Microeconomics Practice: Production, Competition, and Market Equilibrium

Production and Cost Optimization

The first set of problems addresses how firms make long-run input decisions to minimize costs and maximize output subject to budget constraints. A firm's production function describes the technological relationship between inputs (labor and capital) and output.

Problem: Davy Metal Company produces brass fittings using a Cobb-Douglas production function:
Q = 500L^0.6K^0.8
where Q is annual output in pounds, L is labor in person-hours, and K is capital in machine-hours. The marginal products are:
MPL = 300L^−0.4K^0.8
MPK = 400L^0.6K^−0.2 Labor costs $15 per hour, capital rents for $50 per hour, and the firm has an annual budget of $500,000. What combination of labor and capital minimizes cost while producing the optimal output?

Solution: Cost minimization requires that the marginal product per dollar spent on each input be equal:
MPL/w = MPK/r This condition, combined with the budget constraint (15L + 50K = 500,000), yields:
L = 14,286 person-hours
K = 5,714 machine-hours
Q = 157,568,191 pounds This problem demonstrates how Cobb-Douglas production functions and marginal analysis guide real capital-labor allocation decisions in firms.

In a perfectly competitive market, many firms produce identical products, entry and exit are unrestricted, and firms are price takers. Long-run equilibrium occurs when firms earn zero economic profit, eliminating incentives for entry or exit.

Problem 1: An industry contains identical firms operating under pure competition. Given market demand and supply conditions, find the equilibrium price, quantity, firm output, and profit.

Perfect Competition: Long-Run Equilibrium

Solution: At equilibrium:
P = $5 per unit
Q = 500 units (market quantity)
q = 500 units (individual firm output)
Profit = $528 The problem then asks whether entry or exit will occur. Since firms earn positive economic profit ($528), we would expect entry into the industry. New firms are attracted by profitable opportunities. As entry occurs, market supply increases, pushing the equilibrium price downward. Each firm's optimal output falls, and profit per firm declines, until price falls to the point where economic profit equals zero and entry ceases.

Problem 2: What is the lowest long-run price at which a competitive firm will sell?

Solution: The lowest price is $3.80 per unit. At this price, economic profit equals zero. This is the break-even price, equal to minimum average total cost. Any price above $3.80 generates positive profit (attracting entry) and any price below triggers losses (causing exit). The long-run competitive price equals minimum average cost, and perfectly competitive firms earn zero economic profit in the long run.

Problem 3: What is the lowest short-run price at which the firm will produce?

Solution: In the short run, the firm will produce at any positive price. However, profit becomes negative if price falls below $3.80. In the short run, firms have fixed costs they cannot avoid; they continue operating if price exceeds average variable cost (AVC), even if price lies below average total cost (AC). The firm shuts down only if price falls below minimum AVC. Since minimum AC ($3.80) exceeds minimum AVC, the firm remains willing to produce for prices between AVC minimum and AC minimum in the short run, accepting losses.

Problem 4: A competitive firm has a total cost function and marginal cost function. If market price is $115 per unit, find output, profit, and producer surplus.

Solution:
Profit = $800
Producer Surplus = $1,250 The firm maximizes profit by setting price equal to marginal cost. Producer surplus measures the gain from production, equal to the area above the marginal cost curve and below the price line.