III. Theory of the firm: Production and Costs
    A. Production theory
    1. Production in the short run

Diminishing returns

    Behavior of productivity curves depicted above is thought to be quite general:
    Law of diminishing returns: if one input is increased while the others are held constant, the marginal and average productivity of the variable input will eventually decline.
    DMR = diminishing marginal returns
    DAR = diminishing average returns

    Assumes
    (1) a given technology
    (2) other inputs fixed.

Ex: Cobb-Douglas production function:     Q = boKbKLbL , 0 < bK, bL     Q = 18 K1/2L1/2     K = 4 => Q = 36L1/2     APL = 36L1/2/L = 36/L1/2     MPL = dQ/dL = 1/2(36L-1/2) = 18/L1/2 L Q APL=36/L1/2 MPL= 18/L1/2 1       4       9

    Note from formulas:
    --L in denominator: APL and MPL fall as L rises.
    --Therefore, for all L, this C-D production function has DMR and DAR.
    (Verify that it will not have DMR and DAR if exponent on L exceeds 1.)

    2. Production in the long run

Learning objectives: Draw and interpret isoquants, and explain their properties. Explain the meaning, geometry and formula for the marginal rate of technical substitution (MRTS).

    Now all inputs can be varied--here, both K and L

    a. Resource combinations and isoquants

    An isoquant is a curve showing all the various efficient input combinations capable of producing a specified amount of output (Qo).

The utility/production parallel:
U = U(X,Y,...)	Q= Q(K,L,...)
U is not quantifiable, so we never compute its value	Q IS quantifiable, so we do compute it
Constant U => indiff. curves	Constant Q => isoquants

Oxy headbands     Note the difference vs. U's: we attach an actual Q-level to each isoquant.

b. The marginal rate of technical substitution (MRTS)

Consider 2 points on Qo:   L   K   Q a 4  10  100 b 5   7  100 From a to b:    ? DL = ?    ? DK = ?    ? DQ = ?   (1) MRTS meaning: the maximum amount of input Y that a firm is able to give up for another unit of input X without losing any output

    Don't leave the "T" out--it distinguishes inputs from outputs:
    MRS tells you how much a good is worth to a consumer
    MRTS tells you how much an input is worth to producer

    (2) MRTS geometry: the size of the slope of an isoquant.
    DY/DX|_Qo = DK/DL

    But what does that actually tell us about the production process?
    Note from the example, moving from a to b: we see that if we increase L by 1 unit, the move back to Q1 (down) tells us the amount of K we can substitute for and still produce the initial Q. 
    Size of slope Ex: DK/DL = 3 means that the firm could exchange 3K for 1L without losing any output since the 1L gained is just as useful as the 3K lost.
    Here: MRTS = 3 => the firm is able to give up 3K for 1L and still produce Q1

    (3) MRTS formula

    It turns out that this ability to substitute L for K is related to the comparative productivities of the two inputs.

    Ex: Headband productivity: MPL = 6; MPK = 2 headbands

    DL=+1 --> DQ= 6 · +1 = +6
    DK=-3 --> DQ = 2 · -3 = -6
    Net DQ=0

In general:

Along Qo, output is constant =>      +Q from more L must be offset by -Q from less K => MPL·DL   =  -MPK·DK             => |-DK| -------- |  DL|   =  MPL -------- MPK = MRTS   The size of the slope of Qo   =  the marginal productivity of L relative to K (MRTS)

    Isoquants are convex to the origin because of a diminishing MRTS.

Consider 2 points on Qo (b with more L and less K than a):    ? Why would you expect L to be more productive at point a than point b?     Convexity rests on the law of diminishing marginal returns    Convexity => L becomes progressively less effective as a substitute for K

Example of diminishing MRTS:
Cobb-Douglas production function:

    Q = BoL^BLK^BK

    MPL   BL(BoL^BL-1K^BK)    BL  K
    --- = ------------- = --- ---
    MPK   BK(BoL^BLK^BK-1)    BK  L

Example:    Q = 8L1/2K1    ? MRTS formula?   L K Q MRTS a 1 4 32 2 b 4 2 32 1/4