Principles of Economics and Management I

Session 2 · Consumer Choice

Authors

Affiliation

Introduction

🔁 Where We Came From

Session 1 gave the whole logic of a rational choice: surplus \(B - C\), and the opportunity cost that decides between options.

It also drew a budget line once, informally, for coffees and cakes. ☕

Today we write the consumer’s problem down properly and solve it. You already have the mathematics; we are only supplying the economics. 🎯

🗺️ Today’s Map

1. 📜 What we assume about preferences, and what utility is for
2. ⚠️ Why the number utility takes carries no information
3. 💶 The budget set
4. 🧮 \(\max U\) subject to the constraint, straight to the Lagrangian
5. 💡 \(\lambda\) as a shadow price
6. 📉 The solution as a function of prices and income: Marshallian demand

We finish holding a demand function. Next session turns it into a market. 🔧

Part 1 · Preferences and Utility

📜 What We Assume About Preferences

Before any function, four assumptions on the ranking \(\succsim\) over bundles:

Complete: for any two bundles, the consumer can say which they prefer, or that they are indifferent.

Transitive: if \(a \succsim b\) and \(b \succsim c\) then \(a \succsim c\). Without this there is no “best” bundle to find.

Monotone: more is better.

Convex: averages are at least as good as extremes.

🎯 Utility

Given those assumptions, we can represent \(\succsim\) by a utility function \(U(x, y)\).

It is a representation, nothing more: \(U(a) \ge U(b)\) means exactly \(a \succsim b\).

Write \(U_x \equiv \partial U/\partial x\) throughout. Monotonicity is \(U_x > 0\) and \(U_y > 0\); diminishing marginal utility is \(U_{xx} < 0\) and \(U_{yy} < 0\): the tenth coffee is worth less than the first. ☕

Careful with those two: \(U_{xx} < 0\) is a property of the function, while convexity of preferences is a property of its level sets: the set of bundles at least as good as a given one is convex. Only the second is a statement about preferences, for the reason on the next slide. ⚠️

📖 Source: OpenStax Micro 3e, §6.1 “Consumption Choices”, “Total Utility and Diminishing Marginal Utility”.

⚠️ Utility Is Ordinal, Not Cardinal

This point costs one slide and saves a great deal of confusion later.

If \(f\) is strictly increasing, \(U\) and \(f(U)\) describe exactly the same preferences. \(U = xy\) and \(V = \ln x + \ln y\) are the same consumer.

So the number utility takes is meaningless. Only the ranking survives, and with it any ratio like \(U_x/U_y\).

“Ana has 40 utils and Bruno 20, so Ana is twice as happy” is not a statement economics can make. 🚫

🧊 Two Shapes to Keep in Mind

Convexity is an assumption, and two limiting cases break the interior solution we are about to derive:

🔀 Perfect substitutes: \(U = ax + by\). The consumer trades at a fixed rate, and the solution sits at a corner.

🔗 Perfect complements: \(U = \min\{ax, by\}\). Fixed proportions, and the solution sits at the kink, where the derivative does not exist.

In both, the first order conditions we derive next do not hold with equality. Watch for them in the problem sets. ⚠️

📝 Review · Part 1

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 1

\(U = xy\) is replaced by \(V = \ln x + \ln y\). The consumer’s optimal bundle:

A. Does not change at all.

B. Changes, because utility numbers are smaller.

C. Changes only if income changes.

D. Cannot be computed from \(V\).

✅ A. \(V = \ln U\) is a strictly increasing transformation, so it represents the same preferences. Check the ratio: \(V_x/V_y = (1/x)/(1/y) = y/x\), exactly what \(U\) gives.

❓ Multiple Choice 2

For which preferences should you not expect the first order conditions to hold with equality?

A. \(U = x^{0.5}y^{0.5}\)

B. \(U = \ln x + 2\ln y\)

C. \(U = x^{0.3}y^{0.7}\)

D. \(U = \min\{2x,\, y\}\)

✅ D. Perfect complements have a kink, so \(U\) is not differentiable there and the solution sits at the kink rather than where derivatives balance. The other three are smooth.

🧮 Development Exercise

For \(U(x, y) = x^{\alpha} y^{1-\alpha}\), with \(0 < \alpha < 1\):

Show that \(\dfrac{U_x}{U_y} = \dfrac{\alpha}{1-\alpha}\cdot\dfrac{y}{x}\), and say what it depends on.

\(U_x = \alpha x^{\alpha-1} y^{1-\alpha}\) and \(U_y = (1-\alpha) x^{\alpha} y^{-\alpha}\).

Dividing, \(\dfrac{U_x}{U_y} = \dfrac{\alpha}{1-\alpha}\cdot\dfrac{y}{x}\). It depends only on the ratio \(y/x\), not on the level of utility. This is the ratio the first order conditions set against prices. ✅

Part 2 · The Consumer’s Problem

💶 The Budget Set

With income \(M\) and prices \(p_x\), \(p_y\), the consumer can afford any bundle with

\[ p_x\, x + p_y\, y \le M \]

The boundary is the budget constraint, a line of slope \(-p_x/p_y\): the rate at which the market lets you trade \(x\) for \(y\). 📉

Monotone preferences mean the consumer never stops inside the set, so the constraint binds with equality. That is what lets us use a Lagrangian.

📖 Source: OpenStax Micro 3e, §2.1 “How Individuals Make Choices Based on Their Budget Constraint”.

🧮 The Problem, Written Down

Everything so far collapses into one line:

\[\max_{x,\,y} \; U(x, y) \quad \text{subject to} \quad p_x x + p_y y = M\]

A smooth objective, one linear constraint. You have solved this shape many times. With strictly convex preferences the first order conditions are sufficient, and the solution is unique, so there is no second order check to do. 🧠

So we go straight to the Lagrangian, and spend the time on what the solution means rather than on how to find it.

🧮 The Lagrangian

\[\mathcal{L} = U(x, y) + \lambda\,(M - p_x x - p_y y)\]

First order conditions: \[U_x = \lambda p_x, \qquad U_y = \lambda p_y, \qquad p_x x + p_y y = M\]

Three equations, three unknowns \((x, y, \lambda)\). Solving them is the whole problem.

The constraint is written as \((M - p_x x - p_y y)\), so \(\lambda = U_x/p_x > 0\). Write it the other way round and \(\lambda\) changes sign; keep one convention. ⚠️

⚖️ Reading the First Order Conditions

Divide the first two:

\[\frac{U_x}{U_y} = \frac{p_x}{p_y}\]

Rearranged the other way, \(\dfrac{U_x}{p_x} = \dfrac{U_y}{p_y}\): the marginal utility per euro is the same in every direction.

If it were not, move one euro from the low-return good to the high-return one and gain. The condition is exactly “no free improvement left”. 🎯

📖 Source: OpenStax Micro 3e, §6.1, “Choosing with Marginal Utility”, which reaches the same rule arithmetically.

💡 What \(\lambda\) Actually Is

From any first order condition, \(\lambda = U_x/p_x = U_y/p_y\).

Call \(U^{*}(p_x, p_y, M) = U(x^{*}, y^{*})\) the indirect utility function. The envelope theorem differentiates only the explicit \(M\): \[\frac{\partial U^{*}}{\partial M} = \frac{\partial \mathcal{L}}{\partial M} = \lambda > 0\]

\(\lambda\) is the marginal utility of income: what one more euro of budget is worth to this consumer. A shadow price.

Keep it. The same reading returns for the firm in session 4, and for the government’s budget in session 12. 🔁

📝 Review · Part 2

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 3

In \(\mathcal{L} = U(x,y) + \lambda(M - p_x x - p_y y)\), the multiplier \(\lambda\) is:

A. The price ratio \(p_x/p_y\).

B. The marginal utility of income.

C. The consumer’s total utility at the optimum.

D. Always equal to one at an interior optimum.

✅ B. From the first order conditions \(\lambda = U_x/p_x = U_y/p_y\), the marginal utility per euro. By the envelope theorem it is exactly \(\partial U^{*}/\partial M\).

❓ Multiple Choice 4

At the optimum \(U_x/U_y = p_x/p_y\). Suppose instead \(U_x/p_x > U_y/p_y\). The consumer should:

A. Stop: the bundle is already optimal.

B. Buy less of both goods.

C. Move a euro from \(y\) to \(x\), which raises utility.

D. Raise income, the only way to gain.

✅ C. A euro spent on \(x\) buys more utility than a euro spent on \(y\), so shifting spending gains. Only when the two are equal is there nothing left to take.

🧮 Numerical Exercise

\(U(x, y) = x\,y\), with \(p_x = 2\), \(p_y = 1\) and \(M = 20\) euros.

a) Find the optimal bundle.
b) Compute \(\lambda\) and interpret it.
c) Now \(p_x\) falls to 1. Find the new bundle.
d) What share of the budget goes to each good, and why is that no accident?

✅ Solution

a) From the conditions, \(y/x = p_x/p_y = 2\), so \(y = 2x\). In the constraint \(2x + 2x = 20\), giving \(x^{*} = 5\) and \(y^{*} = 10\).

b) \(\lambda = U_x/p_x = y/p_x = 10/2 = 5\). One more euro of income would raise utility by about 5 units.

c) Now \(y/x = 1\), so \(y = x\), and \(x + x = 20\) gives \(x^{*} = y^{*} = 10\). The quantity of \(x\) doubled.

d) \(p_x x^{*} = 10\) and \(p_y y^{*} = 10\): half the budget on each, both times. Cobb-Douglas \(x^{\alpha}y^{1-\alpha}\) always spends the constant shares \(\alpha\) and \(1-\alpha\). ✅

Part 3 · Marshallian Demand

📉 The Solution Is a Function

Solve the three conditions and the answer is not a number but a rule:

\[x^{*} = x(p_x, p_y, M), \qquad y^{*} = y(p_x, p_y, M)\]

Marshallian demand: the optimal quantity, as a function of prices and income.

Every question about how a consumer responds to anything is a question about the derivatives of this function. 🔑

🧮 Cobb-Douglas, Solved

Take \(U = x^{\alpha} y^{1-\alpha}\). The conditions give \(\dfrac{\alpha}{1-\alpha}\cdot\dfrac{y}{x} = \dfrac{p_x}{p_y}\), so \(p_y y = \dfrac{1-\alpha}{\alpha}\,p_x x\).

Substitute into \(p_x x + p_y y = M\): \[x^{*} = \frac{\alpha M}{p_x}, \qquad y^{*} = \frac{(1-\alpha) M}{p_y}\]

Read it off: spending on \(x\) is \(p_x x^{*} = \alpha M\), a constant share of income. That is the Cobb-Douglas signature. 📐

🔎 What the Demand Function Says

📉 Own price: \(\dfrac{\partial x^{*}}{\partial p_x} = -\dfrac{\alpha M}{p_x^{2}} < 0\). Demand slopes down.

💶 Income: \(\dfrac{\partial x^{*}}{\partial M} = \dfrac{\alpha}{p_x} > 0\), a normal good. Where this is negative the good is inferior.

🔀 Other price: here \(\dfrac{\partial x^{*}}{\partial p_y} = 0\). Cobb-Douglas is special; in general this is where substitutes and complements live.

And \(x^{*}\) is unchanged if all prices and income double: demand is homogeneous of degree zero. Only relative prices matter. 🧠

📈 This Is the Demand Curve

Fix \(p_y\) and \(M\), vary \(p_x\), and plot the pairs \((x^{*}, p_x)\).

Not an assumption: the solution to the problem we just solved, drawn. 🎯

📖 Source: OpenStax Micro 3e, §6.2, “The Foundations of Demand Curves”.

📝 Review · Part 3

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 5

All prices and income double. The consumer’s optimal bundle:

A. Doubles, since income doubled.

B. Halves, since everything is dearer.

C. Is unchanged: demand is homogeneous of degree zero.

D. Cannot be determined without knowing \(U\).

✅ C. The budget constraint is unchanged when both sides scale, and so is the price ratio. Only relative prices and real income matter, which is why money illusion is a mistake and not a preference.

❓ Multiple Choice 6

For \(U = x^{\alpha}y^{1-\alpha}\), income rises 10% with prices fixed. Spending on \(x\):

A. Rises by less than 10%, since some goes to \(y\).

B. Rises by exactly 10%.

C. Is unchanged.

D. Depends on \(p_x\) and \(p_y\).

✅ B. Spending on \(x\) is \(\alpha M\), so it moves one for one with income. The share \(\alpha\) stays fixed, which is what makes Cobb-Douglas convenient and restrictive at once.

🧮 Numerical Exercise

\(U(x,y) = x^{0.25} y^{0.75}\), with income \(M\) and prices \(p_x\), \(p_y\).

a) Derive the Marshallian demands \(x^{*}\) and \(y^{*}\).
b) What share of income goes to each good?
c) Is \(x\) normal or inferior? Show it.
d) With \(M = 100\), \(p_x = 5\), \(p_y = 3\), compute the bundle.

✅ Solution

a) \(x^{*} = \dfrac{0.25\,M}{p_x}\) and \(y^{*} = \dfrac{0.75\,M}{p_y}\), the Cobb-Douglas rule with \(\alpha = 0.25\).

b) \(p_x x^{*} = 0.25M\) and \(p_y y^{*} = 0.75M\): a quarter and three quarters, whatever the prices.

c) \(\partial x^{*}/\partial M = 0.25/p_x > 0\), so \(x\) is normal. Cobb-Douglas cannot produce an inferior good.

d) \(x^{*} = 25/5 = 5\) and \(y^{*} = 75/3 = 25\). Check: \(5(5) + 3(25) = 100\). ✅

Wrap-Up

🎯 What to Take From This Session

📜 Preferences come first. Utility is only a way of writing a ranking down, and any increasing transformation of it is the same consumer.

🧮 The problem is \(\max U\) subject to a linear constraint. The Lagrangian solves it, and the first order conditions say marginal utility per euro is equal everywhere.

💡 \(\lambda\) is the marginal utility of income, a shadow price. It returns for the firm and for the government.

📉 The solution is a function, the Marshallian demand \(x(p_x, p_y, M)\). The demand curve is that function drawn.

⚠️ Corners and kinks break the first order conditions. Perfect substitutes and perfect complements are the two you will meet.

👋 Next Session

Demand and Elasticity.

From one consumer to a market, and then the question that decides revenue: how sharply does quantity respond to price? 📊