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Behavior and Choice: EC 102

Andrew Monaco

Exercises 3.5 Practice Problems

Subsection 3.5.1 Production functions and marginal product

  1. What is the distinction between a fixed input and a variable input?
  2. How do economists distinguish between the long run and the short run?
  3. Find an item around you, and make a list of the inputs you think are used in the production function of this item. Which inputs do you think are fixed inputs? Variable inputs?
  4. Consider a service you recently received, and make a list of the inputs you think are used in the production function of this item. Which inputs do you think are fixed inputs? Variable inputs?
  5. Consider assumption P1 - positive productivity.
    1. Define this assumption in words.
    2. How does this assumption influence the shape of the production function?
    3. How can this assumption be expressed in terms of the marginal product of labor?
  6. Consider assumption P2 - diminishing marginal product of labor.
    1. Define this assumption in words.
    2. How does this assumption influence the shape of the production function?
    3. How can this assumption be expressed in terms of the marginal product of labor?
  7. Consider assumption P2B - eventually diminishing marginal product of labor.
    1. Define this assumption in words.
    2. How does this assumption influence the shape of the production function?
    3. How can this assumption be expressed in terms of the marginal product of labor?
  8. In your opinion, how realistic is assumption P1?
  9. In your opinion, how realistic is assumption P2?
  10. In your opinion, how realistic is assumption P2B?
  11. The table below gives short-run production data for a firm:
    labor (\(L\)) output (\(q\)) \(MP_L\)
    0 0 -
    1 40
    2 65
    3 80
    4 90
    1. Fill in the marginal product of labor column in the table.
    2. Graph this production function carefully, with labor on the x-axis and output on the y-axis.
    3. What assumption(s) hold for this production function?
  12. The table below gives short-run production data for a firm:
    labor (\(L\)) output (\(q\)) \(MP_L\)
    0 0 -
    10 50
    20 150
    30 230
    40 300
    1. Fill in the marginal product of labor column in the table.
    2. Graph this production function carefully, with labor on the x-axis and output on the y-axis.
    3. What assumption(s) hold for this production function?
  13. Consider a firm with a production function \(q = \sqrt{KL}\text{.}\) Suppose that in the short run, the level of capital is fixed at \(\bar{K} = 9\text{.}\)
    1. Write the short-run production function.
    2. How much output is produced when \(L = 0\text{?}\) \(L = 1\text{?}\) \(L = 2\text{?}\)
    3. Graph the short-run production function, with labor on the x-axis and output on the y-axis.
    4. What assumptions does the production function satisfy?

Subsection 3.5.2 Firm Cost

  1. Express a firm’s total cost with an equation that relates its total fixed cost and its total variable cost.
  2. Express a firm’s total cost with an equation that includes its price per unit of labor (wage) and its price per unit of capital (rental rate).
  3. Define a firm’s marginal cost. In words, what does a firm’s marginal cost measure?
  4. What is the relationship between production assumption P1 (positive productivity) and the shape of a firm’s total cost curve? What is the relationship between assumption P1 and a firm’s marginal cost?
  5. What is the relationship between production assumption P2 (diminishing marginal product of labor) and the shape of a firm’s total cost curve? What is the relationship between assumption P2 and a firm’s marginal cost?
  6. What is the relationship between production assumption P2B (eventually diminishing marginal product of labor) and the shape of a firm’s total cost curve? What is the relationship between assumption P2B and a firm’s marginal cost?
  7. The table below gives short-run production data for a firm. The firm pays a wage rate of \(w = 40\text{,}\) and uses 3 units of capital (\(\bar{K} = 3\)) at a rental rate of \(r = 30\text{.}\)
    labor (\(L\)) output (\(q\)) \(MP_L\) \(TFC\) \(TVC\) \(TC\) \(MC\)
    0 0 - -
    1 40
    2 65
    3 80
    4 90
    1. Fill in the marginal product of labor column in the table.
    2. Fill in the \(TFC\text{,}\) \(TVC\text{,}\) and \(TC\) columns in the table.
    3. Graph the \(TFC\text{,}\) \(TVC\text{,}\) and \(TC\) curves on the same graph. How do production assumptions shape the curves you graph? (Be specific! Which assumptions affect which curves and how?)
    4. Fill in the \(MC\) column in the table, and graph \(MC\) on its own graph with output on the x-axis and \(MC\) on the y-axis. How do production assumptions shape the \(MC\) curve? (Be specific! Which assumptions affect the curve and how?)
  8. The table below gives short-run production data for a firm. The firm pays a wage rate of \(w = 15\text{,}\) and uses 3 units of capital (\(\bar{K} = 5\)) at a rental rate of \(r = 40\text{.}\)
    labor (\(L\)) output (\(q\)) \(MP_L\) \(TFC\) \(TVC\) \(TC\) \(MC\)
    0 0 - -
    10 50
    20 150
    30 230
    40 300
    1. Fill in the marginal product of labor column in the table.
    2. Fill in the \(TFC\text{,}\) \(TVC\text{,}\) and \(TC\) columns in the table.
    3. Graph the \(TFC\text{,}\) \(TVC\text{,}\) and \(TC\) curves on the same graph. How do production assumptions shape the curves you graph? (Be specific! Which assumptions affect which curves and how?)
    4. Fill in the \(MC\) column in the table, and graph \(MC\) on its own graph with output on the x-axis and \(MC\) on the y-axis. How do production assumptions shape the \(MC\) curve? (Be specific! Which assumptions affect the curve and how?)
  9. Andy’s Apple Orchard is a local Washington business which produces apples in a perfectly competitive market. With information about Andy’s production of bushels of apples (Q), let’s help him decide how to maximize his profit. Andy’s production information is given below, where labor (L) is Andy’s only variable input, and output (Q) is the number of bushels produced by Andy’s business. Andy can hire each worker for \(\$48\)/day, and he has a fixed cost of \(\$64\text{.}\)
    labor (\(L\)) output (\(q\)) \(MP_L\) \(TFC\) \(TVC\) \(TC\) \(MC\)
    0 0 - -
    1 24
    2 40
    3 52
    4 60
    5 66
    1. Does Andy’s orchard experience a diminishing marginal product of labor? Illustrate this by calculating the marginal product of labor.
    2. Draw Andy’s production function, and interpret the property of diminishing marginal product of labor in words. Does this property seem like a reasonable assumption to make on production functions? Justify your response in words.
    3. Complete the table by calculating Andy’s TFC, TVC, TC, and MC at each level of output.
    4. Carefully graph the TFC, TVC, and TC on the same graph, with output on the x axis and cost on the y axis.
    5. What shape are the TVC and TC curves? What property of production drives the shape of these curves?

Subsection 3.5.3 Optimal decision making: marginal analysis

  1. Describe an activity for which total benefit is increasing but marginal benefit is decreasing. Why does this activity fit this description?
  2. Describe an activity for which total cost is increasing and marginal cost is increasing. Why does this activity fit this description?
  3. Describe in words the cost-benefit principle of decision making by marginal analysis.
  4. What is the concept of net benefit? Describe in words how it can be used to determine an optimal decision?
  5. Consider the table below, which shows the total benefit and total cost to studying for an exam.
    minutes studying (\(x\)) \(TB\) \(MB\) \(TC\) \(MC\)
    0 0 - 0 -
    60 120 20
    120 220 60
    180 300 120
    240 360 200
    300 400 360
    1. The total benefit to studying keeps increasing as you study more. Does this mean you should always study the greatest number of minutes for an exam? Explain why or why not.
    2. At 120 minutes of studying, the total benefit is larger than the total cost. Is this the optimal number of minutes to study? Explain.
    3. Graph the total benefit and total cost curves on the same graph. Where is the optimal quantity of minutes to study?
    4. Calculate the marginal benefit (\(MB\)) and marginal cost (\(MC\)) to studying. Determine the optimal quantity of minutes to study.
    5. Graph the marginal benefit and marginal cost curves on the same graph. Where is the optimal quantity of minutes to study?

Subsection 3.5.4 Profit maximization

  1. Give the equation which defines a firm’s profit, \(\pi\text{.}\)
  2. If a firm’s total cost exceeds its total revenue, what happens to the profit earned by the firm?
  3. If a firm’s total cost equals its total revenue, what happens to the profit earned by the firm?
  4. Define a firm’s marginal revenue, \(MR\text{.}\)
  5. What is a firm’s marginal profit? How does marginal profit relate to optimal decision making for the firm?
  6. Consider a firm whose total revenue is \(TR = 20q\text{.}\)
    1. Draw the firm’s total revenue function.
    2. Use a table to calculate total revenue when \(q = 0\text{,}\) \(q = 1\text{,}\) \(q = 2\text{,}\) \(q = 3\text{,}\) and \(q = 4\text{.}\)
    3. On this same table, calculate the marginal revenue when \(q = 1\text{,}\) \(q = 2\text{,}\) \(q = 3\text{,}\) and \(q = 4\text{.}\)
    4. Draw the firm’s marginal revenue curve on a separate graph, with output \(q\) on the x-axis, and \(MR\) on the y-axis.
  7. Consider a firm whose total revenue is \(TR = 12q - q^2\text{.}\)
    1. Draw the firm’s total revenue function.
    2. Use a table to calculate total revenue when \(q = 0\text{,}\) \(q = 4\text{,}\) \(q = 8\text{,}\) \(q = 12\text{,}\) and \(q = 16\text{.}\)
    3. On this same table, calculate the marginal revenue when \(q = 4\text{,}\) \(q = 8\text{,}\) \(q = 12\text{,}\) and \(q = 16\text{.}\)
    4. Draw the firm’s marginal revenue curve on a separate graph, with output \(q\) on the x-axis, and \(MR\) on the y-axis.
  8. Consider the table below, which shows the total revenue and total cost for a firm producing output \(q\text{.}\)
    output (\(q\)) \(TR\) \(MR\) \(TC\) \(MC\)
    0 0 - 50 -
    10 200 100
    20 400 200
    30 600 350
    40 800 600
    50 1000 900
    1. The firm’s total revenue keeps increasing as more output is produced. Does this mean the firm should produce as much quantity as possible? Explain why or why not.
    2. Why would the firm’s total cost be greater than zero when it produces zero output?
    3. Graph the total revenue and total cost curves on the same graph. Where is the optimal quantity of output to produce?
    4. Calculate the marginal revenue (\(MR\)) and marginal cost (\(MC\)) to studying.
    5. What does the marginal cost tell us about the firm’s production function?
    6. Determine the optimal quantity of output to produce.
    7. Graph the marginal revenue and marginal cost curves on the same graph. Where is the optimal quantity of output?
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