Consider a Cobb-Douglas utility function U (X, Y) = Xα Y (1- α), 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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Introduction

The Cobb-Douglas utility function is one of the most widely used utility functions in microeconomics. It models consumer preferences where two goods are consumed in fixed proportions. This problem involves the derivation of the Marshallian demand, indirect utility, and application of specific values to determine maximum utility. We also derive Roy’s identity, which helps relate utility maximization to demand functions.

Given Utility Function

U(X, Y) = X^α Y^(1−α)

Let:
• X and Y = quantities of goods consumed
• Px and Py = prices of goods X and Y
• M = consumer’s income

Part a: Marshallian Demand Functions

The consumer maximizes utility subject to the budget constraint:

Maximize: U(X, Y) = X^α Y^(1−α)

Subject to: PxX + PyY = M

The Marshallian demand functions derived from the Cobb-Douglas utility function are:

X* = (αM) / Px
Y* = ((1−α)M) / Py

This implies the consumer will spend a constant proportion α of their income on good X and (1−α) on good Y.

Part b: Indirect Utility Function

The indirect utility function represents the maximum utility a consumer can achieve given prices and income. It is found by substituting the Marshallian demands into the utility function.

So, substitute X* and Y* into U(X, Y):

U = [(αM)/Px]^α × [((1−α)M)/Py]^(1−α)

Indirect Utility Function:
V(Px, Py, M) = (α^α) × ((1−α)^1−α) × M / (Px^α Py^1−α)

Part c: Compute Maximum Utility

Given:
• α = 1/2
• Px = ₹2
• Py = ₹8
• M = ₹4000

First, compute the demand for X and Y:

X* = (1/2 × 4000)/2 = 2000 / 2 = 1000
Y* = (1/2 × 4000)/8 = 2000 / 8 = 250

Now plug into utility function:

U = X^1/2 × Y^1/2 = √(1000 × 250) = √250000 = 500

Maximum Utility = 500

Part d: Roy’s Identity

Roy’s Identity relates the indirect utility function to Marshallian demand. It is defined as:

X = – (∂V/∂Px) / (∂V/∂M)

To derive Roy’s identity from the indirect utility function:

We have: V(Px, Py, M) = (α^α) × ((1−α)^1−α) × M / (Px^α Py^1−α)

Let A = α^α × (1−α)^1−α for simplicity

V = A × M / (Px^α Py^1−α)

∂V/∂Px = -A × M × α × Px^(−α−1) × Py^−(1−α)
∂V/∂M = A / (Px^α Py^1−α)

So, Roy’s identity becomes:

X = – (∂V/∂Px) / (∂V/∂M)

= – [ -A × M × α × Px^−α−1 × Py^−(1−α) ] / [A / (Px^α Py^1−α)]

= M × α / Px = X* = (αM)/Px

This confirms that Roy’s identity accurately retrieves the Marshallian demand for good X.

Conclusion

In this question, we used a Cobb-Douglas utility function to derive demand functions and utility levels. The Marshallian demand shows how much of each good a consumer buys depending on income and prices. The indirect utility function expresses utility in terms of prices and income. Roy’s identity connects the utility function with consumer demand and has been verified here. These tools are essential in microeconomic theory to understand consumer choice and behaviour.