III. Theory of the firm: production and costs

    C. The producer's optimum

    We can look at a firm's decisionmaking from 2 directions:
    Given Co: How much can the firm produce with that budget--strictly analogous to consumer theory
        --Max Q given Co, PL, PK
    Not really the typical situation

    Alternatively and more common:
    Given an output level to produce: What is the minimum cost of producing that output
    --this is the problem we'll work on. It's a problem which must be solved for any Q which the firm produces.
    --rather than start with some BL and move Uo's around to reach max., start with some Qo and move Co around to find min. TC.
    --but equil. will look the same
    --we'll want to then figure out just how much the final TC is.

    Comparing value and cost

    Example:
    L=6; K=10; Qo = 190 cassettes
    PL=$60; PK=$20
    MPL=35; MPK=7

    ? is this firm at a production optimum?

    The tangecy condition for production:
        MRTS = MPL/MPK = 35/7 = 5
        PL/PK = 60/20 = 3
    MRTS > PL/PK
    => relative prod'y of another L exceeds its relative cost
    => hire L, fire K.

    Substitution process:

    +L, -K --> MRTS falls.

    Continue until MRTS = PL/PK

Geometry     Compare points a and b: The firm gains by reallocating its budget, hiring more L and less K.     Firm goes to tangency between Qo and lowest Co which allows production of Qo   a b L (PL=60) 6 7 K (PK=20) 10 6 MPL 35 30 MPK 7 10 MRTS 5 3 TC $560 $540

    Note the tangency condition:

    Relative productivity of labor = its relative cost
    MPL/MPK = MRTS = PL/PK
    Slope of Qo = Slope of Co

    3 ways to think about the optimum:

    (1) tangency condition:

        MRTS = MPL/MPK = PL/PK
        30/10 = $60/$20 = 3

    (2) Least-cost condition (LCC):

        MPL/PL = MPK/PK

        Works for many inputs:

        ? Can you be more efficient?

        ? How?

        In general, switch to the input with the highest pay-off per $
        Optimum => equal pay-offs per $ for all inputs

    (3) equal marginal cost condition:

        PL/MPL =? PK/MPK =? Psteel/MPsteel
        $60/30 =? $20/10 =? $100/100

        Using L or K, MC = $2
        Using steel, MC = $1

        In general, switch to the input which gives you the lowest MC.
        Optimum => MC of Q is the same, no matter how we try to produce it

    Calculus of a production optimum

    Same justification as before: to formalize quantitatively the validity of this optimization process

    See solving cost minimization problems worksheet

Resume Day 18

    Final exercise to try: Least-cost condition worksheet

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