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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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