Duality and indirect utility function and expenditure function,:Indian Economic Service

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Duality in Consumer Theory, Indirect Utility Function, and Expenditure Function

In microeconomics, duality refers to the relationship between the primal problem (utility maximization) and its dual problem (expenditure minimization). This concept allows us to derive consumer behavior in two equivalent ways:

1. Utility Maximization Problem (UMP): The consumer maximizes utility given income and prices.
2. Expenditure Minimization Problem (EMP): The consumer minimizes expenditure while achieving a given utility level.

Both problems lead to key functions:

• Indirect Utility Function (from UMP) → Tells us the highest utility a consumer can achieve at given prices and income.
• Expenditure Function (from EMP) → Tells us the minimum income needed to reach a certain utility level at given prices.

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1. Duality in Consumer Theory

🔹 Primal Problem: Utility Maximization max⁡U(X1,X2)subject toP1X1+P2X2=M\max U(X_1, X_2) \quad \text{subject to} \quad P_1X_1 + P_2X_2 = M

• The solution to this gives Marshallian (ordinary) demand functions: X1∗=X1(P1,P2,M),X2∗=X2(P1,P2,M)X_1^* = X_1(P_1, P_2, M), \quad X_2^* = X_2(P_1, P_2, M)
• Substituting these into the utility function gives the Indirect Utility Function: V(P1,P2,M)=U(X1∗,X2∗)V(P_1, P_2, M) = U(X_1^*, X_2^*)

🔹 Dual Problem: Expenditure Minimization min⁡E=P1X1+P2X2subject toU(X1,X2)=U0\min E = P_1X_1 + P_2X_2 \quad \text{subject to} \quad U(X_1, X_2) = U_0

• The solution to this gives Hicksian (compensated) demand functions: X1h=X1(P1,P2,U0),X2h=X2(P1,P2,U0)X_1^h = X_1(P_1, P_2, U_0), \quad X_2^h = X_2(P_1, P_2, U_0)
• Substituting these into the budget equation gives the Expenditure Function: E(P1,P2,U0)=P1X1h+P2X2hE(P_1, P_2, U_0) = P_1X_1^h + P_2X_2^h

🔹 Key Duality Relationship: V(P1,P2,M)=Uif and only ifE(P1,P2,U)=MV(P_1, P_2, M) = U \quad \text{if and only if} \quad E(P_1, P_2, U) = M

This means:

• If we know the indirect utility function, we can find the expenditure function.
• If we know the expenditure function, we can find the indirect utility function.

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2. Indirect Utility Function (V)

Definition

The Indirect Utility Function tells us the maximum utility a consumer can achieve at given prices and income: V(P,M)=max⁡U(X)subject toP⋅X=MV(P, M) = \max U(X) \quad \text{subject to} \quad P \cdot X = M

Properties of Indirect Utility Function

1. Non-increasing in prices: If prices rise, purchasing power decreases, lowering utility.
2. Non-decreasing in income: More income means higher possible utility.
3. Homogeneous of degree 0 in (P, M): If we multiply all prices and income by a positive scalar, utility remains unchanged.
4. Quasi-convex in prices: The level sets of V(P,M)V(P, M) are convex.

Example (Cobb-Douglas Utility Function)

For a Cobb-Douglas function: U(X1,X2)=X1αX2βU(X_1, X_2) = X_1^\alpha X_2^\beta

The Marshallian demand functions are: X1=αMP1,X2=βMP2X_1 = \frac{\alpha M}{P_1}, \quad X_2 = \frac{\beta M}{P_2}

Substituting into U(X1,X2)U(X_1, X_2): V(P1,P2,M)=(αMP1)α(βMP2)βV(P_1, P_2, M) = \left( \frac{\alpha M}{P_1} \right)^\alpha \left( \frac{\beta M}{P_2} \right)^\beta

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3. Expenditure Function (E)

Definition

The Expenditure Function tells us the minimum amount of income needed to achieve a given utility level U0U_0 at given prices: E(P,U0)=min⁡Msubject toU(X)≥U0E(P, U_0) = \min M \quad \text{subject to} \quad U(X) \geq U_0

Properties of Expenditure Function

1. Non-decreasing in prices: If prices rise, more income is needed to reach the same utility.
2. Increasing in utility: Higher utility levels require more expenditure.
3. Homogeneous of degree 1 in prices: If we double all prices, the required expenditure also doubles.
4. Concave in prices: The function is concave because higher prices make it harder to achieve the same utility.

Example (Cobb-Douglas Utility Function)

From the Indirect Utility Function: V(P,M)=MαP1αP2βV(P, M) = \frac{M^\alpha}{P_1^\alpha P_2^\beta}

Solving for MM: E(P1,P2,U0)=U0⋅P1αP2βE(P_1, P_2, U_0) = U_0 \cdot P_1^\alpha P_2^\beta

This tells us how much money is needed to reach U0U_0 at given prices.

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4. Duality Relationship Between Indirect Utility and Expenditure Function

1. Expenditure Function from Indirect Utility Function

E(P,U)=Msuch thatV(P,M)=UE(P, U) = M \quad \text{such that} \quad V(P, M) = U

2. Indirect Utility Function from Expenditure Function

V(P,M)=Usuch thatE(P,U)=MV(P, M) = U \quad \text{such that} \quad E(P, U) = M

🔹 Hicksian (compensated) demand can be derived from the expenditure function: Xih=∂E(P,U)∂PiX^h_i = \frac{\partial E(P, U)}{\partial P_i}

🔹 Marshallian (ordinary) demand can be derived from the indirect utility function using Roy’s Identity: Xi=−∂V∂Pi∂V∂MX_i = – \frac{\frac{\partial V}{\partial P_i}}{\frac{\partial V}{\partial M}}

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5. Applications of Duality in Economics

1. Consumer Welfare Analysis: The expenditure function helps measure how much compensation a consumer needs when prices change.
2. Policy Impact on Consumers: Helps in assessing the effects of subsidies, taxes, and inflation.
3. Deriving Demand Functions: Roy’s Identity and Shephard’s Lemma allow us to derive demand functions from utility and expenditure functions.
4. Cost-of-Living Adjustments: Used in real-world applications like price indices.

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6. Conclusion

The duality approach in economics allows us to study consumer behavior from two perspectives:

• Utility maximization → leads to the Indirect Utility Function.
• Expenditure minimization → leads to the Expenditure Function.

Both functions are mathematically linked, making them useful for analyzing consumer demand, welfare changes, and economic policies.