Demand Functions and Duality : Microeconomics

In microeconomics, especially in consumer theory, two important concepts help us understand how people make choices: demand functions and duality. One part of duality is called expenditure minimization.

What Is a Demand Function?

A demand function shows how much of a good a person will buy depending on the price of that good, the prices of other goods, and their income.

There are two main types of demand functions:

i. Marshallian Demand (Ordinary Demand): It comes from the idea that a consumer tries to maximize their utility (happiness) given a budget.

Example: You have $100 to spend on apples and bananas. You want to choose the combination that makes you happiest, without spending more than $100. Marshallian demand depends on prices and income.

ii. Hicksian Demand (Compensated Demand): It comes from the idea that a consumer wants to get a certain level of utility at the lowest cost. This is called expenditure minimization. Hicksian demand depends on prices and utility level.

What Is Duality in Economics?

Duality means that the same consumer problem can be looked at in two different but connected ways:

Utility Maximization (Primal problem) - Given prices and income, find the best bundle that gives the highest utility.

Expenditure Minimization (Dual problem) - Given prices and a target utility level, find the cheapest bundle that achieves that utility.

Both lead to related demand functions:

Utility maximization → Marshallian demand

Expenditure minimization → Hicksian demand

These two approaches are dual to each other, meaning one can be derived from the other.

What Is Expenditure Minimization?

Let's say you want to be at a happiness level of Ū. The question is: What is the cheapest way (least expenditure) to reach this level, given the prices of goods?

You solve the following problem:

Minimize: 𝑝₁x₁ + 𝑝₂x₂, Subject to: 𝑢(x₁, x₂) ≥ Ū

This gives you the Hicksian demand function - it tells you the quantities of goods you would choose just enough to reach utility Ū at the lowest cost.

From this, we get the expenditure function: E(p₁, p₂, Ū) = Minimum cost to reach utility Ū at prices p₁ and p₂

Key Relationships and Properties:

Shephard's Lemma: The derivative of the expenditure function with respect to price gives the Hicksian demand:

∂E(p, Ū)/∂pᵢ = hᵢ(p, Ū)

Link Between Marshallian and Hicksian Demand:

 The Marshallian demand includes income, while Hicksian demand includes utility. But they are linked: Marshallian demand at income I = Hicksian demand at utility level u(x), where x is from Marshallian demand.

Expenditure Function Properties:

• Increasing in utility and prices
• Concave in prices
• Homogeneous of degree 1 in prices (if all prices double, total cost doubles)

Why Is Duality Useful?

It helps economists understand consumer choices from two angles.

Policy analysis: When governments consider price changes or subsidies, duality helps calculate how much compensation is needed to keep consumers at the same utility.

Cost-of-living index: Expenditure functions are used to measure how much more money a person needs to maintain the same lifestyle when prices change.

Example to Tie It All Together:

Suppose you consume only two goods: food and clothing.

Prices: food = $2, clothing = $5. You want to reach a utility level of Ū = 50.

Using expenditure minimization, you find: Optimal bundle: 10 units of food and 6 units of clothing

Minimum expenditure: (10 × 2) + (6 × 5) = 20 + 30 = $50. So, your Hicksian demand is (10, 6), and the expenditure function value is $50.

Understanding demand functions and duality helps us see how people make smart choices with limited money. The Marshallian demand tells us what people choose given income. The Hicksian demand tells us what people need to just reach a target utility. Duality ties both together beautifully and gives economists powerful tools to analyze real-world problems like price changes, taxes, or welfare programs.