Class 6: Choice

Optional textbook reading: Varian, Chapter 5: “Choice.”

Key Ideas

Now we’ll put all the pieces together to form the theory of the consumer: the consumer chooses the most preferred bundle from their budget sets. Class 2 built the budget set, Classes 4 and 5 defined what more and less preferred mean, and Lab 3 taught you to trade your way to the highest curve you can reach. Today the spotlight stays on one family of preferences, Cobb-Douglas ($u = x_1^a x_2^b$): smooth, bowed-in indifference curves where the tangency rule works perfectly. Kinked and straight-line preferences get their turn in Lab 4.

Problems of the Day

Q18: Shopping Is Trading

The food truck from Class 2 is back: tacos cost $2, drinks cost $3, and today two customers, Mika and Ralph, each arrive with $24. Treat both goods as divisible, so an order like 4.8 tacos is allowed.

Draw the budget line. What is the x-intercept, the y-intercept, and the slope? What is the formula to quickly find the slope of any budget line given prices $p_1$ and $p_2$ and income $m$?
Mika (tacos $x_1$, drinks $x_2$, utility $u = x_1 x_2$) plans to spend her whole $24 on the bundle $(3, 6)$. Find her MRS at that bundle. Then argue why the budget line is the same thing as the trade line now. Compare her MRS at that bundle to the slope of the budget line: is her planned order the best order? If not, which direction should she rearrange it?
Mika keeps rearranging her order until no rearrangement helps anymore. Find the bundle where she ends up. State the stopping condition she satisfies there as an equation relating her MRS to the two prices, and then draw the picture: her budget line and the indifference curve through her final bundle. What do the two graphs do at that point? Finally, take notes for next time: write down one sentence you can use to remember how to solve for a consumer’s choices when they have Cobb-Douglas preferences ($u = x_1^a x_2^b$).
Ralph (utility $u = x_1^2 x_2$) also has $24 to spend on tacos and drinks. Make a hypothesis: compared to Mika, will Ralph buy more tacos or more drinks? Then solve for Ralph’s choice.

Q19: The Cobb-Douglas Shortcut

Write a formal proof that if someone’s utility function has the form $u = x_1^a x_2^b$ where $a + b = 1$, they will spend 100 * a % of their income on good 1 and 100 * b % of their income on good 2.

Then, use the Cobb-Douglas shortcut you just found to solve for Mika and Ralph’s choices again.

Answers

Q18: Shopping Is Trading.

The budget line is $2x_1 + 3x_2 = 24$. Spending everything on tacos buys $24/2 = 12$ of them, so the x-intercept is 12; spending everything on drinks buys $24/3 = 8$, so the y-intercept is 8. The slope is $-8/12 = -2/3$: each extra taco costs two thirds of a drink. In general the intercepts are $m/p_1$ and $m/p_2$, and dividing the rise by the run gives the slope $-p_1/p_2$. Income cancels, so the slope depends only on the two prices.
The bundle $(3, 6)$ costs $2 \cdot 3 + 3 \cdot 6 = 24$, so it is on the budget line. Her MRS is $-MU_1/MU_2 = -x_2/x_1 = -6/3 = -2$: she would give up two drinks for one more taco and be exactly as happy.

The budget line is a trade line because a shopper who is already spending everything can only change her order by trading. Putting back one taco frees $2, which buys $2/3$ of a drink, so the market offers drinks for tacos at a rate of $2/3$ drink per taco. That rate is the slope of the budget line, and the budget line is the set of bundles she can trade her way to. In Lab 3 the friends set the rate; today the prices set it.

Compare the two numbers in drinks per taco. Mika values a taco at 2 drinks, and the market charges only $2/3$ of a drink. A taco is worth more to her than it costs, so $(3, 6)$ is not her best order. The trade line is flatter than her indifference curve at $(3, 6)$, so she should rearrange toward more tacos and fewer drinks.

She stops when the two rates agree, that is, when the indifference curve has the same slope as the budget line: \[MRS = -\frac{p_1}{p_2} \quad\Longrightarrow\quad \frac{x_2}{x_1} = \frac{2}{3}.\] This is one equation in two unknowns, and the budget line is the second: $2x_1 + 3x_2 = 24$. Substituting $x_2 = \tfrac{2}{3}x_1$ into the budget gives $2x_1 + 2x_1 = 24$, so $x_1 = 6$ and $x_2 = 4$. The bundle $(6, 4)$ costs $24 and gives utility 24, compared with 18 at $(3, 6)$.

In the picture, the budget line runs from $(0, 8)$ to $(12, 0)$ and the indifference curve $x_1 x_2 = 24$ passes through $(6, 4)$. The two graphs are tangent there: they touch at one point and have the same slope, $-2/3$. Because the curve bows in, every other point on the budget line lies below the curve, so no affordable bundle is on a higher curve.

A sentence for next time: set the MRS equal to the price ratio, then spend the whole budget; two equations, two unknowns. Any wording that names both conditions will do.

A sensible hypothesis is that Ralph buys more tacos than Mika. Tacos enter his utility squared, so he treats a taco as more valuable relative to a drink than Mika does, and the prices they face are the same. To check, compute his MRS: $MU_1 = 2x_1 x_2$ and $MU_2 = x_1^2$, so \[MRS = -\frac{2x_1 x_2}{x_1^2} = -\frac{2x_2}{x_1}.\] Tangency requires $2x_2/x_1 = 2/3$, or $x_2 = x_1/3$. Substituting into the budget line, $2x_1 + 3 \cdot \tfrac{x_1}{3} = 3x_1 = 24$, so $x_1 = 8$ and $x_2 = 8/3$. Ralph buys 8 tacos and $2\tfrac{2}{3}$ drinks: more tacos and fewer drinks than Mika, as predicted. He spends $16 of his $24 on tacos; Mika spent $12.

Q19: The Cobb-Douglas Shortcut.

Claim. If $u = x_1^a x_2^b$ with $a, b > 0$ and $a + b = 1$, then at the consumer’s chosen bundle $p_1 x_1 = a\,m$ and $p_2 x_2 = b\,m$.

Proof. The consumer chooses the bundle on the budget line $p_1 x_1 + p_2 x_2 = m$ that reaches the highest indifference curve. Cobb-Douglas indifference curves are smooth and bowed in, and they never touch the axes, so the best bundle is where an indifference curve is tangent to the budget line: the MRS equals $-p_1/p_2$.

The marginal utilities are $MU_1 = a x_1^{a-1} x_2^b$ and $MU_2 = b x_1^a x_2^{b-1}$. Dividing, \[MRS = -\frac{MU_1}{MU_2} = -\frac{a x_1^{a-1} x_2^{b}}{b x_1^{a} x_2^{b-1}} = -\frac{a x_2}{b x_1}.\] The tangency condition is therefore $\dfrac{a x_2}{b x_1} = \dfrac{p_1}{p_2}$, which rearranges to \[p_2 x_2 = \frac{b}{a}\, p_1 x_1.\] In words, spending on good 2 is a fixed multiple $b/a$ of spending on good 1, whatever the prices. Substitute this into the budget constraint: \[p_1 x_1 + \frac{b}{a}\, p_1 x_1 = m \quad\Longrightarrow\quad p_1 x_1 \cdot \frac{a + b}{a} = m.\] Since $a + b = 1$, this gives $p_1 x_1 = a\,m$. Then $p_2 x_2 = m - a\,m = (1 - a)\,m = b\,m$. The consumer spends the fraction $a$ of income on good 1 and the fraction $b$ on good 2. $\blacksquare$

The shares pin down quantities immediately: $x_1 = a m / p_1$ and $x_2 = b m / p_2$. Notice that the demand for each good depends on its own price and on income, but not on the other good’s price.

Using the shortcut. Mika’s exponents are $a = b = 1$, which sum to 2, not 1. But $u = x_1 x_2$ and $v = \sqrt{u} = x_1^{1/2} x_2^{1/2}$ rank every pair of bundles the same way, since the square root is increasing. The same indifference curves, hence the same choice, so Mika may be treated as $a = b = 1/2$. She spends half of $24 on each good: $12 on tacos buys 6, and $12 on drinks buys 4. That is $(6, 4)$, as in Q18(c).

Ralph’s exponents are 2 and 1, which sum to 3. Taking the cube root gives $x_1^{2/3} x_2^{1/3}$, so $a = 2/3$ and $b = 1/3$. He spends $\tfrac{2}{3} \cdot 24 = 16$ dollars on tacos, buying 8, and $\tfrac{1}{3} \cdot 24 = 8$ dollars on drinks, buying $8/3$. That is $(8, 8/3)$, as in Q18(d), with no tangency algebra required.

Check Your Understanding

The Budget Line

Two Conditions for the Best Bundle

Q18 ended with Mika at a bundle where two things are true at once: her MRS equals $-p_1/p_2$, and $p_1 x_1 + p_2 x_2 = m$. These questions check that you can state each condition, see what each one rules out on its own, and recognize why the best bundle needs both.

The Cobb-Douglas Shortcut

Q18 handed you the most useful trick in consumer theory: a Cobb-Douglas consumer with $u = x_1^a x_2^b$ and $a + b = 1$ spends the share $a$ of her income on good 1 and $b$ on good 2, with no tangency algebra required. When the exponents do not sum to 1, raising $u$ to a positive power fixes that without changing who the consumer is. These questions check both halves: why the rescaling is legitimate, and how to use the shares to jump straight to quantities.