III. THEORY OF THE FIRM: PRODUCTION AND COSTS

    A. PRODUCTION THEORY

    2. PRODUCTION IN THE LONG RUN

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

    a. RESOURCE COMBINATIONS AND ISOQUANTS

    Isoquant: a curve showing all the various efficient input combinations capable of producing a specified amount of output.

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.

    Slope of Qo:
    Geometry: DY/DX = DK/DL

    But what does that actually tell us about the production process?

Consider 2 points on Qo:   L   K   Q a 4  10  50 b 5   7  50 From a to b:    ? DL = ?    ? DK = ?    ? DQ = ?

    Note from the example, moving from a to b: we see that if we increase L by 1 unit, the move back to Qo (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.

This substitution rate = the...
    Marginal rate of technical substitution (MRTS): 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

Example here: mrts=3=> 1l is worth 3k.

    MRTS tells us the trade-off of labor for capital that a firm is willing to make since output is unaffected.

    Here: MRTS = 3 => the firm is able to give up 3K for 1L.

    MRTS tells you the firm's value of input X relative to Y

Resume day 15:

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

    MPL/MPK tells us how productive another L is compared to K

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

    MPL/MPK = 6/2=3
    1 more L is as productive as 3K.

    +1L --> DQ= +6
    - 3K --> DQ = -3x2 = -6
    Net DQ=0

    This suggests that MPL/MPK should determine the slope of Qo

    It does:

    DQ = MPL^.DL + MPK^.DK

    Along Qo, DQ = 0 =>

    -MPK^.DK = MPL^.DL

    Rearrange terms:
        |-DK|      MPL
        -------- = --------  ( = MRTS)
        |  DL|      MPK

    Lefthand side: size of slope of Qo
    Righthand sied: productivity of L compared to K.

    The logic:

	General case:	|	Example here:
	MPL = output gained from an extra L	|	MPL = 6 q
	MPK = rate output falls when giving up K	|	MPK = 2 q
		|
	=> MPL/MPK	|	6q/(2q per K)
	= total K you can give up for 1L w/o losing output	|	3K for 1L
	= |-DK/DL| (the size of the slope of Qo)	|

    The more productive (i.e., valuable) labor is compared to capital, the more capital a firm can give up to get that labor.
    So Qo will be steep when L is valued highly compared to K, flat when L is not so highly valued compared to K.

    A key property of isoquants:
    Isoquants are convex to the origin, reflecting a diminishing marginal rate of technical substitution.

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:

    Q = BoK^BkL^Bl

    MPL   Bl(BoK^BkL^Bl-1)    Bl  K
    --- = ------------- = --- ---
    MPK   Bk(BoK^Bk-1L^Bl)    Bk  L

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

    b. INTERPRETING ISOQUANTS

    Note: Inputs = factors of production

    (1) factor intensities

    Pertains to factor combinations in production

    Example: capital per worker (K/L)

Consider 2 points on Q1 (b with more L and less K than a):     ? which is more k-intensive?     Production technique a is capital intensive relative to b.     ? Rank factor intensities a..e:   Useful in    --comparing industries and -    -considering how production techniques respond changes in resource prices

    (2) RETURNS TO SCALE

    Pertains to movements between isoquants, holding factor intensity constant.
    All inputs are changed in the same proportion => changing the scale of operations.

    3 possibilities:
    %DQ = %DScale => CRTS
    %DQ > %DScale => IRTS
    %DQ < %DScale => DRTS

Example: Consider 3 input cominations:     K   L a  10  20 b  20  40 c  30  60 3 products:     (1) pumpkin pies (1/2/3)     (2) costumes (1/3/6)     (3) jack-o-lanterns (1/1.5/1.8)

    Why does this matter?
    --optimal firm size: feasibility of competition
    --resource productivity: up L productivity => up real wage
        2 sources: +K/L and E/S

Example: Cobb-Douglas:

See worksheet

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