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 showed that a slope can be an opportunity cost. That was the \(MRT\) of the production frontier, for a whole economy. 📐

Today the same reading returns on the consumer’s side, and we write the 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. 📈 Indifference curves, and what their slope is
4. 💶 The budget set, and what moves it
5. 🧮 \(\max U\) subject to the constraint, straight to the Lagrangian
6. 💡 \(\lambda\) as a shadow price
7. 📉 The solution as a function of prices and income: Marshallian demand
8. ➕ A second functional form, and why the shape decides the answers

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.

📜 What We Assume About Preferences

Complete and transitive are what let us speak of a best bundle at all.

Completeness rules out “I cannot say”. Put any two bundles in front of the consumer and there is an answer.

Transitivity rules out cycles. Suppose \(a \succ b\), \(b \succ c\), and \(c \succ a\). 🔁

Then I sell you \(a\) for your \(c\) plus a cent, then \(b\) for your \(a\) plus a cent, then \(c\) for your \(b\) plus a cent. You are holding what you started with, three cents lighter, and I can run it again. 💸

A consumer with intransitive preferences is a machine for making me rich. So we assume it away.

📜 What We Assume About Preferences

Monotone and convex are about shape, and both can fail.

Monotone says more is better. It fails for a bad: nobody wants more noise, more pollution, more risk. 🔊

It also fails at satiation. The fifteenth coffee may be worth less than nothing. ☕

Convex says averages are at least as good as extremes. Half a coffee and half a cake beats two coffees or two cakes, for most people.

It fails when two goods do not mix. Half a portion of rice with half a portion of breakfast cereal is worse than a full portion of either. 🍚

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. 🚫

⚠️ Utility Is Ordinal, Not Cardinal

Take \(U = xy\) and \(V = \ln x + \ln y\), and three bundles:

Bundle	\(U = xy\)	\(V = \ln x + \ln y\)
\((1, 4)\)	4	1.386
\((2, 2)\)	4	1.386
\((3, 3)\)	9	2.197

The numbers are different. The ranking is identical, and the first two bundles are still tied.

And the ratio survives: \(V_x/V_y = (1/x)/(1/y) = y/x\), which is what \(U\) gives. Everything we are about to do runs on that ratio. 🔁

📈 Indifference Curves

If only the ranking survives, draw the ranking. Fix a number and collect every bundle that delivers it.

With \(U = xy\), the bundles worth exactly 4:

Every bundle on the curve is worth the same to this consumer. That is what indifferent means.

📈 Indifference Curves

Change the number and you get another curve. Do it repeatedly and you have the map:

Curves further out are better, because more is better. They never cross: one bundle cannot sit on two different rankings. And \(V = \ln x + \ln y\) draws these same curves, with different numbers on them. 🔁

📐 The Slope of an Indifference Curve

Along one curve, \(U\) does not change. Take the total differential and set it to zero:

\[U_x\,dx + U_y\,dy = 0 \qquad \Longrightarrow \qquad \frac{dy}{dx} = -\frac{U_x}{U_y}\]

So the ratio \(U_x/U_y\) is the slope of the curve you just drew, up to the minus sign. 📐

It answers one question: how much \(y\) will this consumer give up for one more \(x\), and stay exactly as well off?

Hold that question. In Part 2 the market answers the same one, with \(p_x/p_y\). 🔁

🧊 Two Shapes to Keep in Mind

Two limiting cases are convex, but only just:

🔀 Perfect substitutes: \(U = ax + by\). Here \(U_x/U_y = a/b\) at every bundle, so the consumer gives up \(y\) for \(x\) at the same exchange rate whatever they already hold.

🔗 Perfect complements: \(U = \min\{ax, by\}\). The goods are used in fixed proportions, and more \(x\) without more \(y\) adds nothing. At the corner of each curve, \(U_x/U_y\) does not exist.

We come back to both in Part 2, once there are prices. ⚠️

🧊 Two Shapes to Keep in Mind

Straight lines on the left: the slope is \(-a/b\) everywhere, so the trade-off never changes along a curve.

Right angles on the right: starting from the corner, extra \(x\) alone or extra \(y\) alone leaves the consumer on the same curve. There is no slope at the corner at all. 📐

❓ Multiple Choice 1

\(U = xy\) is replaced by \(V = \ln x + \ln y\). The indifference curves of \(V\):

A. Are the same curves as those of \(U\), with different numbers on them.

B. Are different, because the utility numbers are smaller.

C. Are straight lines, because \(V\) is a sum.

D. Cannot be drawn where \(V\) is negative, that is where \(xy < 1\).

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

💬 Discussion Question

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

Show that \(\tfrac{U_x}{U_y} = \tfrac{\alpha}{1-\alpha}\cdot\tfrac{y}{x}\), 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, \(\tfrac{U_x}{U_y} = \tfrac{\alpha}{1-\alpha}\cdot\tfrac{y}{x}\). It depends only on the ratio \(y/x\), not on the level of utility. ✅

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 \(y\). 📉

That rate is an opportunity cost, in the session 1 sense: one more \(x\) costs \(p_x/p_y\) units of \(y\). The \(MRT\) said what the economy gives up; \(p_x/p_y\) says what the consumer does. 🔁

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

💶 The Budget Set

With \(M = 100\), \(p_x = 10\) and \(p_y = 5\): at most 10 of \(x\), at most 20 of \(y\), and the line between them has slope \(-p_x/p_y = -2\).

💶 The Budget Set

Raise income to 150, prices unchanged:

The line moves out, parallel. No price changed, so the rate at which the market lets you swap \(x\) for \(y\) is what it was. You can simply afford more of both.

💶 The Budget Set

Now hold income at 100 and let \(p_x\) fall from 10 to 5:

The line rotates about the \(y\) intercept. Spending everything on \(y\) buys exactly what it bought before, and the slope went from \(-2\) to \(-1\): \(x\) is now cheaper in terms of \(y\).

📖 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 Solution, Drawn

The best affordable bundle is where an indifference curve just touches the budget line.

🎯 The Solution, Drawn

The two grey bundles in that figure are affordable too, and both sit on a lower curve.

Between them the consumer can do better, and the best they can do is the touching point. 🎯

At that point the two slopes agree: \[-\frac{U_x}{U_y} = -\frac{p_x}{p_y}\]

The rate the consumer is willing to swap at equals the rate the market swaps at. Which is the equation we are about to get out of the Lagrangian. 🔁

🧮 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 Lagrangian

Do it once with numbers. \(U = xy\), \(p_x = 10\), \(p_y = 5\), \(M = 100\):

\[\mathcal{L} = xy + \lambda\,(100 - 10x - 5y)\]

\[\mathcal{L}_x = y - 10\lambda = 0, \qquad \mathcal{L}_y = x - 5\lambda = 0, \qquad 100 - 10x - 5y = 0\]

From the first two, \(y = 10\lambda\) and \(x = 5\lambda\). Put them in the third: \(100 = 50\lambda + 50\lambda\), so \(\lambda = 1\).

Then \(x^{*} = 5\) and \(y^{*} = 10\), costing \(10(5) + 5(10) = 100\). ✅

Which is the bundle the figure two slides back was pointing at. 🎯

⚖️ Reading the First Order Conditions

Divide the first two:

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

Rearranged the other way, \(\tfrac{U_x}{p_x} = \tfrac{U_y}{p_y}\): 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”.

⚖️ Reading the First Order Conditions

Suppose it fails. At some affordable bundle, \(U_x/p_x = 3\) and \(U_y/p_y = 1\).

Move one euro off \(y\) and onto \(x\). You lose 1 unit of utility and gain 3. Net: +2. 📈

So that bundle was not the best one. Keep moving euros while the two sides differ.

As \(x\) rises \(U_x\) falls, and as \(y\) falls \(U_y\) rises, so the two ratios close on each other. They meet exactly where the first order conditions hold, and there is nothing left to gain. ✅

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

🧊 Two Shapes to Keep in Mind

Back to the two shapes from Part 1, now with \(p_x = 10\), \(p_y = 5\) and \(M = 100\):

Neither optimum is a point where a curve touches the budget line with the same slope.

🧊 Two Shapes to Keep in Mind

Run the euro-moving argument on \(U = x + y\):

\(U_x/p_x = 1/10\) and \(U_y/p_y = 1/5\), at every bundle. Moving a euro from \(x\) to \(y\) always gains.

The two ratios never meet, so the consumer keeps moving euros until there is no \(x\) left: the corner \(x^{*} = 0\), \(y^{*} = 20\). The first order conditions do not hold with equality there.

For \(U = \min\{x, y\}\), \(U_x/U_y\) does not exist at the kink, so there is nothing to set equal to \(p_x/p_y\). The optimum comes from the kink itself, \(x = y\), and the budget: \(15x = 100\), so \(x^{*} = y^{*} = 20/3\).

Watch for both in the problem sets. ⚠️

💡 What \(\lambda\) Actually Is

The optimum depends on what the consumer faces. Change \(p_x\), \(p_y\) or \(M\) and the conditions give a different \((x^{*}, y^{*})\).

So the utility reached at the optimum is also a function of prices and income: \[U^{*}(p_x, p_y, M) = U\big(x^{*}(p_x, p_y, M),\; y^{*}(p_x, p_y, M)\big)\]

Indirect utility: the utility of the best affordable bundle, as a function of prices and income.

In optimization terms, \(U^{*}\) is the value function of the consumer’s problem: the maximized objective, as a function of the parameters \((p_x, p_y, M)\).

With \(U = xy\), \(p_x = 10\), \(p_y = 5\) and \(M = 100\) we found \(x^{*} = 5\) and \(y^{*} = 10\), so \(U^{*} = 50\). Part 3 looks at \(x^{*}(\cdot)\) itself; for \(\lambda\) we only need \(U^{*}\).

💡 What \(\lambda\) Actually Is

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

By the envelope theorem, the derivative of \(U^{*}\) in \(M\) is the derivative of \(\mathcal{L}\) in the explicit \(M\) only, at the optimum: \[\frac{\partial U^{*}}{\partial M} = \frac{\partial \mathcal{L}}{\partial M} = \lambda > 0\]

\(\lambda\) 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. 🔁

💡 What \(\lambda\) Actually Is

Back to \(U = xy\), \(p_x = 10\), \(p_y = 5\), \(M = 100\), where we found \(\lambda = 1\) and \(U^{*} = 5 \times 10 = 50\).

Raise income to \(M = 101\) and redo it: \(x^{*} = 5.05\), \(y^{*} = 10.1\), so \(U^{*} = 51.005\).

The extra euro bought 1.005 units of utility, against a predicted \(\lambda = 1\). The gap is the second order term: \(\lambda\) is a derivative, so it is exact only in the limit. 📐

Now instead raise \(p_x\) to 20, at \(M = 100\). Then \(x^{*} = 2.5\), \(y^{*} = 10\), and \(\lambda = U_x/p_x = 10/20 = 0.5\).

The same euro of income is worth half as much, because the bundle it buys is worse. That is what makes \(\lambda\) a price. 💶

❓ Multiple Choice 2

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 3

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.

❓ Multiple Choice 4

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.

🧮 Numerical Question

\(U(x, y) = x\,y\), 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 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 \(\tfrac{\alpha}{1-\alpha}\cdot\tfrac{y}{x} = \tfrac{p_x}{p_y}\), so \(p_y y = \tfrac{1-\alpha}{\alpha}\,p_x x\).

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

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

🔀 Other price: here \(\tfrac{\partial x^{*}}{\partial p_y} = 0\). 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 \((x^{*}, p_x)\).

The solution to the problem we just solved, drawn.

📈 This Is the Demand Curve

Where it comes from: three prices, three budget lines, three optima. 👇

Read down: each optimum above hands one point to the panel below. Join them and that is the demand curve.

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

➕ A Second Shape: Quasi-Linear

Cobb-Douglas is one functional form, and a very particular one. Here is another:

\[U(x, y) = v(x) + y\]

Utility is linear in \(y\) and curved only in \(x\). Read \(y\) as “everything else”, measured in euros. 💶

Take \(v(x) = 2\sqrt{x}\), so \(U_x = 1/\sqrt{x}\) and \(U_y = 1\).

The slope of an indifference curve is then \(-U_x/U_y = -1/\sqrt{x}\). It depends on \(x\) alone, and not at all on \(y\). 📐

➕ A Second Shape: Quasi-Linear

Set that slope against the price ratio, with \(p_y = 1\):

\[\frac{1}{\sqrt{x}} = p_x \qquad \Longrightarrow \qquad x^{*} = \frac{1}{p_x^{2}}\]

Look at what is missing: \(M\) does not appear. 🔎

Demand for \(x\) does not respond to income at all. Every extra euro goes to \(y\), and \(y^{*} = M - 1/p_x\) is where it went.

That holds only while \(M\) is large enough to pay for \(x^{*}\). Below that the consumer is at a corner, and the first order conditions stop holding with equality. ⚠️

➕ A Second Shape: Quasi-Linear

One consumer, \(p_x = p_y = 1\), two incomes. Blue: \(M = 2.5\). Red: \(M = 0.5\). Dashed lines are the budgets.

➕ A Second Shape: Quasi-Linear

Whatever \(M\) is, the first order condition asks for \(x^{*} = 1/p_x^{2} = 1\), which costs 1.

A: with \(M = 2.5\) that is affordable. The curve touches the budget line and the rest goes to \(y\), \(y^{*} = 1.5\).

B: with \(M = 0.5\) it is not. So all of it goes to \(x\): \(x^{*} = 0.5\), \(y^{*} = 0\). There the curve is steeper than the budget line, \(1/\sqrt{0.5} \approx 1.41 > 1\), so the consumer would still give up \(y\) for more \(x\)… but there is no \(y\) left.

➕ A Second Shape: Quasi-Linear

Every curve is the same curve shifted vertically. So the slope at a given \(x\) is identical on all of them, which is exactly why \(M\) dropped out.

It matters in session 3: with quasi-linear utility the area under the demand curve is the consumer’s gain in euros. With Cobb-Douglas it is an approximation. 🔁

🔎 Two Forms, Side by Side

	Cobb-Douglas \(x^{\alpha}y^{1-\alpha}\)	Quasi-linear \(2\sqrt{x} + y\)
\(x^{*}\)	\(\alpha M / p_x\)	\(1/p_x^{2}\)
Responds to income	one for one in spending	not at all
Share of income on \(x\)	fixed at \(\alpha\)	falls as \(M\) rises
Effect of \(p_y\)	none	none

Neither one is “the” consumer. Each is a guess about shape, and the guess decides the answers before any data arrives. Session 3 makes the same point again, with elasticities. 🔁

❓ 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 Question

\(U(x,y) = x^{0.25} y^{0.75}\), 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) \(x^{*} = \tfrac{0.25\,M}{p_x}\) and \(y^{*} = \tfrac{0.75\,M}{p_y}\), 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

📜 Utility is a way of writing a preference ranking down. 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.

➕ Cobb-Douglas and quasi-linear answer the same problem differently. The functional form is an assumption, not a finding.

👋 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? 📊