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


    such that . f(x,x)y12

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


    c(y) = c (y) +F. v

    C c (y) s

     c (y) v


     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)



     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

Add n equations:

    c(n) = MC(0) + MC(1) ++ MC(n-1) y

     C MC

     O n y

By integration:


    c(y) = MC(u)duv0

Long-run Costs

    The long-run cost function


    such that f(x, x) = y 12



     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


     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


     SMC SAC 11


     SAC 2

     O y

    Long-Run Marginal Costs

    LMC = dc(y)/dy


     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.


     O y

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


    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.



     O y

    The Supply Decision of a Competitive Firm

Firms revenue: R = py

    Its cost: c(y)

    The profit maximization problem facing a firm


     C R


     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)



     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


    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




     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


    p < AVC

     C MC




     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


     p AVC




     O y output

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

     C MC


     p AVC



     O y y output

    PS = py (S + S) ABy'

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

    S= MC(u)duB'y


    S + S = = c (y) MC(u)duABv0

     C MC


     p AVC

     O y y output


    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*)






     O y* y output


    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.


    pp2 p()122

    2p 14

The producers surplus

    21pp()(). Ap224

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