Micro3 — Utility Maximization & Choice

1. The Budget Constraint

A consumer has income \(I\) and faces prices \(p_1, p_2\) for goods \(x_1, x_2\). The budget set is all affordable bundles:

\[ B = \{ (x_1, x_2) \in \mathbb{R}^2_+ \;:\; p_1 x_1 + p_2 x_2 \le I \} \]

The budget line (the boundary, where all income is spent) is:

\[ p_1 x_1 + p_2 x_2 = I \qquad \Longrightarrow \qquad x_2 = \frac{I}{p_2} - \frac{p_1}{p_2}\, x_1 \]

Intercepts

Set \(x_2=0\): \(\;x_1 = I/p_1\) (horizontal intercept)

Set \(x_1=0\): \(\;x_2 = I/p_2\) (vertical intercept)

Slope of the budget line

\(\text{slope} = -\dfrac{p_1}{p_2}\)

Slope = opportunity cost of good 1 in terms of good 2.

What happens when things change?

Income changes: Line shifts parallel in or out (slope unchanged).

Price \(p_1\) changes: Horizontal intercept moves; slope changes. The line rotates around the vertical intercept.

2. Optimal Choice & The MRS

What is the MRS?

The Marginal Rate of Substitution \(MRS_{12}\) is how many units of good 2 the consumer is willing to give up to get one more unit of good 1, staying on the same indifference curve (same utility).

\[ MRS_{12} = -\frac{dx_2}{dx_1}\bigg|_{U=\text{const}} = \frac{\partial U / \partial x_1}{\partial U / \partial x_2} = \frac{MU_1}{MU_2} \]

The Tangency Condition (Optimality)

Key rule — interior solution

At the optimal bundle, the consumer is willing to trade at the same rate the market allows:

\[ MRS_{12} = \frac{p_1}{p_2} \]

Equivalently (equalising bang-per-buck across all goods):

\[ \frac{MU_1}{p_1} = \frac{MU_2}{p_2} = \cdots = \frac{MU_n}{p_n} = \lambda \]

where \(\lambda\) = marginal utility of income (extra utility from €1 more income).

Why does it work intuitively?

If \(MU_1/p_1 > MU_2/p_2\), good 1 gives more utility per euro than good 2 → buy more good 1, less good 2. Adjust until both ratios are equal.

When does the tangency condition fail?

Corner solutions

Consumer consumes zero of at least one good. The optimal point is at a corner of the budget set — tangency doesn't apply.

Non-differentiable preferences

Kinks in indifference curves (perfect complements). No unique slope → no tangency condition.

Non-convex preferences

If indifference curves are concave (MRS increasing), a tangency point may be a local minimum, not maximum. Always check second-order conditions or compare with corner solutions.

Sufficient vs. Necessary

The tangency condition is sufficient for optimality when the MRS is strictly decreasing (convex indifference curves, interior solution).

It is only necessary (not sufficient) when MRS is not strictly decreasing.

3. The Lagrangian Method — Step by Step

Standard utility maximisation problem:

\[ \max_{x_1, x_2} \; u(x_1, x_2) \quad \text{subject to} \quad p_1 x_1 + p_2 x_2 = I \]

Write the Lagrangian:
\[ \mathcal{L} = u(x_1, x_2) + \lambda\,(I - p_1 x_1 - p_2 x_2) \]

\(\lambda\) is the Lagrange multiplier — it enforces the constraint. At the optimum it equals the MU of income.
First-Order Conditions (FOC) & How Derivatives Work:
To find these, we take the partial derivative of \(\mathcal{L}\) with respect to each variable one by one. When differentiating with respect to one variable, treat all other variables as constants (like regular numbers):
- With respect to \(x_1\): The utility function becomes \(\frac{\partial u}{\partial x_1}\). In the second term, \(\lambda\) and \(p_1\) are constants, and \(x_1\) has a power of 1, so its derivative is just \(-\lambda p_1\). Set to 0.
- With respect to \(x_2\): Same logic! \(\frac{\partial u}{\partial x_2}\) minus the derivative of the constraint term \(-\lambda p_2\). Set to 0.
- With respect to \(\lambda\): Treat \(x_1\) and \(x_2\) as constants. The whole expression inside the parentheses expands to \(\lambda I - \lambda p_1 x_1 - \lambda p_2 x_2\). Taking the derivative with respect to \(\lambda\) leaves just the inner budget constraint \(I - p_1 x_1 - p_2 x_2\). Set to 0.
\[ \frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1 = 0 \] \[ \frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial u}{\partial x_2} - \lambda p_2 = 0 \] \[ \frac{\partial \mathcal{L}}{\partial \lambda} = I - p_1 x_1 - p_2 x_2 = 0 \]
Divide the first two FOCs to eliminate \(\lambda\):
How it works: Rearrange the first two FOCs to isolate \(\lambda\) on one side \(\lambda = \frac{\partial u / \partial x_1}{p_1}\) and \(\lambda = \frac{\partial u / \partial x_2}{p_2}\)
Since both equal \(\lambda\), set them equal to each other. Rearranging them cancels \(\lambda\) out entirely and leaves you with the tangency condition:

\[ \frac{\partial u/\partial x_1}{\partial u/\partial x_2} = \frac{p_1}{p_2} \qquad \Longleftrightarrow \qquad MRS_{12} = \frac{p_1}{p_2} \]

Economically, this means the ratio of your marginal utilities equals the ratio of the prices (your "bang for your buck" is balanced).
Solve the system: Use the tangency condition + budget constraint to get \(x_1^*, x_2^*\) as functions of \(p_1, p_2, I\).
Check: Are both quantities positive? (Interior solution.) If not, check corner solutions.

4. Examples — Solved

A) Cobb-Douglas: \(u(x_1,x_2) = x_1^\alpha x_2^\beta\)

Setup

Marginal utilities: \(\;MU_1 = \alpha x_1^{\alpha-1}x_2^\beta\), \(\;MU_2 = \beta x_1^\alpha x_2^{\beta-1}\)

How exponents simplify during Tangency:

\[ \frac{MU_1}{MU_2} = \frac{\alpha x_1^{\alpha-1}x_2^\beta}{\beta x_1^\alpha x_2^{\beta-1}} = \frac{\alpha}{\beta} \cdot \left(\frac{x_1^{\alpha-1}}{x_1^\alpha}\right) \cdot \left(\frac{x_2^\beta}{x_2^{\beta-1}}\right) \]

Using exponent rules (\(x^a / x^b = x^{a-b}\)):
• For \(x_1\): \((\alpha - 1) - \alpha = -1 \Rightarrow x_1^{-1} = \frac{1}{x_1}\)
• For \(x_2\): \(\beta - (\beta - 1) = 1 \Rightarrow x_2^1 = x_2\)

Tangency condition: \(\;\dfrac{MU_1}{MU_2} = \dfrac{\alpha x_2}{\beta x_1} = \dfrac{p_1}{p_2} \;\;\Rightarrow\;\; x_2 = \dfrac{\beta p_1}{\alpha p_2} x_1\)

Substitute into budget constraint: \(p_1 x_1 + p_2 \cdot \left(\dfrac{\beta p_1}{\alpha p_2} x_1\right) = I \;\;\Rightarrow\;\; p_1 x_1\!\left(1 + \dfrac{\beta}{\alpha}\right) = I\)

Marshallian demand (Cobb-Douglas)

Step-by-step algebra for \(x_1^*\):

Substitute \(x_2\): \(p_1 x_1 + p_2 \left(\frac{\beta p_1}{\alpha p_2} x_1\right) = I \implies p_1 x_1 + \frac{\beta}{\alpha} p_1 x_1 = I\)
Factor out \(p_1 x_1\): \(p_1 x_1 \left(1 + \frac{\beta}{\alpha}\right) = I \implies p_1 x_1 \left(\frac{\alpha + \beta}{\alpha}\right) = I\)
Isolate \(x_1\): Multiply by the reciprocal \(\frac{\alpha}{\alpha + \beta}\) and divide by \(p_1\).

\[ x_1^* = \frac{\alpha}{\alpha+\beta}\cdot\frac{I}{p_1} \qquad x_2^* = \frac{\beta}{\alpha+\beta}\cdot\frac{I}{p_2} \]

Interpretation: the consumer spends a fixed fraction of income on each good — \(\alpha/(\alpha+\beta)\) on good 1, \(\beta/(\alpha+\beta)\) on good 2. This fraction never changes with prices or income.

B) Perfect Complements: \(u(x_1,x_2) = \min(x_1, x_2)\)

Logic (no tangency — use kink condition)

Consumer only gains utility if both goods increase together. Optimal: consume at the kink, where \(x_1 = x_2\).

Substitute into budget: \(p_1 x_1 + p_2 x_1 = I \;\Rightarrow\; x_1^* = x_2^* = \dfrac{I}{p_1+p_2}\)

C) Perfect Substitutes: \(u(x_1,x_2) = x_1 + x_2\)

Logic (corner solution)

MRS = 1 (constant). Compare to \(p_1/p_2\):

Condition	Optimal choice
\(p_1 < p_2\) (good 1 cheaper)	Spend all on good 1: \(x_1^=I/p_1,\;x_2^=0\)
\(p_1 > p_2\) (good 2 cheaper)	Spend all on good 2: \(x_1^=0,\;x_2^=I/p_2\)
\(p_1 = p_2\)	Any point on budget line is optimal

D) Quasi-linear: \(u(x_1,x_2) = x_1^{0.5} + x_2\)

Setup

Definition: Non-linear in \(x_1\), linear in \(x_2\). Demand for \(x_1^*\) is independent of income \(I\) provided your income is high enough to avoid a corner solution!

Marginal utilities: \(\;MU_1 = 0.5\,x_1^{-0.5},\quad MU_2 = 1\)

Step-by-step Tangency:

Set ratio equal to prices: \(\frac{0.5 x_1^{-0.5}}{1} = \frac{p_1}{p_2} \implies x_1^{-0.5} = \frac{2 p_1}{p_2}\)
Square both sides: \(x_1^{-1} = \frac{4 p_1^2}{p_2^2}\)
Invert to solve for \(x_1^*\): \(x_1^* = \frac{p_2^2}{4 p_1^2}\)

Then substitute \(x_1^*\) into budget constraint \(p_1 x_1 + p_2 x_2 = I\):
\(x_2^* = \dfrac{I}{p_2} - \dfrac{p_1 x_1^*}{p_2} = \dfrac{I}{p_2} - \dfrac{p_2}{4p_1}\)

Check constraint: Must have \(x_2^* \ge 0\). If income \(I\) is too low (\(I < \frac{p_2^2}{4p_1}\)), we get a corner solution: set \(x_2^*=0\) and spend all income on good 1 (\(x_1^*=I/p_1\)).

E) CES: \(u(x_1,x_2) = (x_1^\rho + x_2^\rho)^{1/\rho}\), \(\; 0\ne\rho<1\)

The "Master" Utility Function

CES = Constant Elasticity of Substitution. It unifies all utility functions into one formula based on \(\rho\):

\(\rho \to 1\): Perfect Substitutes
\(\rho \to 0\): Cobb-Douglas
\(\rho \to -\infty\): Perfect Complements

Step-by-Step Derivative & Tangency:

Chain rule gives \(MU_1 = (x_1^\rho + x_2^\rho)^{\frac{1-\rho}{\rho}} x_1^{\rho-1}\).
Taking the ratio \(\frac{MU_1}{MU_2}\) cancels the outer term completely: \(\left(\frac{x_1}{x_2}\right)^{\rho-1} = \frac{p_1}{p_2}\).
Solve ratio: \(\frac{x_1}{x_2} = \left(\frac{p_1}{p_2}\right)^{\frac{1}{\rho-1}}\). Substituting into budget constraint yields demand.

How the "Infinity" Exponent Cancels Out:
Both marginal utilities share the exact same outer component: \[ MU_1 = \underbrace{(x_1^\rho + x_2^\rho)^{\frac{1-\rho}{\rho}}}_{\text{Term with } \infty \text{ exponent}} \cdot x_1^{\rho-1}, \qquad MU_2 = \underbrace{(x_1^\rho + x_2^\rho)^{\frac{1-\rho}{\rho}}}_{\text{Term with } \infty \text{ exponent}} \cdot x_2^{\rho-1} \] When calculating the ratio \(\frac{MU_1}{MU_2}\): \[ \frac{MU_1}{MU_2} = \frac{\cancel{(x_1^\rho + x_2^\rho)^{\frac{1-\rho}{\rho}}} \cdot x_1^{\rho-1}}{\cancel{(x_1^\rho + x_2^\rho)^{\frac{1-\rho}{\rho}}} \cdot x_2^{\rho-1}} = \frac{x_1^{\rho-1}}{x_2^{\rho-1}} = \left(\frac{x_1}{x_2}\right)^{\rho-1} \] Because the problematic term appears identically in both numerator and denominator, it divides out completely before taking any limit. As \(\rho \to 0\), the remaining ratio becomes: \[ \lim_{\rho \to 0} \left(\frac{x_1}{x_2}\right)^{\rho-1} = \left(\frac{x_1}{x_2}\right)^{-1} = \frac{x_2}{x_1} \] which gives the exact Cobb-Douglas MRS: \(\frac{x_2}{x_1} = \frac{p_1}{p_2}\).

Let \(r = \frac{\rho}{\rho-1}\) to simplify exponents:

\[ x_1^* = \frac{p_1^{r-1}\,I}{p_1^r + p_2^r} \qquad x_2^* = \frac{p_2^{r-1}\,I}{p_1^r + p_2^r} \]

5. Marshallian (Ordinary) Demand Function

After solving the utility maximisation problem, the optimal bundle depends on prices and income. This function is called Marshallian demand (also "uncompensated demand"):

\[ x_i^* = x_i^M(p_1, p_2, I) \]

Properties of Marshallian demand

Homogeneous of degree 0 in \((p_1, p_2, I)\): doubling all prices and income leaves demand unchanged.
Satisfies the budget constraint with equality: \(p_1 x_1^* + p_2 x_2^* = I\).
Income and price effects are mixed together (unlike Hicksian demand).

Normal good

\(\partial x_i^M / \partial I > 0\)

Demand rises with income.

Inferior good

\(\partial x_i^M / \partial I < 0\)

Demand falls as income rises (e.g. instant noodles).

6. Indirect Utility Function

Plug the optimal demands back into the utility function. The result tells you the maximum achievable utility given prices and income:

\[ v(p_1, p_2, I) = u\!\left(x_1^M(p_1,p_2,I),\; x_2^M(p_1,p_2,I)\right) \]

Properties of \(v(p,I)\)

Non-increasing in prices: higher prices can't make you better off.
Non-decreasing in income: more money, at least as much utility.
Homogeneous of degree 0 in \((p,I)\): proportional changes cancel.
Quasi-convex in prices
Continuous for \(p\gg0, I>0\).

Example — Cobb-Douglas \(\alpha=\beta=0.5\)

Marshallian demands: \(x_1^*=I/(2p_1)\), \(x_2^*=I/(2p_2)\)

\[ v(p_1,p_2,I) = \left(\frac{I}{2p_1}\right)^{0.5}\!\left(\frac{I}{2p_2}\right)^{0.5} = \frac{I}{2\,p_1^{0.5}p_2^{0.5}} \]

Roy's Identity (bonus — connects indirect utility to Marshallian demand)

\[ x_i^M(p,I) = -\frac{\partial v/\partial p_i}{\partial v/\partial I} \]

Useful shortcut: if you have \(v\), differentiate to recover demand without re-solving.

CES indirect utility

\[ v(\mathbf{p}, I) = I\,(p_1^r + p_2^r)^{-1/r} \qquad r = \frac{\rho}{\rho-1} \]

7. The Expenditure Function

The dual problem: instead of maximising utility given income, minimise expenditure to reach a target utility level \(U\):

\[ \min_{x_1,x_2} \; p_1 x_1 + p_2 x_2 \quad \text{subject to} \quad u(x_1,x_2) \ge U \]

The minimum cost of reaching utility \(U\) at prices \(p\) is the expenditure function:

\[ e(p_1, p_2, U) = \min_{x:\,u(x)\ge U} p\cdot x \]

Properties of \(e(p,U)\)

Non-decreasing in prices (\(\partial e/\partial p_i \ge 0\)).
Homogeneous of degree 1 in \(p\): \(e(tp, U) = t\,e(p,U)\). Double all prices → double required spending.
Concave in prices — consumers can re-optimise when prices change.
Continuous in \(p\) for \(p\gg0\).
Shephard's Lemma: \(\partial e/\partial p_i = h_i(p,U)\) (gives Hicksian demand).

Solving via the Lagrangian (expenditure minimisation)

\[ \mathcal{L} = p_1 x_1 + p_2 x_2 + \mu\,(U - u(x_1,x_2)) \]

FOCs: \(\;p_i = \mu\,\partial u/\partial x_i\). Dividing again gives the same tangency condition \(MRS = p_1/p_2\). So the geometry is identical — the same tangency point solves both problems (at appropriate parameter values).

Example — Cobb-Douglas \(\alpha=\beta=0.5\)

\[ e(p_1,p_2,U) = 2\,p_1^{0.5}\,p_2^{0.5}\,U \]

Derivation: from the tangency condition \(x_2/x_1 = p_1/p_2\), substitute into the utility constraint \(x_1^{0.5}x_2^{0.5}=U\) to get \(x_i^h\), then compute \(p_1 x_1^h + p_2 x_2^h\).

CES expenditure function

\[ e(\mathbf{p}, U) = U\,(p_1^r + p_2^r)^{1/r} \]

8. Hicksian (Compensated) Demand Function

The solution to the expenditure minimisation problem — what bundle achieves utility \(U\) at minimum cost? This is Hicksian demand:

\[ h_i(p_1, p_2, U) = \text{argmin}_{x:\,u(x)\ge U} p\cdot x \]

Why "compensated"?

When prices change, we imagine the consumer is compensated with extra/less income to stay on the same utility level. Hicksian demand shows only the substitution effect of a price change, with no income effect.

Two ways to find Hicksian demand

Method 1 — Shephard's Lemma

\[ h_i(p, U) = \frac{\partial e(p, U)}{\partial p_i} \]

Differentiate the expenditure function with respect to \(p_i\).

Method 2 — Substitute expenditure into Marshallian demand

\[ h_i(p, U) = x_i^M(p,\; e(p, U)) \]

Replace income \(I\) with \(e(p,U)\) in the Marshallian demand.

Example — Cobb-Douglas \(\alpha=\beta=0.5\)

Via Shephard's Lemma on \(e = 2p_1^{0.5}p_2^{0.5}U\):

\[ h_1 = \frac{\partial e}{\partial p_1} = p_1^{-0.5}p_2^{0.5}U \qquad h_2 = p_1^{0.5}p_2^{-0.5}U \]

CES Hicksian demands

\[ x_1^h = \frac{p_1^{r-1}\,U}{(p_1^r+p_2^r)^{1-1/r}} \qquad x_2^h = \frac{p_2^{r-1}\,U}{(p_1^r+p_2^r)^{1-1/r}} \]

Property — Hicksian demand is downward sloping in own price

\(\partial h_i / \partial p_i \le 0\). Always. Unlike Marshallian demand (which can slope up for Giffen goods), Hicksian demand always slopes down because it captures only the substitution effect.

9. Equivalence (Duality)

The utility maximisation and expenditure minimisation problems are duals of each other. They have the same optimal bundle at matching parameters.

The four equivalence relations — memorise these

\[ v(p,\, e(p,U)) = U \] \[ e(p,\, v(p,I)) = I \] \[ x_i^M(p,I) = h_i(p,\, v(p,I)) \] \[ h_i(p,U) = x_i^M(p,\, e(p,U)) \]

The first two just say that \(v\) and \(e\) are inverses of each other (in their second argument). The last two say that Marshallian and Hicksian demands coincide when you feed them consistent parameters.

Slutsky Equation (bonus — comes up in next topics)

\[ \underbrace{\frac{\partial x_i^M}{\partial p_j}}_{\text{total price effect}} = \underbrace{\frac{\partial h_i}{\partial p_j}}_{\text{substitution effect}} - \underbrace{x_j \frac{\partial x_i^M}{\partial I}}_{\text{income effect}} \]

This is the Slutsky decomposition. Useful context even if not in this chapter's exams.

10. Exam Toolkit — How to Solve Any Problem

Step-by-step roadmap for exam questions

Identify the utility function type (C-D, perfect complements, substitutes, quasi-linear, CES) — this tells you which method to use.
Check corner solutions first if relevant (perfect substitutes, quasi-linear).
Set up the Lagrangian for interior solutions and derive FOCs.
Apply tangency condition \(MRS = p_1/p_2\) to get a relation between \(x_1\) and \(x_2\).
Substitute into the budget constraint to solve for \(x_1^*\) and \(x_2^*\) → Marshallian demands.
Indirect utility: substitute \(x_1^*, x_2^*\) into \(u\).
Expenditure function: invert \(v(p,I)=U\) for \(I\) (solve for \(I\) as a function of \(U\)).
Hicksian demands: use Shephard's Lemma \(h_i = \partial e/\partial p_i\), or substitute \(e(p,U)\) for \(I\) in Marshallian demand.
Verify equivalence if asked: check \(h_i(p,U) = x_i^M(p, e(p,U))\).

Quick-reference table

Function	Depends on	Answers the question	How to get it
Marshallian demand \(x^M(p,I)\)	\(p, I\)	"What do I buy given my income?"	Lagrangian or MRS = price ratio
Indirect utility \(v(p,I)\)	\(p, I\)	"How happy can I be?"	Plug \(x^M\) into \(u\)
Expenditure function \(e(p,U)\)	\(p, U\)	"How much do I need to spend to reach utility \(U\)?"	Invert \(v\), or solve expenditure min problem
Hicksian demand \(h(p,U)\)	\(p, U\)	"What's the cheapest way to reach utility \(U\)?"	Shephard's Lemma (\(\partial e/\partial p_i\)) or substitute \(e\) into \(x^M\)

Lump-Sum Principle (policy context)

The key insight

For the same tax revenue, a lump-sum income tax leaves the consumer better off than a per-unit tax on a specific good.

Why: A good-specific tax distorts prices, forcing the consumer away from her preferred bundle. An income tax reduces purchasing power but lets her re-optimise freely → reaches a higher indifference curve for the same government revenue.

Formally: Let good 1 have a tax \(t\), so effective price becomes \(p_1+t\). The consumer now faces distorted prices and achieves some utility \(U_\text{tax}\). An income tax of equal revenue \(T = t \cdot x_1^*(p_1+t, p_2, I)\) would give income \(I-T\) at undistorted prices → utility \(U_\text{lump} \ge U_\text{tax}\).

Summary of Common Utility Functions

Utility function	Type	MRS	Optimal condition	Notes
\(x_1^\alpha x_2^\beta\)	Cobb-Douglas	\(\dfrac{\alpha x_2}{\beta x_1}\)	Tangency; fixed expenditure shares	Always interior solution
\(\min(ax_1, bx_2)\)	Perfect complements	Undefined (kink)	\(ax_1 = bx_2\) at optimum	No tangency — use kink
\(ax_1 + bx_2\)	Perfect substitutes	\(a/b\) (constant)	Compare MRS to \(p_1/p_2\)	Corner solutions typical
\(x_1^{0.5} + x_2\)	Quasi-linear	\(0.5 x_1^{-0.5}\)	Tangency if \(x_2^*>0\)	Check corner in \(x_2\)
\((x_1^\rho+x_2^\rho)^{1/\rho}\)	CES	\((x_2/x_1)^{1-\rho}\)	Tangency; use \(r=\rho/(\rho-1)\)	Generalises all cases

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