2. Consider a Cobb-Douglas production function Q = A . L^a . K^b . Answer the following in terms of L, K, a, b
(a) What is the marginal product of labour ?
(b) What is the marginal product of capital ?
(c) What is the rate of technical substitution (RTS L for K)?
(d) From the above what is the relation between K L and RT SL,K?
(e) What is the relation between ∆ K L ∆RT SL,K (f) What is the value of RT SL,K [ K L ] ?
(g) If elasticity of substitution σ is defined as σ = ∆ K L ∆RT SL,K RT SL,K K L , what is the value of σ
Ans. Production function, Q = A*L^a * K^b
a) Marginal product of L, MPL = dQ/dL = Aa*L^(a-1) * K^b
b) Marginal product of K, MPK = dQ/dK = Ab*L^a * K^(b-1)
c) Marginal Rate of Technical Substitution, RTS = dK/dL = MPL/MPK = (a/b)*(K/L)
d) The relationship between K/L and RTS,
RTS/(K/L) = a/b
So, with each unit increase in K/L, the RTS increases by a/b. Thus, if a/b is positive, K/L and RTS have a direct relationship but if the a/b is negative, K/L and RTS have an inverse relationship.
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