Firm Theory
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Cost Minimization
The profit-maximization problem can be broken into two stages:
- minimize the costs of producing any level of output y;
- derive the optimal out y that maximizes profit
If a firm is maximizing profits and if it chooses to supply output
y, then it must be minimizing the cost of producing y.
Firm Theory
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I. Cost minimization problem
- minimize cost to produce some given level of output:
- geometric solution: slope of isoquant equals slope of isocost
curve.
- equation is:
- optimal choices of factors are the conditional factor
demand functions
- optimal cost is the cost function
Firm Theory
II. Examples
- if
- if
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, then
, then
III. Revealed cost minimization
- suppose we hold output fixed and observe choices at
different factor prices.
- when prices are
, choice is
, and when prices
, choice is
.
are
- if choices minimize cost, then we must have
- this is the Weak Axiom of Cost Minimization (WACM)
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- add these two inequalites:
- roughly speaking, “factor demands move opposite to changes
in factor prices”. In particular, factor demand curves must
slope downward.
IV. Returns to scale and the cost function
- increasing returns to scale implies decreasing average cost
- constant returns implies constant average cost
- decreasing returns implies increasing average cost
Firm Theory
V. Long-run and short-run costs
- long run: all inputs variable
- short run: some inputs fixed
VI. Fixed and quasi-fixed costs
- fixed: must be paid, whatever the output level
- quasi-fixed: only paid when output is positive (heating,
lighting, etc.)
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Firm Theory
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Cost Curves
I. Family of cost curves
- total cost:
or
- marginal cost is the change in cost due to change in output
a) marginal cost equals AVC at zero units of output
b) goes through minimum point of AC and AVC.
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- this is negative (for example) when
c) fundamental theorem of calculus implies that
d) geometrically: the area under the marginal cost curve gives
the total variable costs. Figure 21.3.
e) intuitively: the marginal cost curve measures the cost of each
additional unit, so adding up the MCs gives the variable cost
II. Example:
- AC =
- AVC =
- MC =
- Figure 21.4.
Firm Theory
III. Long-run cost from short-run cost
- average costs: Figure 21.8.
- marginal costs: Figure 21.9.
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Firm Theory
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Firm Supply
I. Firms face two sorts of constraints
- technological constraints：summarize in cost function
- market constraints：how will consumers and other firms
react to a given firm's choice?
II. Pure competition
- formally takes market price as given, outside of any particular
firm's control
- example: many small price takers
- demand curve facing a competitive firm: Figure 22.1.
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III. Supply decision of competitive firm
- maxy py − c(y)
, i.e. price equals marginal cost
- first-order condition:
determines supply as function of price
- second-order condition:
, i.e. only upward-sloping
part of marginal cost curve matters
- is it profitable to operate at all?
a) compare
with −F
b) profits from operating will be greater when
c) operate when price covers average variable costs
IV. So supply curve is the upward-sloping part of MC curve
that lies above the AVC curve
- Figure 22.3.
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V. Inverse supply curve
measures the marginal cost curve directly
VI. Example:
- p = 2y gives the (inverse) supply curve
- is
a) yes, since 2y > y for all y > 0
VII. Producer's surplus
- producer's surplus is defined to be py − cv(y)
- since cv(y) = area under marginal cost curve
- producer's surplus is also the area above the marginal cost
curve. we can also use the “rectangle" for part of PS and the
“area above MC” for the rest. see Figure 22.5.
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VIII. Long-run supply: use long-run MC. In long run, price
must be greater than AC