Production Theory 1

Short-Run v. Long-Run u Fixed input/factor of production: quantity of input is fixed regardless of required output level, e.g. capital or specialized labour u Variable input/factor of production: quantity of input used depends on the level of output u Short run: at least one input/factor is fixed u Long run: all inputs/factors are variable

Production Function u A technology is a process by which inputs (e.g. labour and capital) are converted into output. u The output level is denoted by y. u The technology’s production function states the maximum amount of output possible from an input bundle.

Production Function y = f(x) is the production function x’x Input Level Output Level y’ y’ = f(x’) is the maximum output level obtainable from x’ input units. One input

Technology Set u The collection of all feasible production plans is the technology set.

Technology Set y = f(x) is the production function. x’x Input Level Output Level y’ y” One input y” = f(x’) is an output level that is feasible from x’ input units.

Technology Set x’x Input Level Output Level y’ One input y” The technology set

Technology Set x’x Input Level Output Level y’ One input y” The technology set Technically inefficient plans Technically efficient plans

Technology: Multiple Inputs u What does a technology look like when there is more than one input? u The two input case: Input levels are x 1 and x 2. Output level is y. u Example of production function is

PREVIEW: ISOQUANT u An isoquant is the set of all combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output. u The slope of the isoquant = the marginal rate of technical substitution (MRTS) = the technical rate of substitution (TRS) u MRTS (TRS): The number of units of K that we can dispose of if one more unit of L becomes available while remaining on the original isoquant.

Technologies with Multiple Inputs u The complete collection of isoquants is the isoquant map. u The isoquant map is equivalent to the production function. u Example

Isoquants with Two Inputs Y=20 Y=40 L K

Isoquants with Two Inputs u Properties  Y/  K>0,  Y/  L>0  2 Y/  K 2 <0,  2 Y/  L 2 <0 Diminishing marginal product (Diminishing marginal utility)

x2x2 x1x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technology

Marginal (Physical) Product u The marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed.

Marginal (Physical) Product then the marginal product of input 1 is

Marginal (Physical) Product then the marginal product of input 1 is

Marginal (Physical) Product then the marginal product of input 1 is and the marginal product of input 2 is

Marginal (Physical) Product then the marginal product of input 1 is and the marginal product of input 2 is

Marginal (Physical) Product u The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if

Technical Rate-of-Substitution x2x2 x1x1 y  The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution.

Technical Rate-of-Substitution u How is a technical rate-of-substitution computed?

Technical Rate-of-Substitution u How is a technical rate-of-substitution computed? u The production function is u A small change (dx 1, dx 2 ) in the input bundle causes a change to the output level of

Technical Rate-of-Substitution Along an individual isoquant, dy = 0, therefore the changes dx 1 and dx 2 must satisfy the following,

Technical Rate-of-Substitution which rearranges to or

Technical Rate-of-Substitution is the rate at which input 2 must be given up as input 1 increases so as to keep the output level constant. It is the slope of the isoquant = MRTS = TRS.