Fixed input/factor of production: quantity of input is fixed regardless of required

Transcription

1 Production Theory

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

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

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

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

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

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

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

9 Technology: Multiple Inputs What does a technology look like when there is more than one input? The two input case: Input levels are x and x 2. Output level l is y. Example of production function is y f ( x, x ) 2 x /3 x /3 2 2

10 PREVIEW: ISOQUANT An isoquant is the set of all combinations of inputs and 2 that are just sufficient to produce a given amount of output. The slope of the isoquant = the marginal rate of technical substitution (MRTS) = the technical rate of substitution (TRS) 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.

11 Technologies with Multiple Inputs The complete collection of isoquants is the isoquant map. The isoquant map is equivalent to the production function. Example y f / 3 ( x, x ) 2x x 2 2 / 3 2

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

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

14 Cobb-Douglas Technology x 2 All isoquants are hyperbolic, asymptoting to, but never touching any axis. a a 2 y x x 2 x

15 Marginal (Physical) Product y fx (,, x n ) 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. MP i y x i

16 Marginal (Physical) Product / 2 / y f( x, x ) x x then the marginal product of input is

17 Marginal (Physical) Product / 2 / y f( x, x ) x x then the marginal product of input is y MP x 2 / 3 x2 2 / 3 x 3

18 Marginal (Physical) Product / 2 / y f( x, x ) x x then the marginal product of input is y MP x 2 / 3 x2 2 / 3 x 3 and dthe marginal product of finput t2i is

19 Marginal (Physical) Product / 2 / y f( x, x ) x x then the marginal product of input is y MP x 2 / 3 x2 2 / 3 x 3 and dthe marginal product of finput t2i is y 2 /3 MP2 x x 2 /3. x 3 2

20 Marginal (Physical) Product The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if MP x i i x i y x i 2 y x 2 0 i

21 Technical Rate-of-Substitution x 2 x 2 ' The slope is the rate at which input 2 must be given up as input s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution. y x ' x

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

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

24 Technical Rate-of-Substitution dy y x dx y x dx 2. Along an individual isoquant, dy = 0, therefore the changes dx and ddx 2 must satisfy the following, 2 0 y x dx y 2 x dx. 2

25 Technical Rate-of-Substitution y 0 x dx 0 y x dx which rearranges to y x dx y 2 x dx or 2 2 dx2 y x / dx y / x 2. 2

26 Technical Rate-of-Substitution dx2 y x / dx y / x 2 is the rate at which h input 2 must be given up as input increases so as to keep the output level constant. It is the slope of the isoquant = MRTS = TRS.