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Profit-Maximizing Behavior in Factor Markets - Microeconomics

  1. A profit-maximizing firm uses two variable inputs, labor (L) and materials (M), in a perfectly competitive market. The production function exhibits constant returns to scale. If the firm doubles both inputs and finds that its total cost increases by exactly the same proportion as its total revenue, which of the following must be true about the firm's current input usage?
    Answer: The firm is operating at the minimum point of its long-run average cost curve and earning zero economic profit When you encounter a question combining production functions, cost behavior, and profit maximization, focus on the relationship between returns to scale and long-run cost curves. The key insight is that constant returns to scale creates a flat long-run average cost curve. Since the firm has constant returns to scale and operates in perfect competition, doubling inputs doubles output. The critical information is that total cost and total revenue increase by the same proportion when inputs double. This means the firm's profit remains unchanged when it scales up production. In perfect competition, this scenario only occurs when the firm operates at the minimum point of its long-run average cost curve, where price equals minimum long-run average cost. At this point, the firm earns zero economic profit—any deviation from this output level would either create losses or missed profit opportunities. Answer A incorrectly assumes constant returns to scale implies infinite optimal firm size. While the technology allows any scale, profit maximization still determines optimal output where marginal revenue equals marginal cost. Answer B is partially correct about cost minimization but misses that the equal proportional changes in cost and revenue indicate the firm is at the profit-maximizing output level. Answer D contradicts the given information about constant returns to scale—the described cost and revenue behavior is perfectly consistent with constant returns. Remember: In perfect competition with constant returns to scale, long-run equilibrium occurs where firms operate at minimum long-run average cost, earning zero economic profit. This is the only sustainable outcome in such markets.
  2. A perfectly competitive firm hires labor to produce good X. Which of the following scenarios best illustrates that the demand for labor is a derived demand?
    Answer: A major marketing campaign significantly increases consumer demand for good X, causing the firm to hire more workers. The concept of derived demand means that the demand for a factor of production (like labor) is derived from the demand for the final good or service it produces. The scenario in which an increase in consumer demand for the product (good X) leads directly to an increase in the firm's demand for the input (labor) is the best illustration of this core concept.
  3. A firm produces corn and hires farmhands in perfectly competitive markets. What is the most likely effect on the firm's demand for farmhands if a widespread drought simultaneously decreases the marginal product of farmhands and increases the market price of corn?
    Answer: The effect on the demand for farmhands is indeterminate. The demand for farmhands is their marginal revenue product (MRPL), calculated as MRPL = MPL * P. The drought causes the marginal product of labor (MPL) to decrease, which, by itself, would shift the MRPL curve to the left (decreasing demand). Simultaneously, the drought increases the market price of corn (P), which, by itself, would shift the MRPL curve to the right (increasing demand). Since these two effects work in opposite directions, the net effect on the demand for farmhands is indeterminate without knowing the relative magnitudes of the change in MPL and the change in P.
  4. A firm operating in perfectly competitive input and output markets has the production function Q=4L0.5K0.5Q = 4L^{0.5}K^{0.5}. The price of output is $8, the wage rate is $32, and the rental rate of capital is $18. If the firm is currently using 25 units of labor and 16 units of capital, which of the following statements is correct?
    Answer: The firm should decrease labor usage and increase capital usage to maximize profit in the short run To determine the optimal input usage, we need to check if the value of marginal products equals the input prices. The marginal products are: MPL = 2L−0.52L^{-0.5}K0.5K^{0.5} and MPK = 2L0.52L^{0.5}K−0.5K^{-0.5}. With L=25 and K=16: MPL = 2(25)−0.5(25)^{-0.5}(16)0.5(16)^{0.5} = 2(1/5)(4) = 1.6, so VMPL = 8 × 1.6 = $12.8. MPK = 2(25)0.5(25)^{0.5}(16)−0.5(16)^{-0.5} = 2(5)(1/4) = 2.5, so VMPK = 8 × 2.5 = $20. Since VMPL = $12.8 < w = $32, the firm is using too much labor. Since VMPK = $20 > r = $18, the firm is using too little capital. Therefore, the firm should decrease labor and increase capital usage.
  5. Consider two firms, A and B, operating in the same perfectly competitive output market but in different geographic labor markets. Firm A faces a wage of $20 and Firm B faces a wage of $30. Both firms have identical production functions Q=10L0.6Q = 10L^{0.6}. If the output price is $5 and both firms are profit maximizers, what is the ratio of Firm A's employment to Firm B's employment?
    Answer: (3020)2.5=4.22(\frac{30}{20})^{2.5} = 4.22 Each firm maximizes profit by setting VMPL = w. The marginal product of labor is MPL = 6L−0.46L^{-0.4}. For Firm A: 5 × 6L_A^(-0.4) = 20, so 30L_A^(-0.4) = 20, giving LA−0.4L_A^{-0.4} = 2/3, so LAL_A = (3/2)2.5(3/2)^{2.5}. For Firm B: 5 × 6L_B^(-0.4) = 30, so 30L_B^(-0.4) = 30, giving LB−0.4L_B^{-0.4} = 1, so LBL_B = 1. Therefore, the ratio is LAL_A/LBL_B = (3/2)2.5(3/2)^{2.5} = (30/20)2.5(30/20)^{2.5}, showing how the employment ratio relates to the wage ratio raised to the power 2.5.
  6. A firm is producing a target level of output using labor and capital. The wage rate is $15 per hour, and the rental rate of capital is $45 per hour. The marginal product of the last worker hired is 3 units of output. To produce its current output at the minimum possible cost, what must be the marginal product of the last unit of capital employed?
    Answer: 9 units of output The condition for least-cost production is that the ratio of marginal product to input price is equal for all inputs: MPL / w = MPK / r. We are given MPL = 3, w = $15, and r = $45. Plugging these values into the formula: 3 / 15 = MPK / 45. This simplifies to 1/5 = MPK / 45. Solving for MPK gives MPK = 45 / 5 = 9 units of output. A common mistake is to set MPL w = MPK r, which would yield MPK = 1.
  7. A firm uses labor and capital in a perfectly competitive market. The production function is Q=ALαKβQ = AL^{\alpha}K^{\beta} where α+β=1\alpha + \beta = 1. The firm faces output price P, wage rate w, and capital rental rate r. If the government imposes a payroll tax of t per unit of labor (paid by the employer), what happens to the firm's optimal capital-to-labor ratio in the long run?
    Answer: The capital-to-labor ratio increases by exactly the factor (w+t)/w, which equals the ratio of the new effective wage to the old wage With constant returns to scale (α + β = 1), the optimal input ratio for a Cobb-Douglas production function is K/L = (α/β)(w/r). After the payroll tax, the effective wage becomes (w+t), so the new optimal ratio is K'/L' = (α/β)((w+t)/r). The ratio of the new capital-to-labor ratio to the old one is [(α/β)((w+t)/r)]/[(α/β)(w/r)] = (w+t)/w. This result holds regardless of the specific values of α and β, as long as α + β = 1. Choice A is incorrect because it says 'regardless of production function parameters' but this result specifically requires the Cobb-Douglas form with constant returns. Choice B is incorrect because with Cobb-Douglas production, the elasticity of substitution is always 1, and the result doesn't depend on varying elasticity. Choice C is incorrect because the payroll tax only affects labor, not capital.
  8. A perfectly competitive firm is maximizing its profit. It hires labor at a wage of $20 per hour and sells its output at a price of $5 per unit. What must be the marginal product of the last worker the firm hired?
    Answer: 4 units of output A profit-maximizing firm in perfectly competitive markets will hire labor up to the point where the wage (w) equals the marginal revenue product of labor (MRPL). The formula for MRPL is the marginal product of labor (MPL) multiplied by the product price (P). So, w = MPL P. Plugging in the given values: $20 = MPL $5. To find the MPL, we rearrange the equation: MPL = $20 / $5 = 4 units of output.
  9. A firm's short-run production function is given by Q=60L−L2Q = 60L - L^2, where L is the number of workers and Q is total output. The firm sells its output in a perfectly competitive market for $2 per unit and hires workers in a perfectly competitive labor market for a wage of $40 per day. To maximize profit, how many workers should the firm hire?
    Answer: 20 The firm should hire workers until the marginal revenue product of labor (MRPL) equals the wage (w). 1. Find the marginal product of labor (MPL) by taking the derivative of the production function with respect to L: MPL = dQ/dL = 60 - 2L. 2. Calculate MRPL by multiplying MPL by the product price (P): MRPL = MPL P = (60 - 2L) $2 = 120 - 4L. 3. Set MRPL equal to the wage (w) and solve for L: 120 - 4L = $40. This simplifies to 80 = 4L, so L = 20. A common mistake is to set MPL = w (60 - 2L = 40), which would yield L = 10.
  10. Two firms, Innovate Corp. and Standard Inc., hire from the same perfectly competitive pool of engineers and have identical production technologies. Innovate Corp. sells its product in a perfectly competitive market, while Standard Inc. is a pure monopolist in its product market. Which statement correctly compares their hiring decisions?
    Answer: Innovate Corp. will hire more engineers than Standard Inc. because its marginal revenue equals the product price. Both firms hire where MRPL = wage. For Innovate Corp. (perfect competitor), MRPL = MPL P. For Standard Inc. (monopolist), MRPL = MPL MR. Since a monopolist must lower its price to sell more, its marginal revenue (MR) is always less than its price (P). Because both firms have the same MPL curve, but Standard Inc. multiplies it by a smaller value (MR < P), its MRPL curve will lie below Innovate Corp.'s MRPL curve. Therefore, at any given wage, Innovate Corp. will hire more engineers.
  11. In a perfectly competitive labor market, a firm's demand for labor is derived from its marginal revenue product. If the firm faces a perfectly elastic supply of labor at a wage of $20 per hour and currently employs 100 workers, but the marginal revenue product of the 100th worker is $25, what should the firm do to maximize profit?
    Answer: Hire additional workers until the marginal revenue product equals $20, assuming diminishing marginal returns Since the marginal revenue product of the 100th worker ($25) exceeds the wage rate ($20), the firm can increase profit by hiring more workers. Under the assumption of diminishing marginal returns, as the firm hires additional workers, the marginal revenue product will decline. The firm should continue hiring until MRP_L = w = $20. Choice B is incorrect because the firm should hire more, not fewer workers. Choice C is incorrect because the firm should continue adjusting employment, not stop at 100 workers. Choice D is incorrect because the firm is a wage-taker in a perfectly competitive labor market and cannot set the wage rate.
  12. A single firm in a perfectly competitive industry develops a new technology that doubles the marginal product of its labor at all levels of employment. Assuming no other firms adopt this technology, what is the short-run effect on this firm's quantity of labor hired and the market wage rate?
    Answer: The quantity of labor hired increases, and the market wage rate remains unchanged. When the firm's marginal product of labor (MPL) doubles, its marginal revenue product (MRPL = MPL * P) also doubles. This causes the firm's individual labor demand curve to shift to the right. At the existing market wage, the firm will find it profitable to hire more workers. However, because the firm is a single employer in a perfectly competitive labor market, it is a wage taker. Its individual decision to hire more workers is not large enough to affect the overall market wage. Therefore, the firm hires more labor at an unchanged market wage.
  13. A firm's demand curve for labor will be more elastic if
    Answer: it is easy for the firm to substitute capital for labor in the production process. The price elasticity of demand for labor measures how responsive the quantity of labor demanded is to a change in the wage rate. Demand will be more elastic (more responsive) if it is easy for the firm to substitute other inputs (like capital) for labor. If the wage rate rises, a firm with high substitutability can easily switch to using more capital and significantly reduce its labor usage. In contrast, inelastic product demand or a small labor cost share would make labor demand less elastic. The elasticity of labor supply affects the labor market equilibrium but not the elasticity of the demand curve itself.
  14. A company produces apple pies and sells them in a perfectly competitive market. It hires bakers in a perfectly competitive labor market. If a new health study reveals that apples have significant health benefits, leading to an increase in the market demand for apple pies, what is the short-run effect on the company's demand for bakers and the equilibrium wage for bakers?
    Answer: The demand for bakers will increase, and the equilibrium wage will increase. The demand for an input (labor) is a derived demand, determined by its marginal revenue product (MRPL = MPL * P). The increased market demand for apple pies will cause the price of apple pies (P) to rise. This increase in P will increase the MRPL for bakers at every level of employment, shifting the firm's demand curve for bakers to the right. As all firms in the industry experience this, the market demand for bakers shifts right, leading to a higher equilibrium wage, assuming an upward-sloping labor supply curve.
  15. A furniture company uses both carpenters and automated saws. Saws are a substitute for carpenters. If the price of automated saws falls significantly and the substitution effect of this price change is stronger than the output effect, what is the likely impact on the company's demand for carpenters and the equilibrium wage for carpenters?
    Answer: Demand for carpenters will decrease, and their wage will fall. When the price of a substitute input (saws) falls, two effects occur. The substitution effect leads the firm to use more of the cheaper input (saws) and less of the relatively more expensive input (carpenters), thus decreasing the demand for carpenters. The output effect leads the firm to increase production due to lower overall costs, which would increase demand for all inputs, including carpenters. The question states that the substitution effect dominates. Therefore, the net effect is a decrease in the demand for carpenters (a leftward shift of the demand curve), which leads to a lower equilibrium wage.
  16. A profit-maximizing firm uses both labor and capital. The marginal revenue product of the last unit of labor is $40, and the wage rate is $10. The marginal revenue product of the last unit of capital is $75, and the rental rate of capital is $25. Which of the following actions should the firm take to maximize its profit?
    Answer: Employ more labor and rent less capital. To maximize profit (or minimize cost for a given output), a firm should allocate its spending on inputs such that the marginal revenue product per dollar is equal across all inputs: MRPL/w = MRPK/r. For labor: MRPL/w = $40 / $10 = 4. For capital: MRPK/r = $75 / $25 = 3. Since MRPL/w > MRPK/r, the firm is getting more marginal revenue product per dollar spent on labor than on capital. Therefore, the firm should reallocate its resources by employing more labor and renting less capital until the ratios are equal.
  17. A firm in a perfectly competitive factor market hires labor to the point where the marginal revenue product of labor equals the wage rate. This hiring rule is sufficient to ensure the firm is
    Answer: minimizing its losses or maximizing its profits. The rule MRPL = w identifies the level of input usage that will result in the highest possible profit or the lowest possible loss. It is the optimal choice given that the firm decides to operate. However, following this rule does not guarantee a positive economic profit. If the market price for the output is too low, the firm could still be making a loss, even when hiring the 'correct' number of workers. In that case, hiring where MRPL = w ensures that the loss is minimized.
  18. A firm assembles bicycles, selling them for $200 each in a competitive market. It hires assemblers at a wage of $160 per day. The firm is currently employing 5 assemblers, and its total daily output is 10 bicycles. If it hires a 6th assembler, total daily output will rise to 11.5 bicycles. Should the firm hire the 6th assembler, and why?
    Answer: Yes, because the marginal revenue product of the 6th worker exceeds the wage. The decision to hire an additional worker depends on a marginal analysis. First, calculate the marginal product (MPL) of the 6th worker: MPL = 11.5 - 10 = 1.5 bicycles. Next, calculate the marginal revenue product (MRPL): MRPL = MPL Price = 1.5 $200 = $300. Finally, compare the MRPL to the wage (w). Since the MRPL of $300 is greater than the wage of $160, hiring the 6th worker will add more to revenue than to cost, thus increasing profit. The fall in average product is irrelevant to the marginal decision.
  19. A firm hires its sole variable input, labor, in a perfectly competitive market. When the firm hires 10 workers, the marginal product of the 10th worker is 25 units. The marginal revenue product of the 10th worker is $50. What is the price of the firm's product if it sells its product in a perfectly competitive market?
    Answer: $2.00 The marginal revenue product of labor (MRPL) is equal to the marginal product of labor (MPL) multiplied by the price of the output (P). The formula is MRPL = MPL * P. We are given MRPL = $50 and MPL = 25 units. We can rearrange the formula to solve for P: P = MRPL / MPL. Plugging in the values, we get P = $50 / 25 = $2.00.
  20. In the perfectly competitive market for electricians, the wage is $50 per hour. A single firm, Sparky's Electrical, finds that its last hired electrician has a marginal product of 2 service calls per hour. The firm charges a price of $30 per service call. To maximize its profit, Sparky's Electrical should
    Answer: hire more electricians because their marginal revenue product exceeds their wage. To check for profit maximization, the firm must compare the marginal revenue product (MRPL) of the last worker to the wage (w). The MRPL is the marginal product (MPL) times the product price (P). Here, MRPL = 2 service calls/hour * $30/service call = $60/hour. Since the MRPL of $60 is greater than the wage of $50, the last electrician hired added more to revenue than to cost. Therefore, the firm should hire more electricians, continuing to do so until the MRPL of the last one hired equals the wage.

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