Question 1 Mpho is a utility maximizing consumer. Her Cobb Douglas Utility function is given by $$ U(X, Y)=X^{\frac{1}{4}} Y^{\frac{3}{4}} $$ (a) Derive the Marshallian demand curves for good $$ X $$ and $$ Y $$. (b) (i) Derive the indirect utility function (ii) Derive the expenditure function from (b) (i). (iii) Derive the Hicksian demand curve for good $$ X $$. (iv) Why are these demand curves termed compensated demand curves? (c) You are given that the price of $$ X, P_{x} $$ is R 15 , whilst the price of $$ Y, P_{y} $$ is R 45 . The consumer's budget M is fixed at $$ \mathrm{R} 540(M=540) $$. (i) Use the Marshallian demand curves from (a) to find the values of $$ X $$ and $$ Y $$. (ii) Compute and comment on Mpho's marginal rate of substitution.

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Rojas Mathis · Accepted Answer

(a) To find the Marshallian demand curves for goods $$ X $$ and $$ Y $$, set up the Lagrange multiplier equation with the budget constraint and solve for the optimal consumption of $$ X $$ and $$ Y $$. (b) (i) The indirect utility function is the utility function expressed in terms of prices and income. (ii) The expenditure function is the minimum expenditure required to achieve a certain level of utility. (iii) The Hicksian demand curve for good $$ X $$ is derived by holding utility constant and changing prices. (iv) Compensated demand curves represent optimal consumption when the consumer is compensated for income or price changes. (c) (i) Use the Marshallian demand curves with given prices and budget to find $$ X $$ and $$ Y $$. (ii) Calculate the marginal rate of substitution and comment on it based on the values of $$ X $$ and $$ Y $$.

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