"Output as a function of a single, labor

The short-run production function of a competitive firm is given by f(L)=6L⅔, where L is the amount of labor it uses. The cost per unit of labor is w=6 and the price per unit of output is p=3.

a. Plot this firm's production function for labor inputs of 0,1,8,and 27 using orange points (square symbol). Line segments will automatically connect the points. Next, using blue lines (circle symbols), position the isoprofit line that passes through the point (0,12),the isoprofit that passes through (0,8), and the isoprofit line that passes through the point (0,4). What is the slope of each of the isoprofit lines?

Which answers would u choose?

1. They all have a slope of 2.

2. They all have a slope of 1/4.

3. They all have a slope of 1.

4. They all have a slope of 1/3.

5. They all have a slope of 4.

6. They all have a slope of 3.

7. They all have a slope of 1/2.

How many points on the isoprofit line(0,12) consist of input-output points that are actually possible?

Which answers would u choose?

1. 1

2. 3

3. 4

4. 5

5. None

6. 6

7. 2

Which parts of the isoprofit line through (0,4) consist that are actually possible? Which answers would u choose?

1. All points on the line above the production function

2. All points on the line with output greater than 4 only

3. Noone

4. All points on the line

5. All points on the line with output less than 4 only

6. All points on the line with output less than 8 only

b. How many units of labor will the firm hire? Which answers would u choose?

1. 6

2. 8

3. 12

4. 24

5. 3

6. 48

7. 72

How much output will it produce? Which answers would u choose?

1. 24

2. 8

3. 12

4. 72

5. 3

6. 48

If the firm has no other costs, how much will its total profits be? Which answers would u choose?

1. 24

2. 48

3. 8

4. 6

5. 72

6. 3

7. 12

c. Suppose that the wage of labor falls to 4, and the price of output remains at p. On the graph, place a purple line(diamond symbols) to show the new isoprofit line for the firm that passes through its old choice of input and output. Will the firm increase its output at the new price?

Which answers would you choose?

1. No

2. Yes

Explain why, referring to your diagram.

Which answers would you choose?

1. As the graph shows, the firm can reach a lower isoprofit line by increasing output.

2. As the graph shows, the firm can reach a higher isocost line by decreasing output.

3. As the graph shows, the firm can reach a higher isocost line by increasing output.

4. As the graph shows, the firm can reach a lower isoprofit line by decreasing output.

5. As the graph shows, the firm can reach a lower isocost line by decreasing output.

6. As the graph shows, the firm can reach a higher isoprofit by decreasing output.

Brother Jed and the Heathens

Brother Jed takes heathens and performs them into righteous individuals. There are two inputs needed in this process: heathens (who are widely available) and preaching. The production function has the following form: r subscriptof p=min{h,p}, where r subscript ofp is the number of righteous persons produced, h is the number of heathens who attend Jed’s sermons, p is the number of hours of preaching. For every person converted, Jed receives a payment of s from the grateful convert. Sad to say, heathens do not flock to Jed’s sermons of their own accord. Jed must offer heathers a payment of w to attract them to his sermons. Suppose the amount of preaching is fixed at p and that Jed is a profit-maximizing prophet.

a. If h<p, what is the marginal product of heathens? Which answers would u choose?

• S

• P

• 1

• W

• 0

• P

What is the value of the marginal product of an additional heathen?

• 0

• P

• S

• 1

• H

• P

• w

b. If h>p, what is the marginal product of heathens?

• S

• 0

• 1

• W

• P

• H

• P

What is the value of the marginal product of an additional heathen in this case?

• S

• 0

• P

• W

• P

• 1

C. Use blue points (circle symbol) to plot this production function in the graph below. Line segments will automatically connect the points. Assume just for this graph that p = 20.

d. If w < s, how many heathens will be converted?

• P

• S

• 0

• 1

• H

• W

• P

If w>s, how many heathens will be converted?

• H

• P

• S

• 0

• 1

• P

• W

1. If the value of the marginal product of labor exceeds the wage rate, then a competitive, profit-maximizing firm would want to hire less labor. Is this statement TRUE or FALSE? Justify the choice that you make?

2. For each of the following production functions, comment on the ability to substitute capital for labor, i.e. having identified the type of production function in each case, comment on the ability of the firm to substitute capital for labor.

a. Y=K+L

b. Y=Ksubscript of 0.5 Lsubscript of 0.5

c. Y=min(K,L)

3. For each of the following production functions, draw a diagram showing the general shape of its corresponding isoquant. Comment on the ease at which labor and capital can be submitted for one another relative to the other two production functions.

a. Y=K+L

b. Y=Ksubscript of 0.5 Lsubscript of 0.5

c. Y=min(K,L)

4. A competitive firm has a production function described as follows. “Weekly out put is the square root of the minimum of the number of units of capital and the number of units of labor employed per week.” Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely.

a. Write down a formula that describes the marginal product of labor in the short run as a function of the amount of the labor used.

MP=1/(2Lsubscript of1/2) if L<16, MP =0 if L>16.

b. If the wage is w=$1 and the price of output is p=$4, how much labor will the firm demand in the short run? Hint: Solve for L in p*MP=w using the given values for p,w and MP.

c. What if w=$1 and p=$10?

d. Write down an equation for the firm’s short-run demand for labor(L) as a function of w and p.

"

The short-run production function of a competitive firm is given by f(L)=6L⅔, where L is the amount of labor it uses. The cost per unit of labor is w=6 and the price per unit of output is p=3.

a. Plot this firm's production function for labor inputs of 0,1,8,and 27 using orange points (square symbol). Line segments will automatically connect the points. Next, using blue lines (circle symbols), position the isoprofit line that passes through the point (0,12),the isoprofit that passes through (0,8), and the isoprofit line that passes through the point (0,4). What is the slope of each of the isoprofit lines?

Which answers would u choose?

1. They all have a slope of 2.

2. They all have a slope of 1/4.

3. They all have a slope of 1.

4. They all have a slope of 1/3.

5. They all have a slope of 4.

6. They all have a slope of 3.

7. They all have a slope of 1/2.

How many points on the isoprofit line(0,12) consist of input-output points that are actually possible?

Which answers would u choose?

1. 1

2. 3

3. 4

4. 5

5. None

6. 6

7. 2

Which parts of the isoprofit line through (0,4) consist that are actually possible? Which answers would u choose?

1. All points on the line above the production function

2. All points on the line with output greater than 4 only

3. Noone

4. All points on the line

5. All points on the line with output less than 4 only

6. All points on the line with output less than 8 only

b. How many units of labor will the firm hire? Which answers would u choose?

1. 6

2. 8

3. 12

4. 24

5. 3

6. 48

7. 72

How much output will it produce? Which answers would u choose?

1. 24

2. 8

3. 12

4. 72

5. 3

6. 48

If the firm has no other costs, how much will its total profits be? Which answers would u choose?

1. 24

2. 48

3. 8

4. 6

5. 72

6. 3

7. 12

c. Suppose that the wage of labor falls to 4, and the price of output remains at p. On the graph, place a purple line(diamond symbols) to show the new isoprofit line for the firm that passes through its old choice of input and output. Will the firm increase its output at the new price?

Which answers would you choose?

1. No

2. Yes

Explain why, referring to your diagram.

Which answers would you choose?

1. As the graph shows, the firm can reach a lower isoprofit line by increasing output.

2. As the graph shows, the firm can reach a higher isocost line by decreasing output.

3. As the graph shows, the firm can reach a higher isocost line by increasing output.

4. As the graph shows, the firm can reach a lower isoprofit line by decreasing output.

5. As the graph shows, the firm can reach a lower isocost line by decreasing output.

6. As the graph shows, the firm can reach a higher isoprofit by decreasing output.

Brother Jed and the Heathens

Brother Jed takes heathens and performs them into righteous individuals. There are two inputs needed in this process: heathens (who are widely available) and preaching. The production function has the following form: r subscriptof p=min{h,p}, where r subscript ofp is the number of righteous persons produced, h is the number of heathens who attend Jed’s sermons, p is the number of hours of preaching. For every person converted, Jed receives a payment of s from the grateful convert. Sad to say, heathens do not flock to Jed’s sermons of their own accord. Jed must offer heathers a payment of w to attract them to his sermons. Suppose the amount of preaching is fixed at p and that Jed is a profit-maximizing prophet.

a. If h<p, what is the marginal product of heathens? Which answers would u choose?

• S

• P

• 1

• W

• 0

• P

What is the value of the marginal product of an additional heathen?

• 0

• P

• S

• 1

• H

• P

• w

b. If h>p, what is the marginal product of heathens?

• S

• 0

• 1

• W

• P

• H

• P

What is the value of the marginal product of an additional heathen in this case?

• S

• 0

• P

• W

• P

• 1

C. Use blue points (circle symbol) to plot this production function in the graph below. Line segments will automatically connect the points. Assume just for this graph that p = 20.

d. If w < s, how many heathens will be converted?

• P

• S

• 0

• 1

• H

• W

• P

If w>s, how many heathens will be converted?

• H

• P

• S

• 0

• 1

• P

• W

1. If the value of the marginal product of labor exceeds the wage rate, then a competitive, profit-maximizing firm would want to hire less labor. Is this statement TRUE or FALSE? Justify the choice that you make?

2. For each of the following production functions, comment on the ability to substitute capital for labor, i.e. having identified the type of production function in each case, comment on the ability of the firm to substitute capital for labor.

a. Y=K+L

b. Y=Ksubscript of 0.5 Lsubscript of 0.5

c. Y=min(K,L)

3. For each of the following production functions, draw a diagram showing the general shape of its corresponding isoquant. Comment on the ease at which labor and capital can be submitted for one another relative to the other two production functions.

a. Y=K+L

b. Y=Ksubscript of 0.5 Lsubscript of 0.5

c. Y=min(K,L)

4. A competitive firm has a production function described as follows. “Weekly out put is the square root of the minimum of the number of units of capital and the number of units of labor employed per week.” Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely.

a. Write down a formula that describes the marginal product of labor in the short run as a function of the amount of the labor used.

MP=1/(2Lsubscript of1/2) if L<16, MP =0 if L>16.

b. If the wage is w=$1 and the price of output is p=$4, how much labor will the firm demand in the short run? Hint: Solve for L in p*MP=w using the given values for p,w and MP.

c. What if w=$1 and p=$10?

d. Write down an equation for the firm’s short-run demand for labor(L) as a function of w and p.

"

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