Utility Maximization

Consumer’s problem: \[\max_{x,y} U(x,y) \quad \text{subject to} \quad p_x x + p_y y = I\]

Set up the Lagrangian: \[L = U(x,y) + \lambda(I - p_x x - p_y y)\]

First-order conditions → demand functions: \[x^*(p_x, p_y, I) \quad \quad y^*(p_x, p_y, I)\]

Note: These are called Marshallian (uncompensated) demand functions.

Indirect Utility Function

Definition: Maximum utility as a function of prices and income

\[V(p_x, p_y, I) = \max_{x,y} U(x,y) \text{ s.t. } p_x x + p_y y = I\]

Or equivalently: \[V(p_x, p_y, I) = U(x^*(p_x, p_y, I), y^*(p_x, p_y, I))\]

Properties:

1. Increasing in \(I\): \(\frac{\partial V}{\partial I} > 0\)
2. Decreasing in prices: \(\frac{\partial V}{\partial p_i} \leq 0\)
3. Homogeneous of degree zero: \(V(tp_x, tp_y, tI) = V(p_x, p_y, I)\)

Application: Gasoline Tax

• Indirect utility as a tool for welfare analysis of price/income changes

• Example: Government raises gas tax by $0.50/gallon. To compensate, gives everyone $200 cash transfer.

  • To answer if people are better or worse off, compare indirect utility before and after policy.
    • Before: \(V(p_{\text{gas}}, p_{\text{other}}, I)\)
    • After: \(V(p_{\text{gas}} + 0.50, p_{\text{other}}, I + 200)\)

Question: Would it be better if we could measure welfare in dollars rather than abstract “utils”?

The Dual Problem

Two ways to think about consumer choice.

I. Utility Maximization: Given prices and income, maximize utility

\[\max_{x,y} \ U(x,y) \quad \text{s.t.} \quad p_x x + p_y y = I\]

\(\rightarrow\) Marshallian (uncompensated) demand and Indirect Utility Function.

Visualizing Duality

Expenditure Minimization

Consumer’s problem: \[\min_{x,y} p_x x + p_y y \quad \text{subject to} \quad U(x, y) = \bar{U}\]

Set up the Lagrangian: \[L = p_x x + p_y y + \mu(\bar{U}-U(x, y))\]

FOCs give us Hicksian (compensated) demand functions: \[x^h(p_x, p_y, \bar{U}) \quad \quad y^h(p_x, p_y, \bar{U}) \]

Expenditure Function

Definition: Minimum expenditure needed to reach \(\bar{U}\) at prices \((p_x, p_y)\)

\[E(p_x, p_y, \bar{U}) = \min_{x,y} \ p_x x + p_y y \text{ s.t. } U(x,y) = \bar{U}\]

Or equivalently: \[E(p_x, p_y, \bar{U}) = p_x x^h + p_y y^h\]

Properties:

1. Increasing in \(\bar{U}\): \(\frac{\partial E}{\partial \bar{U}} > 0\)
2. Increasing in prices: \(\frac{\partial E}{\partial p_i} \geq 0\)
3. Homogeneous of degree 1 in prices: \(E(tp_x, tp_y, \bar{U}) = t \cdot E(p_x, p_y, \bar{U})\)
4. Concave in prices

Duality: Inverse Relationship

The indirect utility and expenditure functions are inverses.

\[V(p_x, p_y, E(p_x, p_y, \bar{U})) = \bar{U}\]

\[E(p_x, p_y, V(p_x, p_y, I)) = I\]

Gasoline Tax Revisited

Example: Government raises gas tax by $0.50/gallon.

How much will the consumer need to be compensated to reach their original utility level \(\bar{U}\)?

\[ E(p_{\text{gas}} + 0.50, p_{\text{other}}, \bar{U}) - E(p_{\text{gas}}, p_{\text{other}}, \bar{U}) \]

Now we have a way of measuring how much worse off the consumer is in dollar terms.

Two Types of Demand

Marshallian (uncompensated) demand: \(x(p_x, p_y, I)\)

• From utility maximization
• Holds income fixed as prices change
• Reflects both substitution and income effects

Relationship Between Two Demands

At optimal consumption, the two demand functions give the same quantity:

\[x(p_x, p_y, I) = x^h(p_x, p_y, V(p_x, p_y, I))\]

where \(V(p_x, p_y, I)\) is the maximum utility achieved at income \(I\).

Graphical: Marshallian Demand

Decomposing Price Effects

When \(p_x\) changes, quantity demanded changes for two reasons:

1. Substitution effect: Relative prices change → reallocate consumption
2. Income effect: Real purchasing power changes → change consumption

Hicksian demand isolates the substitution effect by holding utility constant.

Graphical: Hicksian Demand

The Slutsky Equation

\[\underbrace{\frac{\partial x}{\partial p_x}}_{\text{Total effect}} = \underbrace{\frac{\partial x^h}{\partial p_x}}_{\text{Substitution effect (SE)}} - \underbrace{x \cdot \frac{\partial x}{\partial I}}_{\text{Income effect (IE)}}\]

Substitution effect:

• Always negative (or zero)
• Pure effect of relative price change
• Movement along indifference curve

Income effect:

• Sign depends on whether good is normal or inferior
• Effect of change in real purchasing power
• Movement to different indifference curve

Three Cases

\[\underbrace{\frac{\partial x}{\partial p_x}}_{TE} = \underbrace{\frac{\partial x^h}{\partial p_x}}_{SE \leq 0} - \underbrace{x \cdot \frac{\partial x}{\partial I}}_{IE}\]

1. Normal Good

• \(\partial x/\partial I > 0\) (buy more as income rises)
• TE: Strongly negative

2. Inferior Good

• \(\partial x/\partial I < 0\) (buy less as income rises)
• |IE| < |SE| → TE: Negative (but smaller than normal good)

Three Cases (Contd.)

\[\underbrace{\frac{\partial x}{\partial p_x}}_{TE} = \underbrace{\frac{\partial x^h}{\partial p_x}}_{SE \leq 0} - \underbrace{x \cdot \frac{\partial x}{\partial I}}_{IE}\]

3. Giffen Good

• \(\partial x/\partial I < 0\) (inferior good)
• |IE| > |SE| → TE: Positive (buy less when price falls)

Demand Elasticities

Own-price elasticity (Marshallian): \[\varepsilon_{x,p_x} = \frac{\partial x}{\partial p_x} \cdot \frac{p_x}{x}\]

Income elasticity: \[\varepsilon_{x,I} = \frac{\partial x}{\partial I} \cdot \frac{I}{x}\]

Cross-price elasticity: \[\varepsilon_{x,p_y} = \frac{\partial x}{\partial p_y} \cdot \frac{p_y}{x}\]

Compensated Elasticities

Compensated own-price elasticity: \[\varepsilon_{x,p_x}^c = \frac{\partial x^h}{\partial p_x} \cdot \frac{p_x}{x^h}\]

Compensated cross-price elasticity: \[\varepsilon_{x,p_y}^c = \frac{\partial x^h}{\partial p_y} \cdot \frac{p_y}{x^h}\]

The Slutsky Equation in Elasticities

Dividing by \(x\) and multiplying by \(p_x\):

\[\boxed{\varepsilon_{x,p_x} = \varepsilon_{x,p_x}^c - s_x \cdot \varepsilon_{x,I}}\]

where:

• \(\varepsilon_{x,p_x}\) = Marshallian price elasticity
• \(\varepsilon_{x,p_x}^c\) = compensated price elasticity
  
• \(s_x = \frac{p_x x}{I}\) = budget share of good x
• \(\varepsilon_{x,I}\) = income elasticity

Interpretation: Total elasticity = substitution elasticity - budget share × income elasticity

Gross (Marshallian) Substitutes and Complements

• Two goods \(x\) and \(y\) are gross substitutes if an increase in the price of \(y\) leads to an increase in the demand for \(x\): \[\frac{\partial x}{\partial p_y} > 0\]
• They are gross complements if an increase in the price of \(y\) leads to a decrease in the demand for \(x\): \[\frac{\partial x}{\partial p_y} < 0\]

Net (Hicksian) Substitutes and Complements

• Two goods \(x\) and \(y\) are net substitutes if an increase in the price of \(y\) leads to an increase in the compensated demand for \(x\): \[\frac{\partial x^h}{\partial p_y} > 0\]
• They are net complements if an increase in the price of \(y\) leads to a decrease in the compensated demand for \(x\): \[\frac{\partial x^h}{\partial p_y} < 0\]

Measuring Welfare Changes

Question: How much is a consumer hurt by a price increase?

Three approaches:

1. Compensating Variation (CV): Money needed to restore original utility after price change

2. Equivalent Variation (EV): Money equivalent to price change (willingness to pay to avoid it)

3. Consumer Surplus (CS): Area under demand curve

All three approximate welfare changes, but differ in treatment of income effects.

Compensating Variation

Definition: Income needed to compensate for price increase to maintain original utility

For price increase from \(p_0\) to \(p_1\):

\[CV = E(p_1, p_y, U_0) - E(p_0, p_y, U_0)\]

where \(U_0\) is utility before the price change.

Equivalent Variation

Definition: Income change equivalent to price change in terms of utility

For price increase from \(p_0\) to \(p_1\): \[EV = E(p_1, p_y, U_1) - E(p_0, p_y, U_1)\]

where \(U_1\) is utility after the price change.

(Change in) Consumer Surplus

Definition: Area under Marshallian demand curve between two prices \[ \Delta CS = \int_{p_0}^{p_1} x(p, p_y, I) \, dp\]