this Production Function, answer the following questions.

 a.Does Carpenter’s Production Function represent increasing, decreasing, or constant returns to scale?

 Constant Returns To Scale
 

b. Determine Carpenter’s equations for MPL & APL . (K=500)

 

 MPL = Q/ L

= 4K.6/L.6

= 166.51/L.6

 

APL = Q/L

= 10K.6/L.6

= 10(41.63)/L.6

= 416.27/L.6

 
c. Is Carpenter’s Productions Function consistent with the Law of Diminishing Returns (K=500)?

 

Yes, based on the following table MPL decreases as the input of labor increases.

 
 

L

Q

APL

MPL(D Q/D L)

¶ Q/¶ L

0

0

0

0

-

1

416.27

416.27

416.27

166.51

2

549.27

274.00

133.00

109.90

3

645.99

215.33

96.72

86.00

 

d. Fixing K at 500, and variable amounts of labor, determine the optimal amounts of labor, if labor costs $20/unit and price for the golf clubs is $50.

 

 MRPL = MCL; MRPL= MPL(P) and MCL= $20

 

166.51/L.6 (50) = $20

8326/L.6 = $20

20L.6 = 8326

L.6 = 416

L = 23,187

 
e.Find the MRTSLK, when K=500 and L=50.

 

 MRTSLK = MPL/MPK

 

MPL = 4K.6L-.6, MPK = 6L.4K-.4

 

= 4K.6L-.6/6L.4K-.4

= .6667K/L

= .6667(500/50)

= 6.667

 

 f. Determine the slope of the budget line when the budget is set at $5,000, PL = $20/hr, PK = $10/hr.

 TC = (PLL) + (PKK)

K = TC/PK – PL/PK(L)

K = 5000/10 – (20/10)L

K = 500 – 2L

Slope = -2

 

-If the budget is increased to $6,000, what will be the new slope?

 

Answer: K = 6000/10 – (20/10)L

K = 600 –2L

Slope = -2

Slope does not change with a budget increase, the line shifts to the right. 

 

g.If the budget is set at $ 5,000, what is Carpenter’s maximum level of output? (PL = 20, PK = 10)

 

Answer: MPL/MPK = PL/PK

 

.667(K/L) = 20/10

K/L = 3

3L = K

L = .333K

TC = PLL + PKK

5000= 20(.333K) + 10(K)

5000= 6.666K +10K

16.66K= 5000

K= 300

L= 100

Q = 10 L.4K.6

Q = 10(100).4 (300).6

Q= 1,933

 
 
h.If Carpenter’s decides to limit the output of the golf clubs to 1200 units, what is their total costs?

Q = 10L.4K.6

1200 = 10(.333K).4K.6

1200 = 10 *.333.4 K.4K.6

120 = .644K

K = 186, L = 62

TC = (PLL) + (PKK)

TC = 20(62) + 10 (186)

TC = 1240 + 1860

TC = $3,100

 

i. What is Carpenter’s likely to do if the cost of labor is increasing compared to the cost of a new high-tech ultra efficient robot that has just been purchased?

They would substitute more capital for labor.

 

4. The Production Function for the Midget Widget Manufacturing Company is: Q = 50 L .3 K .4

 

a. What is the Marginal Product of Labor equation for Midget Widget?

MPL = .3 (50) L .3-1 K .4 = 15 L -.7 K .4

 

b. If K is fixed at 32, what is the Average Product of Labor?

APL = Q/L = 50 L -.7 (32) .4 = 200 L -.7

 

c. If the Price of a unit of output is $20, what is the Marginal Revenue Product of Labor?

 

MRPL = MPL * P = 15 L -.7 (20) .4 * $20 = 994.34 L -.7

 

d. If the Cost of a unit of labor is $100, what would be the optimum amount of labor and output?

 

Let MRPL = MCL;

 

994.34 L -.7 = 100

L -.7 = .100

L = 26.6

Q = 50 (26.6).3 * (20) .4 = 443.45

 

e. What type of Returns to Scale does Midget Widget exhibit?

 

Adding the exponents, .3 + .4 = .7 < 1.0

Therefore, it exhibits decreasing returns to scale

 

f. Describe the firm's long-run average cost curve (related to Chap. 7).

 

With (substantial) decreasing returns to scale, the firm experiences (substantial) diseconomies of scale.

 

4. A wheel manufacturer produces a single product but has the capability of using 2 different plants. Total demand for the product is equal to 10,000 units. The corporation uses aluminum to manufacture the wheels

key equations

Q=Q(a)+Q(b)

M(a)+M(b)=10,000

~2=denotes square

 

Production Functions

Plant A Q(a)=15-m(a)-.5M~2(a)

Plant B Q(b)=11M(b)-M~2(b)

 

a. What is the marginal product for wheel output? -in thousands of units

Plant A MP(a)=15-M(a)

Plant B MP(b)=11-2M(b)

 

b. Should they each produce 5,000 units? NO-They are not optimizing marginal products.

 

c. Draw the declining marginal product curve for each plant.

-see attached

 

d. Solve algebraically for optimal output at each plant

 

M(a)+M(b)=10

15-M(a)=11-2M(b)

M(a)=10-M(b)

M(b)=2

M(a)=8

 

Plant A should produce 8 thousand units and Plant B should produce 2,000.

 

 7. Suppose a firm has the following production function:

 

Q = 20L - L2 + 48K - 2K2

 

a) Assuming capital is constant with a value of 10 units, what is the

marginal product of labor?

 

MPL = 20 - 2L

 

b) With this scenario, what would be the greatest total output the firm

can achieve?

 

Set MPL=0 and solve: L=10 and Q=380

 

c) The firm can get $3 for each additional unit sold. Suppose it costs

the firm $15/hour for each additional worker. What is the optimum

amount of labor ?

 

Set MRPL=MCL. L=7.5

 

d) What is the firm's operating profit?

 

P = 1009.50

 

 

e) Describe the Returns to Scale for this problem.

 

With L=5 and K=5, Q=265; with L=10 and K=10, Q=380.

Thus a doubling of each input does not double the output. Th production function has decreasing returns to scale (over this range of inputs).

 

8. A dirty Politician has just received word that the police are on their way to arrest him. Therefore, he must destroy the files that will be used as evidence in a trial. The politician has two shredding machines at his disposal. The relationship between the amount of files to be destroyed and the amount of time it will take the shredders is as follows:

 

D1=14+36T1-0.5T12 and D2=10+120T2-3T22

Where, D=destroyed files and T=time (in minutes)

 

a. Determine the maximum number of files that can be destroyed with each machine (before it breaks down).

Set the 1st derivatives equal to zero and solve: D1 = 662 and D2 = 1210. 

b. If the politician has 500 files to destroy the files, determine the minimum time it will take to shred them?

 To minimize time using both machines, the work should stop at the same time; otherwise there is a misallocation. Thus let T1=T2=T and set D1+D2=500. Solving a quadratic for T results in 2 solutions (41.3 and 3.3). Select the smaller value of 3.3 minutes and allocate about 128 files to machine 1 and 372 to machine 2.