A firm in a competitive market has fixed costs of 20 and the following marginal costs for item #N: Item #

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1 Microeconom ics m odule 11: m arginal costs, value, and revenue (practice problem s) ** Exercise 11.1: Shutdown Price A firm in a competitive market has fixed costs of 0 and e following marginal costs for item #N: Item # Marginal cost A. W hat is e firm s shut-down price? B. W hat is e firm s total profit at its short-run shutdown price? Part A: The firm s shut-down price is e minimum average variable cost. The variable cost is e sum of e marginal costs of e items produced. For example, if ree items are produced, variable costs are = 18. Average variable costs are variable costs divided by e number of items produced. Item # Marginal cost Variable cost Avg variable cost The minimum average variable cost is 5 when 5 items are produced, so e shut down price is 5. Part B: At e shut down price, average variable costs equal e price of e good, so e firm s total profit is e negative of its fixed costs, or 0.

2 **Exercise 11.: Shutdown Price A firm in a competitive market has fixed costs of 0 and e following marginal costs for item #N: Item # Marginal cost A. W hat is e firm s shut-down price? B. W hat is e firm s total profit at its short-run shutdown price? Part A: The firm s shut-down price is e minimum average variable cost. The variable cost is e sum of e marginal costs of e items produced. For example, if ree items are produced, variable costs are = 16. Average variable costs are variable costs divided by e number of items produced. Item # Marginal cost Variable cost Avg variable cost The minimum average variable cost is 4.50 when 6 items are produced, so e shut down price is Part B: At e shut down price, average variable costs equal e price of e good, so e firm s total profit is e negative of its fixed costs, or 0.

3 ** Exercise 11.3: Slope of Marginal Cost Curve In e short run, as more output is produced, how does a manufacturing firm s marginal cost change? Solution 11.3: At very low output, m arginal cost m ay be high, since ere is too little output for e optim al ratio of labor to capital. In e short run, as quantity increases, e firm runs its factories at greater an capacity, using overtime to produce more goods. The higher cost of labor (overtime pay) and e increased cost from using a ratio of labor to capital at is higher an optimal causes marginal cost to rise. This pattern is true for many firms, ough it is clearest for manufacturers.

4 ** Exercise 11.4: Study Time The value of passing an exam is 30,000. The probability of passing (P) e exam depends on e hours e 0.5 candidate studies (S) as P = S / 30, wi a maximum of 100%. For example, if e candidate studies 400 hours, e chance of passing is 0 / 30 = 66.7% The cost of S hours of study is S. For example, e cost of 400 hours of study is 400 = 16,000. How many hours does e candidate study? Solution 11.4: W e derive marginal cost and benefit curves and we equate marginal cost to marginal benefit The marginal cost is 1.5 S = 3 S. The marginal benefit of anoer hour of study is 0.5 (30, / 30) S = 500 S. Equating marginal cost and marginal benefit gives S = 500 / 3 = hours. ** Exercise 11.5: Marginal cost, revenue, and profit The demand curve is P = 40 5Q, and e cost curve is Total Cost = 3Q + Q. The units are not divisible: e firm must produce an integral number of units (1,, 3, ). A. W hat is e marginal cost of producing e ird unit? B. W hat is e marginal revenue from selling e fif unit? C. How many units should e firm produce to maximize its profits? D. W hat is e maximum profit is firm can earn? Part A: The marginal cost of e ird unit is e cost of ree units minus e cost of two units:! Two units total cost: 3 + = 14! Three units total cost: = 30 The marginal cost of e ird unit = = 16. Part B: The marginal revenue from e N unit is e revenue from N units minus e revenue from N-1 units:! Total revenue from 5-1 units: Q P = 4 (40 5 4) = 4 0 = 80! Total revenue from 5 units: Q P = 5 (40 5 5) = 5 15 = 75 The marginal revenue from e fif unit is = $5. Parts C and D: W e form a table wi marginal cost, marginal revenue, and profit for each level of output: Units Total Revenue Total Cost Marginal Revenue Marginal Cost Profit To maximize its profits, e firm should produce two units, earning 46.

5 **Exercise 11.6: Marginal cost, m arginal revenue, and profit for continuous functions The demand curve is P = 37 5Q, and e cost curve is Total Cost = 3Q + Q. The units are in millions: e firm may produce 1 unit, units, and so for. Solve e problem by integration, assuming e units are perfectly divisible. A. W hat is e marginal cost of producing e ree million unit? B. W hat is e marginal revenue from selling e five million unit? C. How many units should e firm produce to maximize its profits? D. W hat is e maximum profit is firm can earn? Part A: The marginal cost of e N unit is e partial derivative of e total cost wi respect to quantity: TC/ Q = (3Q + Q)/ Q = 6Q + 1 = 19. Part B: The marginal revenue is e partial derivative of e total revenue wi respect to quantity:! Total revenue = P Q = (37 5Q) Q = 37Q 5Q! Marginal revenue = (37Q 5Q )/ Q = 37 10Q The marginal revenue at e five million unit is = $13. Part C: Marginal cost is 6Q + 1, and marginal revenue is 37 10Q. Setting em equal gives Part D: Profit is total revenue minus total cost: 6Q + 1 = 37 10Q 16Q = 36 Q = 36/16 =.5 million! Total revenue is.5 (37 5.5) = $57.94 million.! Total cost is = $17.44 million Profit is $57.94 million $17.44 million = $40.50 million