Production Theory 2
Returns-to-Scaleto Marginal product describe the change in output level as a single input level changes. (Short-run) Returns-to-scale describes how the output level changes as all input levels change, e.g. all input levels doubled. (Long-run)
Returns-to-Scale If, for any input bundle (x 1,,x n ), f ( tx, tx,, tx ) t. f ( x, x,, x ) 1 2 n 1 2 n then the technology described by the production function f exhibits constant returns-to-scale, to scale e.g. doubling all input levels doubles the output level (t=2). Note: Books often (confusingly) replace ( g y) p t with k.
Returns-to-Scaleto One input Output Level 2y y y = f(x) Constant returns-to-scaleto scale x x 2x Input Level
Returns-to-Scaleto If, for any input bundle (x 1,,x n ), f ( tx, tx,, tx ) tf ( x, x,, x ) 1 2 n 1 2 n then the technology exhibits decreasing returns-to-scale, e.g. doubling all input levels less than doubles the output level (t=2).
Returns-to-Scaleto One input Output Level 2f(x ) f(2x ) f(x ) y = f(x) Decreasing returns-to-scale to scale x 2x x Input Level
Returns-to-Scaleto If, for any input bundle (x 1,,x n ), f ( tx, tx,, tx ) tf ( x, x,, x ) 1 2 n 1 2 n then the technology exhibits increasing returns-to-scale, e.g. doubling all input levels more than doubles the output level (t=2).
Returns-to-Scaleto One input Output Level f(2x ) Increasing returns-to-scale y = f(x) 2f(x ) f(x ) x 2x x Input Level
Returns-to-Scale: Example The Cobb-Douglas production function is a 1 2 a a 1 2 y x 1 x 2 x. a n a n n a a a ( kx ) 1( kx ) 2( kx ) n k 1 ny. The Cobb-Douglas technology s returnsto-scale is constant if a 1 + + a n = 1 increasing if a 1 + + a n > 1 decreasing if a 1 + + a n < 1.
Short-Run: Marginal Product A marginal product is the rate-of- change of output as one input level increases, holding all other input levels fixed. Marginal product diminishes because the other input levels are fixed, so the increasing i input s units each have less and less of other inputs with which h to work.
Long-Run: Returns-to-Scaleto When all input levels are increased proportionately, there need be no such crowding out as each input will always have the same amount of other inputs with which to work. Input productivities need not fall and so returns-to-scale can be constant or even increasing.
Homogenous Production Function A production function is homogeneous of degree if F(tK, tl) = t F(K,L) for all t. If = 1 CRS If > 1 IRS If < 1 DRS N N ll d i f i Note: Not all production functions are homogeneous. (Y = 1 + L + K)
Perfect Substitutes Constant Returns to Scale: Show K Y=aK + bl MRTS= (-) b/a L
Perfect Complements Constant Returns to Scale: Show K Y = Min {L, K} Y=Y 1 Y=Y o L
Cobb-Douglas Homogeneous of degree ( + ) K Y=AK L Y=Y 1 Y=Y o L
Properties of Cobb-Douglas Production Function Y=AK L The Cobb-Douglas is homogeneous of degree = (+ ).
Properties of Cobb-Douglas Production Function Proof: Given Y=K L Y=(tK) (tl) Y= t K t L Y=t + K L Y= t + Y Y=t Y as =+ now introduce t If =1 (+=1) then CRS If >1 (+>1) then IRS If <1 (+<1) then DRS
Properties of Cobb-Douglas Production Function Output Elasticity Y=AK L Y. K K Y For Capital (show) Y. L L Y For Labour (show)
Properties of Cobb-Douglas Production Function Y=AK L Marginal Product of Capital (show). AP k Marginal Product of Labour (show).ap L
Properties of Cobb-Douglas Production Function Y=AK L Marginal Rate of Technical Substitution (MRTS = TRS) K L Show
Properties of Cobb-Douglas Production Function Y=AK L Euler s theorem: MP K K MPL L ( ) Y Where is the degree of homogeneity Where is the degree of homogeneity (show)
Elasticity of Substitution The Elasticity of Substitution is the ratio of the proportionate p change in factor proportions to the proportionate p change in the slope of the isoquant. Intuition: If a small change in the slope of the isoquant leads to a large change in the K/L ratio then capital and labour are highly substitutable.
Elasticity of Substitution = % Change in K/L % Change in Slope of Isoquant = % Change in K/L % Change in MRTS
Elasticity of Substitution K L A small change in the MRTS Large change in K/L High K and L are highly substitutable for each other
Elasticity of Substitution K L A large change in the MRTS Small change in K/L Low K and L are not highly substitutable for each other
Elasticity of Substitution K L K / K K L L L
Properties of Cobb-Douglas Production Function Y=AK L The elasticity it of substitution = 1 Show K L K / L K KK LL L
Properties of Cobb-Douglas Production Function In equilibrium,mrts = w/r and so the formula for reduces to, % in K % in w L r Useful for Revision Purposes: Useful for Revision Purposes: Not Obvious Now
Properties of Cobb-Douglas Production Function For the Cobb-Douglas, =1 means that a 10% change in the factor price ratio leads to a 10% change in the opposite direction i in the factor input ratio. U f l F R i i P Useful For Revision Purposes: Not Obvious Now
Well-Behaved Technologies A well-behaved technology is monotonic, and convex.
Well-Behaved Technologies - Monotonicity Monotonicity: More of any input generates more output. y monotonic y not monotonic x x
Well-Behaved Technologies - Convexity Convexity: If the input bundles x and x both provide y units of output then the mixture tx + (1-t)x provides at least y units of output, for any 0 < t < 1.
x 2 Well-Behaved Technologies - Convexity x 2 ' x 2 " x 1 ' x 1 " y x 1
x 2 Well-Behaved Technologies - Convexity x 2 ' x 2 " x 1 ' 1 ' " 1 ' 2 " 2 tx ( 1t) x, tx ( 1t) x x 1 " y x 1
Well-Behaved Technologies - Convexity x 2 x 2 ' Convexity implies that the MRTS/TRS decreases as x 1 increases. x 2 " x 1 ' x 1 " x 1