Skip to main content

Behavior and Choice: EC 102

Andrew Monaco

Exercises 5.9 Practice Problems

Subsection 5.9.1 Profit maximization in perfect competition

  1. Draw a residual demand curve for a firm with no market power. Underneath the residual demand curve, shade in two total revenue areas, and describe how the lack of market power influences a firm’s incentive to increase its output.
  2. On one graph, indicate marginal revenue and marginal cost (under the assumption of diminishing marginal product of labor.) Indicate the optimal quantity of output on this graph, and label it \(q^*\text{.}\)
  3. For a perfectly competitive firm, is marginal revenue equal to the price at every level of output? Explain.
  4. For a perfectly competitive firm, is marginal revenue equal to marginal cost at every level of output? Explain.
  5. Consider a firm who looks to choose a level of output \(q\) to maximize its profit in a perfectly competitive market.
    1. According to our model of marginal analysis, what condition (an equality) will be true at the firm’s profit-maximizing level of output, \(q^*\text{?}\)
    2. Complete the table below. If \(TFC = 100\text{,}\) calculate the firm’s \(TC\) and \(MC\text{.}\)
      output (\(q\)) \(TFC\) \(TVC\) \(TC\) \(MC\)
      0 0 —
      20 100
      40 240
      60 400
      80 600
    3. Carefully graph the firm’s marginal cost curve, with \(q\) on the x-axis and \(MC\) on the y-axis. Does this curve give you any information about whether this firm experiences diminishing marginal product of labor or eventually diminishing marginal product of labor? Explain in words.
    4. Since a perfectly competitive firm cannot influence the price in the market, \(P\text{,}\) it will earn exactly \(P\) dollars per units sold. Therefore, its marginal revenue is equal to the price: \(MR = P\) for a perfectly competitive firm. Use this - and the optimal decision making rule - to find the firm’s optimal quantity of output \(q^*\) when the price is 7.
    5. Does the optimal quantity of output change when the price increases to 8? Find \(q^{**}\text{.}\)
  6. Consider the table below, which shows the total revenue and total cost for a firm producing output \(q\text{.}\)
    output (\(q\)) \(TR\) \(MR\) \(TC\) \(MC\) \(\pi\)
    0 0 - 50 -
    10 200 100
    20 400 200
    30 600 350
    40 800 600
    50 1000 900
    1. Graph the total revenue and total cost curves on the same graph. Where is the optimal quantity of output to produce?
    2. Calculate the marginal revenue (\(MR\)) and marginal cost (\(MC\)) to studying.
    3. Use marginal analysis to determine the optimal quantity of output to produce.
    4. Graph the marginal revenue and marginal cost curves on the same graph. Where is the optimal quantity of output?
    5. Confirm your solution by completing the profit (\(\pi\)) column in the table.
  7. Consider a firm with a total cost curve of \(TC(q) = 20 + 2q^2\) and a total revenue curve of \(TR(q) = 12q\text{.}\)
    1. Graph the total revenue and total cost curves on the same graph.
    2. What is the firm’s marginal revenue?
    3. If the firm’s marginal cost is \(MC(q) = 4q\text{,}\) find the firm’s profit-maximizing choice of output, \(q^*\text{.}\) Illustrate on your graph from part a..
    4. If the price in the market increases to $16, how does this change your graph from part a.? Redraw the graph, and include both \(P = 12\) and \(P = 16\) scenarios on the same graph.
    5. If the price in the market increases to $16, how does this change \(q^*\text{?}\)
  8. Consider a firm with a marginal cost curve of \(MC(q) = 10q\) and a marginal revenue curve of \(MR = 60\text{.}\)
    1. Graph the marginal revenue and marginal cost curves on the same graph.
    2. What price does the firm receive for its product?
    3. Find the firm’s profit-maximizing choice of output, \(q^*\text{.}\) Illustrate on your graph from part a..
    4. If the price in the market increases to $80, how does this change your graph from part a.? Redraw the graph, and include both \(P = 60\) and \(P = 80\) scenarios on the same graph.
    5. How does the price increase change \(q^*\text{?}\) Recalculate and show on your graph from d..

Subsection 5.9.2 Market power and price markups

  1. How does MP3 provide an alternate way of defining a firm’s market power?
  2. Will a firm competing in a perfectly competitive market markup its price above its marginal cost when maximizing its profit? Explain using the concept of marginal revenue and the \(q^*\) profit-maximization condition.
  3. Will a firm competing in a perfectly competitive market markup its price above its marginal cost when maximizing its profit? Explain using a graph of marginal revenue and marginal cost.
  4. A perfectly competitive firm charges the same price for each of its units. Is its price markup equal to zero for every unit? Or only the last unit? Explain (using a graph is likely helpful!).
  5. Suppose a firm can charge a price of $30 while its marginal cost is $15.
    1. Does this firm have market power? Yes or no - and why?
    2. What is the firm’s price markup?
    3. What is the value of the firm’s Lerner Index?
  6. Suppose a firm can charge a price of $75 while its marginal cost is $75.
    1. Does this firm have market power? Yes or no - and why?
    2. What is the firm’s price markup?
    3. What is the value of the firm’s Lerner Index?
  7. Suppose a firm can charge a price of $10 while its marginal cost is $20.
    1. Does this firm have market power?
    2. What is the firm’s price markup?
    3. What is the value of the firm’s Lerner Index?
    4. Under what circumstances might a firm find themselves in this situation?

Subsection 5.9.3 Firm supply and market supply

  1. Consider a firm who looks to choose a level of output \(q\) to maximize its profit in a perfectly competitive market.
    1. According to our model of marginal analysis, what condition (an equality) will be true at the firm’s profit-maximizing level of output, \(q^*\text{?}\)
    2. Complete the table below. If \(TFC = 100\text{,}\) calculate the firm’s \(TC\) and \(MC\text{.}\)
      output (\(q\)) \(TFC\) \(TVC\) \(TC\) \(MC\)
      0 0 —
      20 100
      40 240
      60 400
      80 600
    3. Since a perfectly competitive firm cannot influence the price in the market, \(P\text{,}\) it will earn exactly \(P\) dollars per units sold. Therefore, its marginal revenue is equal to the price: \(MR = P\) for a perfectly competitive firm. Use this - and the optimal decision making rule - to find the firm’s optimal quantity of output \(q^*\) when the price is 7.
    4. Does the optimal quantity of output change when the price increases to 8? Find \(q^{**}\text{.}\)
    5. Plot these price-quantity combinations on your graph from c.). What curve does the marginal cost curve represent? Explain your answer in words.
  2. Consider a firm with a marginal cost curve of \(MC(q) = 10q\) and a marginal revenue curve of \(MR = 60\text{.}\)
    1. Graph the marginal revenue and marginal cost curves on the same graph.
    2. Find the firm’s profit-maximizing choice of output, \(q^*\text{.}\) Illustrate on your graph from part a..
    3. If the price in the market increases to $80, how does this change your graph from part a.? Redraw the graph, and include both \(P = 60\) and \(P = 80\) scenarios on the same graph.
    4. How does the price increase change \(q^*\text{?}\) Recalculate and show on your graph from d..
    5. What curve are you illustrating on this graph?
  3. What curve is a perfectly competitive firm’s marginal cost curve identical to?
  4. If a firm’s supply curve is \(Q_S = 10P\text{,}\) what is the firm’s inverse supply curve? Graph this curve.
  5. Is it common for a firm’s supply curve to begin at the origin? What is the significance of the supply curve beginning at the origin?
  6. Firm A has a supply curve of \(Q_A = 2P\text{,}\) and firm B has a supply curve of \(Q_B = 3P\text{.}\)
    1. How much does each firm supply to the market when the price is 15?
    2. Calculate the market supply curve, \(Q_S = Q_A + Q_B\text{.}\)
    3. How much is the quantity supplied in the market?
    4. Graph the two individual supply curves on one graph, and the market supply curve on a second graph.
  7. How does a change in the number of firms in a market shift a market supply curve? Explain this using a numerical example of supply functions.

Subsection 5.9.4 Supply and Demand - 3-PART FINAL EXAM QUESTION

  1. A consumer wants to choose an optimal consumption bundle of good \(x\) and good \(y\text{.}\)
    1. What two mathematical conditions should be satisfied at this optimal bundle? (Write the equations and explain.)
    2. Find the optimal bundle \((x^*, y^*)\) when \(I = 200\text{,}\) \(P_x = 20\text{,}\) \(P_y = 10\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    3. Find the optimal bundle \((x^*, y^*)\) when \(I = 200\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) \(MU_y = \frac{1}{y}\text{,}\) and where \(P_x\) and \(P_y\) are unknown prices.
    4. What familiar curve is your \(x^*\) function? Name the function, and draw this function on a graph with price on the vertical axis and units of \(x\) on the horizontal axis. Label at least 2 points on the graph.
    5. If the market has 40 identical consumers with identical levels of income and consumer preferences, how many units of \(x\) are demanded in the market? Give a mathematical function, and draw this function on a new graph. Make sure price is on the vertical axis and units of \(x\) is on the horizontal axis. Label at least 2 points on the graph.
  2. Consider a firm who competes in a perfectly competitive market to produce units of output \(q\text{.}\) The firm has a cost function \(TC(q) = 5q^2\text{,}\) which means the firm’s marginal cost function is \(MC = 10q\text{.}\)
    1. What mathematical condition should be satisfied at the firm’s profit-maximizing choice of output? (Write an equation and explain.)
    2. If the firm is able to sell for a price of \(P = 20\text{,}\) what is its profit-maximizing choice of output? What if \(P = 40\text{?}\)
    3. In general, if the firm is able to sell for a general price \(P\text{,}\) what is its profit-maximizing choice of output? Find a function for \(q^*\) that depends on \(P\text{.}\)
    4. What familiar curve is your \(q^*\) function? Name the function, and draw this function on a graph with price on the vertical axis and units of \(q\) on the horizontal axis. Label at least 2 points on the graph.
    5. If the market has 100 identical sellers with identical costs, how many units of \(q\) are supplied to the market? Give a mathematical function, and draw this function on a new graph. Make sure price is on the vertical axis and units of \(q\) is on the horizontal axis. Label at least 2 points on the graph.
  3. Let’s combine questions 1 and 2 to explore how equilibrium works and adjusts in markets. Let’s assume good \(x\) from question 1 and output \(q\) from question 2 are the same good! (Let’s call it \(q\) here to make the notation consistent.) Suppose the 40 consumers in question 1 (purchasing good \(x\)) now interact with the 100 sellers in question 2 (producing good \(q\)) to exchange goods in the market.
    1. Use your answers from 1e and 2e to calculate the equilibrium price and quantity in the market for the good. Show your work. Indicate the equilibrium price and quantity in a graph showing both curves!
    2. How many units does each individual consumer purchase at the equilibrium? How many units does each individual firm sell at the equilibrium? For questions c.) and d.), you’ll be asked to complete a comparative statics exercise. You’ll need to trace the impact of a change in parameter on individual choice and on the market equilibrium. The number of consumers (40) and the number of firms (100) is unchanged.
    3. Suppose each individual consumer (from question 1) receives an increase in income to 300. Assume nothing changes for firms. Calculate individual demand for good \(x\text{,}\) market demand for good \(x\text{,}\) and the new equilibrium price and quantity. (Hint: equilibrium price and quantity may not be round numbers!) Include two graphs: the first (1) should show a shift by indicating the individual demand function before and after the income change; the second (2) should show a shift in market demand and indicate the change in equilibrium price and quantity.
    4. Suppose each individual firm (from question 2) faces a higher wage rate of 10, making their TC function \(TC(q) = 10q^2\) and their marginal cost function \(MC = 20q\text{.}\) Assume nothing changes for consumers (income back to 200). Calculate individual supply for output \(q\text{,}\) market supply for \(q\text{,}\) and the new equilibrium price and quantity. (Hint: equilibrium price and quantity may not be round numbers!) Include two graphs: the first (1) should show a shift by indicating the individual supply function before and after the wage change; the second (2) should show a shift in market supply and indicate the change in equilibrium price and quantity.

Subsection 5.9.5 Market outcomes and competitive equilibrium

  1. Under what condition does a market experience a surplus?
  2. Under what condition does a market experience a shortage?
  3. Give the market-clearing condition on quantity supplied and quantity demanded.
  4. How does the equilibrium price relate to the market-clearing condition?
  5. Is every market outcome an equilibrium? Explain why or why not.
  6. Consider a perfectly competitive market with demand curve given by \(Q_D = 100 - 2P\) and supply curve given by \(Q_S = 50 + 3P\text{.}\)
  7. Consider a perfectly competitive market for milk (where 1Q = 100 gallons), with demand curve given by \(Q_D = 200 - 2P\) and supply curve given by \(Q_S = 2P\text{.}\)
    1. Find the equilibrium price and quantity of milk exchanged, using the supply function and demand function given above.
    2. Draw this equilibrium very carefully, and label each axis; equilibrium price and quantity.
    3. When economists say that this market is efficient, what does this mean? Use your graph to explain.
    4. Label consumer surplus and producer surplus in the market on your graph.
    5. Calculate the value for consumer surplus and producer surplus.
    1. At a price of 20, what is \(Q_D\text{?}\) What is \(Q_S\text{?}\) Does the market clear? If not, is there a shortage or a surplus? Illustrate this on a graph of supply and demand.
    2. At a price of 5, what is \(Q_D\text{?}\) What is \(Q_S\text{?}\) Does the market clear? If not, is there a shortage or a surplus? Illustrate this on a graph of supply and demand.
    3. At what price does the market clear? Solve for the equilibrium price and quantity, \(P^*\) and \(Q^*\text{.}\) Illustrate this on a graph of supply and demand.
  8. Consider equilibrium \((P^*, Q^*)\) in the market from the problem above, with demand curve given by \(Q_D = 100 - 2P\) and supply curve given by \(Q_S = 50 + 3P\text{.}\) Examine what happens when there is a supply increase such that the new supply curve in the market is given by \(Q_S' = 60 + 3P\text{.}\)
    1. Is the market still in equilibrium at the price \(P^*\text{?}\) Show this numerically and with a graph.
    2. If there is a new equilibrium with the new supply curve, call it \((P^{**}, Q^{**})\) and find it.
    3. What does this show about the impact of an increase in supply on equilibrium price and quantity?
  9. Even though market transactions are legally prohibited, what if human kidneys could be exchanged in a perfectly competitive market? Let the supply of kidneys in the market be given by \(Q_S = 80 + 4P\text{,}\) so even at a price of zero, 80 kidneys are supplied to the market. Demand for kidneys (representing patients in need of a transplant) is given by \(Q_D = 160 - 4P\text{.}\)
    1. Find the equilibrium price and quantity of kidneys exchanged if market exchanges are legal, using the supply function and demand function given above.
    2. Draw this equilibrium very carefully, and label each axis; equilibrium price and quantity; consumer surplus; producer surplus; and deadweight loss (if it exists).
    3. Calculate the value for consumer surplus, producer surplus, and deadweight loss.

Subsection 5.9.6 Dynamic stability

  1. What does it mean for equilibrium to be dynamically stable?
  2. In words, describe how the presence of a surplus or shortage leads to price adjustment in the direction of equilibrium.
  3. Discuss two factors which make it challenging for a market to arrive at market equilibrium.

Subsection 5.9.7 Market efficiency

  1. Describe in words how efficiency of market equilibrium is represented through the presence (or absence) of mutual beneficial transactions.
  2. Describe in words how efficiency of market equilibrium is measured via total surplus.
  3. Describe in words how efficiency of market equilibrium is represented through the presence (or absence) of deadweight loss.
PrevTopNext