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# micvar13

By Phillip Simmons,2014-03-02 21:46
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micvar13

13 COST CURVES

Cost function:

c(w,w,y) = min C = wx +wx 1 2 1122

subject to f(x, x) = y 12

Take the factor prices to be fixed, cost function: c(y).

Short-run cost

c(y,x)minwx;wxs21122x2

such that . f(x,x)y12

The total costs as the sum of the variable costs and the fixed

costs:

c(y) = c (y) +F. v

C c (y) s

c (y) v

F

O y

Average Costs

Average cost function(平均成本):

AC(y) = c(y)/y

By the construction:

c(y)c(y)FvAC(y);AVC(y);AFC(y) yyy

AVC (y): average variable costs(平均可变成本) AFC (y): average fixed costs (平均不变成本)

C MC AC(y)

AVC(y)

AFC(y)

O y

Marginal Costs

If we change output by some amount ?y, the change in cost

?c(y) = c(y +?y) c(y)

Marginal cost (边际成本)

c(y)c(y;y)c(y) MC(y).yy

The derivative form

dc(y)MC(y) dy

Relation between MC and AC

dc(y)dAC(y)dc(y)1=(y c(y) ) .2dydydyyy

c(y)dc(y)1=( ) dyyy

1= (MC(y) AC(y)) y

When MC is less than AC, AC is decreasing. If MC is greater than AC, AC is rising. MC must intersect AC at latter’s minimum point.

Marginal costs and variable costs

Marginal cost in terms of the variable cost function:

[c(yΔy)F][(c(y)F]c(y);;;vvvMC(y)= Δyy

The derivative form

dc(y)dc(y)v MC(y).dydy

When MC is less than AVC, AVC is decreasing. If MC is greater than AVC, AVC is rising. MC must intersect AVC at latter’s minimum point.

Take y = 1

MC(0) = c(1) c(0) yv

MC(1) = c(2) c(1) yv

MC(2) = c(3) c(2) yv

……

MC(n 1) = c(n) c(n1) yv

c(n) = MC(0) + MC(1) ++ MC(n-1) y

C MC

O n y

By integration:

y

c(y) = MC(u)duv0

Long-run Costs

The long-run cost function

c(y)minwx;wx1122xx,12

such that f(x, x) = y 12

C

c(y)

O y y* y

Fixed factor: k (= x ) , 2

The firms short-run cost function: c(y,k), s

For any given y, the optimal k: k(y)

What would be c(y, k(y))? s

The long-run cost function of production of y: c(y) = c(y, k(y)). s

***Pick y, and let k = k(y).

*c(y) c(y, k) s***c(y) = c(y, k) s

for all levels of y.

x k(y) 2

y 3

k*

y 2

y 1

O x 1

Long run average cost

AC(y) = c(y)/y

Long run and short run average costs *AC(y) AC(y, k) s***AC(y) = AC(y, k). s

C SMC 2

LMC

SMC SAC 11

LAC

SAC 2

O y

Long-Run Marginal Costs

LMC = dc(y)/dy

C

O y

The long-run marginal cost at any output level y has to equal the

short-run marginal cost associated with the optimal level of plant size to produce y.

Discrete levels of plant size

The long-run average cost curve will be the lower envelope of the short-run average costs.

C

O y

The long-run marginal cost curve consists of the appropriate pieces of the short-run marginal cost curves.

14 FIRM SUPPLY

Market Environments

Every firm faces two important decisions:

----choosing how much it should produce;

----choosing what price it should set.

Constraints facing the firm:

Technological constraints: summarized by the production function.

Market constraint: it can only sell as much as people are willing to buy.

Demand curve facing the firm: the relationship between the price a firm sets and the amount that it sells.

Pure competition.

Purely competitive market: each firm assumes that the market

price is independent of its own level of output.

An industry composed of many firms that produce an identical product,

Each firm is a small part of the market.

Firm is a price taker.

Firm will sell nothing if it charges a price higher than the market price.

If firm sells below the market price, it will get the entire market demand at that price.

If it sells at the market price, it can sell whatever amount it wants.

p

p*

O y

The Supply Decision of a Competitive Firm

Firms revenue: R = py

Its cost: c(y)

The profit maximization problem facing a firm

maxRc(y)y

C R

c(y)

O y y y 12

What level of output will a firm choose to produce?

MR = MC (y).

For a competitive firm, p doesnt change when y changes:

MR = p

A competitive firm will choose a level of output y:

p = MC(y)

C

p

O y y y 12

An exception

At y: p = MC(y). 2

It is not the firms best choice.

The best choices must be the points on the MC(y) with positive

slope.

MR =MC is a necessary condition for profit maximization.

Another Exception

Will firm keeps producing when it is facing py c(y) < 0?

In short-tun, when AC > p, the firms profits

c(y) F py v

= y(p AVC(y)) F

C MC

AC

AVC

p

O y y

The firm is better off going out of business when

F > py c(y) F v

That is

py < c(y) v

or

p < AVC

C MC

AC

AVC

p

O y

The Inverse Supply Curve

The supply of a competitive firm: price equals marginal cost

p = MC(y) if p ( min AVC

y = 0 if p < min AVC

---- inverse supply curve”.

Profits and Producers Surplus

Form’s profits in short-run

Profits = py c(y) F v

C MC

AC

p AVC

Profits

FC

VC

O y output

Producer’s surplus: the area to the left of the supply curve.

C MC

AC

p AVC

B

A

O y y output

PS = py (S + S) ABy'

S = AVC(y)y = c (y) = MC(u)duA v0y

S= MC(u)duB'y

y

S + S = = c (y) MC(u)duABv0

C MC

AC

p AVC

O y y output

y

PS = py MC(u)du0

Producer’s surplus is the revenues minus the area beneath the

MC curve.

PS = py c (y) v

=y(p AVC(y)).

Producer’s surplus is equal to revenues minus variable costs.

Because c(y) = c(y) + F, v

Producers surplus = py (c(y) F)

= (y) + F

The change in producers surplus When price changes from p* to p, output changes from y* to y.

The change in producers surplus

py (c(y) F ) (p*y* (c(y*) F )

= (y) (y*)

C

p

p*

B

A

O y* y output

Example

For a cost function 2 c(y) = y+ 1

Then 2 vc(y) = y

avc(y) = y

Price equals marginal costs:

p = 2y.

Solving for output as a function of price we have

pS(p)y 2

The maximum profits for each price p.

(p)pyc(y)

pp2 p()122

2p 14

The producers surplus

21pp()(). Ap224

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