The consumer's problem

A preference ranking on its own chooses nothing, because more is always better and the best bundle is infinitely far away.

What stops it is that bundles have to be paid for. Put a budget in front of the ranking from the previous lesson and the problem becomes exactly the constrained maximisation the first lesson promised: an objective function, a constraint, and a first-order condition where they meet. This lesson solves it in general and then in enough special cases to show where the general solution stops applying.

The budget set

A consumer with income $m$ facing prices $p_{x}$ and $p_{y}$ can afford any bundle satisfying

p_{x} x + p_{y} y \leq m

That inequality defines the budget set, a triangle with its right angle at the origin, and its hypotenuse is the budget line where the whole income is spent. The intercepts are $m / p_{x}$ and $m / p_{y}$ , the most of each good that could be bought alone, and rearranging to $y = m / p_{y} - (p_{x} / p_{y}) x$ shows the slope is $- p_{x} / p_{y}$ .

That slope is the important object, and it is worth naming what it is. It is the relative price, the rate at which the market will convert $x$ into $y$ . The previous lesson produced a rate at which the consumer will convert $x$ into $y$ , the marginal rate of substitution. Two exchange rates for the same trade is the whole problem in one sentence: if they differ, there is a trade worth making.

Since monotonicity says more is better, the consumer spends everything, so the inequality holds with equality at the optimum and the budget line is where the search happens. That is a modelling decision with a cost attached, since it defines saving out of existence. The repair is to read $y$ as consumption in a later period, which turns the budget line into an intertemporal one with an interest rate in its slope, and nothing else changes.

Two comparative statics come free. Raising $m$ shifts the line outward without tilting it, since neither intercept ratio changes. Raising $p_{x}$ pivots the line inward about the $y$ intercept, which stays put because a consumer buying no $x$ is unaffected by its price. And multiplying $p_{x}$ , $p_{y}$ and $m$ all by the same factor leaves the budget set identical: demand is homogeneous of degree zero, so only relative prices and real income matter. Pure inflation, in this model, does nothing at all, which is a sharp and testable claim and one reason the model is worth taking seriously.

The tangency condition

Maximise $u (x, y)$ subject to $p_{x} x + p_{y} y = m$ . The direct method is substitution: solve the constraint for $y = (m - p_{x} x) / p_{y}$ , put it in the objective, and differentiate with respect to $x$ alone. The chain rule gives

\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} \cdot (- \frac{p_{x}}{p_{y}}) = 0

and rearranging,

MRS = \frac{{MU}_{x}}{{MU}_{y}} = \frac{p_{x}}{p_{y}}

The consumer's rate of exchange equals the market's. Geometrically the indifference curve is tangent to the budget line, which is where the argument becomes obvious: if the curve cut the line rather than touching it, points on the line to one side would lie on a higher curve, so the crossing point could not have been the best.

The condition is worth reading as an arbitrage argument, because that is what it is. Suppose the MRS is 3 while $p_{x} / p_{y}$ is 2. The consumer will give up three units of $y$ for one more $x$ , and the market only asks two. Buying one more $x$ and paying for it with two $y$ leaves them strictly better off, and this stays true until convexity has pushed the MRS down to 2. No gap between the two rates can survive at an optimum, which is the same "equalise at the margin" logic from the first lesson with prices attached.

The same result from Lagrange

Substitution gets clumsy with three goods and impossible with twenty, so the standard route is a Lagrange multiplier. Form

L = u (x, y) + λ (m - p_{x} x - p_{y} y)

and set the partial derivatives to zero. From $x$ : ${MU}_{x} = λ p_{x}$ . From $y$ : ${MU}_{y} = λ p_{y}$ . From $λ$ : the budget constraint. Dividing the first two recovers the tangency condition, so nothing new has been proved, but the intermediate form is more informative than the ratio:

\frac{{MU}_{x}}{p_{x}} = \frac{{MU}_{y}}{p_{y}} = λ

The marginal utility per pound spent is the same on every good. Hermann Heinrich Gossen stated this in 1854, before the machinery existed to derive it, and the reasoning is the allocation argument from the first lesson: if the last pound spent on $x$ bought more than the last pound spent on $y$ , move a pound. The multiplier $λ$ is that common value, the marginal utility of income, and it inherits utility's arbitrariness: relabel $u$ and $λ$ changes with it. The ratios do not.

Demand functions

Solving the two conditions together gives the quantities as functions of prices and income, which is what a demand function is. The Cobb-Douglas case is worth doing once in general because it recurs throughout the course.

Take $u = x^{a} y^{b}$ . The marginal utilities are ${MU}_{x} = a x^{a - 1} y^{b}$ and ${MU}_{y} = b x^{a} y^{b - 1}$ , so the MRS is $(a / b) (y / x)$ . Setting that equal to $p_{x} / p_{y}$ gives $p_{y} y = (b / a) p_{x} x$ : the two expenditures are in a fixed ratio, whatever the prices are. Substituting into the budget constraint, $p_{x} x (1 + b / a) = m$ , so

x^{*} = \frac{a}{a + b} \cdot \frac{m}{p_{x}}, y^{*} = \frac{b}{a + b} \cdot \frac{m}{p_{y}}

Cobb-Douglas preferences spend constant shares of income on each good, $a / (a + b)$ on $x$ regardless of every price. That is a strong prediction and a convenient one, and it is also the functional form's main weakness: real budget shares move with income and with prices, so Cobb-Douglas is a first approximation rather than a description.

Example. A consumer has $u = x^{0.4} y^{0.6}$ , income £120, $p_{x} = £ 4$ and $p_{y} = £ 3$ . Find the optimal bundle and verify the tangency condition.

The shares are 0.4 and 0.6, so $x^{*} = 0.4 \times 120 / 4 = 12$ and $y^{*} = 0.6 \times 120 / 3 = 24$ . Spending checks out: $4 \times 12 + 3 \times 24 = 48 + 72 = £ 120$ . The MRS at that bundle is $(0.4 / 0.6) (24 / 12) = 4 / 3$ , and the price ratio is $4 / 3$ . The consumer will trade 4 units of $y$ for 3 of $x$ , and so will the market.

Now you. A consumer has $u = x^{0.25} y^{0.75}$ , income £200, $p_{x} = £ 5$ and $p_{y} = £ 2$ . Find the bundle and check the tangency.

Answer

$x^{*} = 0.25 \times 200 / 5 = 10$ and $y^{*} = 0.75 \times 200 / 2 = 75$ , costing $50 + 150 = £ 200$ . The MRS is $(0.25 / 0.75) (75 / 10) = 2.5$ , matching $p_{x} / p_{y} = 5 / 2$ .

Notice one feature of these demand functions that will matter in the next lesson: $x^{*}$ does not contain $p_{y}$ at all. A Cobb-Douglas consumer does not change their spending on $x$ when the price of $y$ moves, which is a special property of this functional form and emphatically not a general result.

When the price of a good is not the only thing that changes

Quasilinear utility, $u = g (x) + y$ , gives a demand function of a different shape and is worth a worked case because the surplus arguments later in this course lean on it. Its MRS is $g' (x)$ , which depends on $x$ alone.

Example. A consumer has $u = 10 \sqrt{x} + y$ , with $p_{x} = £ 2$ , $p_{y} = £ 1$ and income £50. Find the optimum.

The MRS is $d (10 \sqrt{x}) / d x = 5 / \sqrt{x}$ . Setting it equal to $p_{x} / p_{y} = 2$ gives $\sqrt{x} = 2.5$ , so $x^{*} = 6.25$ . That costs £12.50, leaving $y^{*} = 37.50$ , and total utility is $10 \times 2.5 + 37.5 = 62.5$ . The demand for $x$ came out of the tangency alone, with no reference to income: raise the income to £80 and $x^{*}$ is still 6.25, with every extra pound going to $y$ .

Now you. Same consumer and same prices, but income falls to £10. What now?

Answer

The tangency still says $x = 6.25$ , which costs £12.50 and cannot be afforded. The answer is the corner: spend everything on $x$ , giving $x^{*} = 5$ and $y^{*} = 0$ , with utility $10 \sqrt{5} = 22.36$ . Checking a nearby interior point, $x = 4$ and $y = 2$ gives $20 + 2 = 22$ , which is worse. Income independence held only while the interior solution was affordable.

Corners, and why tangency can be the wrong answer

The first-order condition assumes an interior optimum. Three situations break that, and all three are common enough to recognise on sight.

Perfect substitutes. With $u = a x + b y$ the MRS is the constant $a / b$ , so it equals $p_{x} / p_{y}$ only by coincidence. Otherwise the consumer spends everything on one good, whichever delivers more utility per pound, and there is no tangency to find.

Example. A consumer has $u = 3 x + 2 y$ with $p_{x} = £ 2$ , $p_{y} = £ 1$ and income £60. What do they buy?

Compare utility per pound: $x$ gives $3 / 2 = 1.5$ and $y$ gives $2 / 1 = 2$ . Good $y$ wins, so $y^{*} = 60$ and $x^{*} = 0$ , for utility 120. The all- $x$ alternative buys 30 units for utility 90, which is worse. A small change in $p_{x}$ moves nothing at all, and if $p_{x}$ ever falls below $£ 1.50$ the consumer switches the entire budget across at once. Perfect substitutes give demand curves that are flat, then vertical, with no smooth response anywhere.

Now you. Same utility function, but now $p_{x} = p_{y} = £ 1$ and income £60.

Answer

Utility per pound is 3 for $x$ and 2 for $y$ , so the consumer buys $x^{*} = 60$ and $y^{*} = 0$ , for utility 180. The switch happened because the price of $x$ crossed the threshold $p_{y} \times a / b = 1.5$ .

Perfect complements. With $u = \min (a x, b y)$ there is no MRS at the kink, since the indifference curve has no derivative there. The optimum is always at the corner of the L, so $a x = b y$ , and combining that with the budget constraint solves the problem with no calculus at all.

Non-convex preferences. If an indifference curve bows the wrong way, the tangency point is a minimum rather than a maximum along the budget line, and the true optimum is at an end. This is why convexity was assumed in the previous lesson: it is what makes the first-order condition sufficient rather than merely necessary.

What the model can and cannot be asked

Three things are worth being clear about before demand curves are built out of this machinery.

The consumer never appears. Nothing in the derivation required deliberation, arithmetic or awareness of the word "utility". The claim is that choices look as if they maximise a ranking, which is a claim about the pattern of choices and not about what happens in anyone's head. That is a much weaker and much more defensible claim than it first sounds, and it is the one being made.

Prices are taken as given. The consumer here cannot haggle, and the whole apparatus is a description of someone facing a posted price. Where the buyer is large enough to move the price, this is the wrong model and the bargaining lessons later are the right one.

The framework does deliver testable predictions, and one is worth stating now because it is used in policy constantly. An in-kind transfer of a good and a cash transfer of the same value are equivalent for any consumer who was already buying more than the transferred amount, since the budget set is identical over the relevant region. That is why United States experiments in the late 1980s that paid some food stamp recipients in cash instead found only small changes in food spending: most recipients were spending more on food than the benefit was worth, so for them the two transfers were the same budget set wearing different labels.

The bundle is now pinned down at one set of prices. What the model has not yet done is say how it moves when a price changes, which is where the demand curve comes from and where the one genuinely surprising result in consumer theory is hiding.