. If Q = 2K3 + 3L2 K, show that K(MPK) + L(MPL) = 3Q. .The demand functions for two commodities, A and B, are given by QA = AP−0.5Y0.5 and QB = BP−1.5Y1.5 where A and B are positive constants (a) Find the price elasticity of demand for each good and hence comment on the relative sensitivity of demand due to changes in price. (b) Find the income elasticity of demand for each good. Which good is normal and which is superior? Give a reason for your answer.
1 Approved Answer
Manish K
5 Ratings,(9 Votes)
If Q = 2K^3 + 3L^2 K, show that K(MPK) + L(MPL) = 3Q. Sol: MPK = dQ /dK = 6K^2 + 3L^2 MPL = dQ /dL = 6LK To show that K(MPK) + L(MPL) = 3Q. Proof: K(6K^2 + 3L^2) + L(6LK) = 6K^3 + 3L^2K + 6L^2K = 6K^3 + 9L^2K = 3(2K^3 + 3L^2K) = 3Q (as given that Q = 2K^3 + 3L^2 K) = RHS Hence, proved The demand functions for two commodities, A and B, are given by QA = AP^-0.5Y^0.5 and QB = BP^-1.5Y^1.5 where A and B are positive constants (a) Find the price elasticity of demand for each good and hence comment on the relative sensitivity of demand due to changes in price. Sol: We know the price elasticity of demand = (dQA/QA)/(dP/P) = (dQA/dP)/(P/QA) Demand function from equation QA = AP^-0.5Y^0.5 Differentiating QA with respect to P dQA/dP = -0.5AP^-1.5Y^0.5 = -0.5AP^-0.5 P^-1Y^0.5 = -0.5AP^-0.5Y^0.5/P = -0.5QA/P (as given that QA = AP^-0.5Y^0.5) Price elasticity of demand = -0.5QA/P*(P/QA) = -0.5 Note: the negative sign can be ignored as price and quantity demanded is inversely related. Since, price elasticity of demand for good QA is (0.5) is less than one, it is inelastic. Similarly, we calculate price elasticity of demand for good QB QB = BP^-1.5Y^1.5 Demand function from equation QB = BP^-1.5Y^1.5 Differentiating QB with respect to P dQB/dP = -1.5BP^-2.5Y^1.5 = -1.5BP^-1.5 P^-1Y^0.5 = -1.5BP^-1.5Y^1.5/P = -1.5QB/P (as given that QB = BP^-1.5Y^1.5) Price elasticity of demand = -1.5QB/P*(P/QB) = -1
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