Questions: Factor Demand and Input Cost Minimization

Question 1

A firm is producing output Q. At its current input mix, MPL/MPK = 3 and w/r = 2 (wages are twice the rental rate of capital). What should the firm do to minimize the cost of producing Q?

• A. Hire more capital and less labor — capital is currently producing more output per dollar
• B. Hire more labor and less capital — labor produces more output per dollar at current prices
• C. Maintain the current input mix — MRTS is above the price ratio, which is optimal
• D. Increase total spending on both inputs to shift the isocost line outward

Answer: (B)

The cost-minimizing condition is MRTS = w/r, i.e., MPL/MPK = w/r. Here MPL/MPK = 3 > w/r = 2, which means labor's marginal product per dollar (MPL/w) exceeds capital's (MPK/r). The firm can produce the same output more cheaply by substituting labor for capital: hire more labor and reduce capital until the marginal products per dollar equalize. This is the no-arbitrage logic: keep substituting toward the input that delivers more output per dollar until the advantage disappears.

Question 2

What does the factor demand curve for labor represent in the cost minimization framework?

• A. The total amount of labor the firm hires as its output level increases over time
• B. The quantity of labor that minimizes cost for a given output level, as the wage rate varies
• C. The marginal product of labor as a function of the total amount of labor employed
• D. The share of total costs attributable to labor at the profit-maximizing output level

Answer: (B)

The factor demand curve traces the cost-minimizing labor quantity at each wage level, holding output constant. As the wage rises, the isocost line steepens and the tangency point shifts along the isoquant toward more capital and less labor — the firm substitutes away from the now-expensive input. Plotting these wage-quantity pairs produces a downward-sloping factor demand curve. It is 'derived demand' because the firm doesn't want labor for its own sake — it demands labor to produce output that consumers demand.

Question 3

At the cost-minimizing input combination, no reallocation of a dollar from one input to another can produce more output at the same total cost.

• T. True
• F. False

Answer: True

This is the economic meaning of the tangency condition MRTS = w/r. When MPL/w = MPK/r, the last dollar spent on labor and the last dollar spent on capital both yield the same additional output. If they were unequal, the firm could reallocate spending from the lower-productivity input to the higher-productivity one and produce more output at the same cost — contradicting cost minimization. The tangency is precisely the point where this no-arbitrage condition holds, so no profitable reallocation is possible.

Question 4

Cost minimization analysis tells the firm how much output to produce — it determines the profit-maximizing quantity.

• T. True
• F. False

Answer: False

Cost minimization is a constrained optimization at a fixed output level: it finds the cheapest way to produce Q units, not how much Q to produce. The output decision is a separate profit-maximization step (choose Q where MR = MC). Cost minimization runs 'inside' profit maximization: for each candidate output level Q, cost minimization determines the minimum cost of achieving it, generating the cost function C(Q). Profit maximization then selects the Q that maximizes TR(Q) − C(Q). The two steps are conceptually and mathematically distinct.

Question 5

Explain why cost minimization and profit maximization are separate decisions, and how they relate to each other in the firm's overall optimization.

Model answer: Cost minimization asks: given that I want to produce Q units, what is the cheapest input combination? It produces the cost function C(Q) — the minimum cost at every output level. Profit maximization then asks: which Q maximizes profit? It compares revenue and cost across output levels, using C(Q) from the cost minimization step. Cost minimization is a prerequisite for profit maximization: you cannot choose the best output level without knowing how much each output level costs. The firm runs cost minimization 'everywhere' (for all Q) to build C(Q), then uses C(Q) in the profit calculation.

A firm could produce 100 units inefficiently by overpaying for inputs, or efficiently at minimum cost. Profit maximization only makes sense when you assume efficient production — otherwise you are comparing apples to oranges. The separation into two steps also has analytical value: cost minimization is a purely technological-economic question (how to produce), while profit maximization is a market question (how much to sell). The isoquant-isocost analysis lives entirely in the cost minimization step.