Given a Cobb-Douglas utility function U (X, Y) = X1/2 Y1/2, Where X and y are the two goods that a consumer consumes at per unit prices of Px and Py respectively. Assuming the income of the consumer to be ₹M, determine: a. Marshallian demand function for goods X and Y. b. Indirect utility function for such a consumer. c. The maximum utility attained by the consumer where α =1/2, Px =₹ 2, Py = ₹ 8 and M= ₹ 4000. d. Derive Roy’s identity.

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a. Marshallian Demand Functions:
The Marshallian demand functions represent the optimal quantities of goods that a consumer will purchase given prices and income. For the Cobb-Douglas utility function (U(X, Y) = X^{1/2}Y^{1/2}), the demand functions can be derived by maximizing the utility subject to the budget constraint:

[ \max_{X, Y} U(X, Y) \text{ subject to } Px \cdot X + Py \cdot Y = M]

Solving this optimization problem will yield the Marshallian demand functions for goods X and Y.

b. Indirect Utility Function:
The indirect utility function represents the maximum utility attainable at given prices and income. It is derived by substituting the optimal quantities (Marshallian demand functions) into the utility function.

[V(Px, Py, M) = U(X^(Px, Py, M), Y^(Px, Py, M))]

c. Maximum Utility:
To find the maximum utility, substitute the given values into the indirect utility function:

[V(Px, Py, M) = U(X^, Y^)]

d. Roy’s Identity:
Roy’s Identity relates the compensated and uncompensated price elasticities of demand. For a Cobb-Douglas utility function, Roy’s Identity is given by:

[ \frac{\partial x_i}{\partial p_j} = -\frac{x_i}{p_j} \frac{\partial U/\partial x_i}{\partial U/\partial M} ]

This identity expresses how the change in the demand for a good with respect to its price is related to the income and substitution effects.

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