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)
Lx X (Output) MPL (Marg.Product)
1 10 10
2 14.1 4.1
3 17.3 3.2
4 20.0 2.7
5 22.4 2.4
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)
• w Lx w X2 / 100
• Or if the wage rate is 10 an hour,
• VC 0.1 X2

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
  youve had calculus this might help you see the
  relationship.)