Costs of a Competitive Firm

1. Costs of a Competitive Firm Production functions, Total Costs, Variable Costs and Marginal Costs

2. Cost Concepts • Total cost = Fixed cost + Variable cost • Fixed costs = same regardless of the level of output (for example, interest payments) • Variable costs = vary with level of output, such as total wages and the total cost of raw materials. • For simplicity, we treat variable costs as equivalent to total wages.

3. Production Functions • Show the relation between inputs and outputs. Our examples assume the only variable input is labor. • Example: X = 3 Lx, where • X is the output of good X (say apples) • Lx is the amount of labor employed per unit of time (say an hour). • 3 is the PRODUCTIVITY COEFFICIENT, which tells you in that one hour of work, a worker will pick 3 bushels of apples.

4. Activity requirements The production function X = 3 Lx also tells us how many workers we need to pick a given number of apples. Rewrite the equation as: Lx = 1/3 X The 1/3 is known as the activity requirement, and tells us we need 1/3 of an hour of work to pick one bushel of apples. In order to pick 30 bushels of apples, you need: Lx = 1/3 (30) = 10 workers.

5. Variable Cost • Remember that we identify variable cost and wage cost • Let w = the hourly wage rate • Then the total labor cost or variable cost is: VC = w Lx = w (1/3 X) = w/3 X If w = $12 per hour, then the variable cost involved in producing X units of good X is: VC = 12/3 X = 4 X

6. Total Cost and Marginal Cost • Total Cost = Fixed Cost + Variable Cost • So Total Cost = $ 200 + $4 X if w = $12 and fixed costs are $200 per unit of time. • To find marginal cost, find the total cost at X units of output and subtract the total cost at X – 1 units of output. • MC = ($200 + $4 X) – ($200 + $4 (X – 1)) or MC = $ 4.

7. Marginal Cost • Note that MC is constant in this example; whatever the value of X, MC = $4. MC at 10 units of output = MC at 100 units of output • Note also that MC = w / MPL or MC = wage divided by the Marginal Product of Labor MC = $12 hourly wage / 3 bushels of apples per hour

8. Diminishing marginal returns • Most production situations involve diminishing marginal returns to any single factor of production. • Hence the previous example does not apply to most production situations. • But a simple modification does: let the production function be X = 10 (sqrt Lx)

9. X = 10 (sqrt Lx)

10. Graphing • Graph the production function data above by putting Lx on the horizontal axis and X on the vertical axis. • Graph the marginal product of labor by putting Lx on the horizontal axis and MPL on the vertical axis. • Note that total product keeps on rising, but marginal product falls.

11. Production function and Costs • Since X = 10 (sqrt Lx), X2 = 100 Lx and Lx = X2 / 100 Hence (multiplying by the wage rate w) wLx = w X2 / 100 Or if the wage rate is $ 10 an hour, VC = 0.1X2

12. Total and Marginal Costs • Total Cost = Fixed Cost + Marginal Costs • Total Cost = $ 200 + 0.1 X2 • Calculate TC at 10 units of output: TC = $ 200 + .1 (100) = $ 210 Then calculate TC at 11 units of output: TC = $ 200 + .1 (121) = $ 212.10 The MARGINAL COST is $ 12.10

13. Increasing Marginal Costs Next, calculate total costs at 100 and 101 units of output. You should find that: TC at 100 units = $ 1,200.00 TC at 101 units = $ 1,220.10 Marginal cost = $ 20.10, up from $12.10 Note that approximately MC = 0.2 X

14. Marginal Costs and Marginal Product As previously, MC = w / MPL This is harder to show unless you know from calculus that MPL = dX / dL = 5 / (sqrt Lx) or MPL = 50 / X(substitute X / 10 for sqrt Lx) Since MC = d TC / dX = 0.2 X, we can see that 0.2 X = $ 10 X / 50 or MC = w / MPL as claimed. (No, the calculus will not be on the test. But if you’ve had calculus this might help you see the relationship.)