The Cobb-Douglas Production Functions Lecture 40 Section 7.1 Robb T. Koether Hampden-Sydney College Mon, Apr 10, 2017 Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 1 / 9
Objectives Objectives Define the Cobb-Douglas production functions. Explore their properties of scaling and elasticity. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 2 / 9
The Cobb-Douglas Production Functions Definition (The Cobb-Douglas Production Functions) Let K the capital investment, L be the labor investment, Q be the output in units, A, α, and β be positive constants with α + β = 1. The Cobb-Douglas production functions are functions of the form Q(K, L) = AK α L β. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 3 / 9
Constant Returns to Scale Theorem If Q(K, L) = AK α L β, then the model predicts constant returns to scale. That is, if capital investment and labor investment are both scaled by a factor s, then output will also scale by the factor s. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 4 / 9
Constant Returns to Scale Replace K with sk and L with sl for some scale factor s. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 5 / 9
Constant Returns to Scale Replace K with sk and L with sl for some scale factor s. Then Q(sK, sl) = A(sK ) α (sl) β Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 5 / 9
Constant Returns to Scale Replace K with sk and L with sl for some scale factor s. Then Q(sK, sl) = A(sK ) α (sl) β = A(s α K α )(s β L β ) Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 5 / 9
Constant Returns to Scale Replace K with sk and L with sl for some scale factor s. Then Q(sK, sl) = A(sK ) α (sl) β = A(s α K α )(s β L β ) = AK α L β s α+β Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 5 / 9
Constant Returns to Scale Replace K with sk and L with sl for some scale factor s. Then Q(sK, sl) = A(sK ) α (sl) β = A(s α K α )(s β L β ) = AK α L β s α+β = s(ak α L β ) Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 5 / 9
Constant Returns to Scale Replace K with sk and L with sl for some scale factor s. Then Q(sK, sl) = A(sK ) α (sl) β = A(s α K α )(s β L β ) = AK α L β s α+β = s(ak α L β ) = sq(k, L). Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 5 / 9
Theorem If Q(K, L) = AK α L β, then α is the capital investment elasticity of production, and β is the labor investment elasticity of production. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 6 / 9
Consider labor investment L to be held fixed, so that Q(K ) = AK α L β. The formula for capital investment elasticity of production, in this case, is E(Q) = K Q(K ) Q (K ). (We dropped the minus sign because production goes up, not down, as investment goes up.) Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 7 / 9
We compute the derivative Q (K ) = A(αK α 1 )L β. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 8 / 9
We compute the derivative Q (K ) = A(αK α 1 )L β. Then E(Q) = K Q(K ) Q (K ) Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 8 / 9
We compute the derivative Q (K ) = A(αK α 1 )L β. Then E(Q) = K Q(K ) Q (K ) = K AK α L β A(αK α 1 )L β Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 8 / 9
We compute the derivative Q (K ) = A(αK α 1 )L β. Then E(Q) = K Q(K ) Q (K ) = K AK α L β A(αK α 1 )L β = K K α (αk α 1 ) Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 8 / 9
We compute the derivative Q (K ) = A(αK α 1 )L β. Then E(Q) = K Q(K ) Q (K ) = K AK α L β A(αK α 1 )L β = K K α (αk α 1 ) = αk α K α Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 8 / 9
We compute the derivative Q (K ) = A(αK α 1 )L β. Then E(Q) = K Q(K ) Q (K ) = K AK α L β A(αK α 1 )L β = K K α (αk α 1 ) = αk α K α = α. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 8 / 9
Elasticity It is convenient, but not necessary, that α + β = 1. Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 9 / 9
Elasticity It is convenient, but not necessary, that α + β = 1. What behavior would you predict if α + β < 1? Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 9 / 9
Elasticity It is convenient, but not necessary, that α + β = 1. What behavior would you predict if α + β < 1? What behavior would you predict if α + β > 1? Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon, Apr 10, 2017 9 / 9