We know from Lecture 6 that indifference curves are downward sloping — to stay equally happy, getting more of one good requires giving up some of the other.

The key question: How much of good 2 is the consumer willing to give up to get one more unit of good 1?

Tourism intuition: Imagine you have 1 beach day and 8 city tours. You’d happily trade several city tours for another beach day.

But if you have 7 beach days and 1 city tour, you’d need a lot of beach days to compensate for losing that last city tour!

👉 This is what makes indifference curves convex (bowed toward the origin).

👉 Diminishing MRS reflects the idea that consumers prefer variety.

In Lecture 6, we described preferences using the symbols \(\succ\), \(\sim\), \(\succsim\).

A utility function translates those preferences into numbers.

Important: Utility is ordinal, not cardinal. Only the ranking matters, not the size of the numbers. If \(U(A) = 10\) and \(U(B) = 5\), we know \(A \succ B\), but we cannot say “A is twice as good.”

For this course, we will mostly work with Cobb-Douglas preferences.

Example: \(U(x_1, x_2) = x_1 \cdot x_2\)

Example: For \(U(x_1, x_2) = \sqrt{x_1 \cdot x_2}\):

\[MU_1 = \frac{\partial (\sqrt{x_1 \cdot x_2})}{\partial x_1} = \frac{1}{2}\sqrt{\frac{x_2}{x_1}} \qquad MU_2 = \frac{\partial (\sqrt{x_1 \cdot x_2})}{\partial x_2} = \frac{1}{2}\sqrt{\frac{x_1}{x_2}}\]

👉 Diminishing marginal utility: As you consume more of a good, each additional unit adds less satisfaction (same idea from the ice cream cone example in the Lecture Notes availabe in Canvas.).

There is a powerful connection between the MRS (from indifference curves) and marginal utilities (from the utility function):

Intuition: If good 1 gives you a lot of extra utility (\(MU_1\) is high) and good 2 gives you little (\(MU_2\) is low), you are willing to give up a lot of good 2 for one more unit of good 1 — so MRS is high.

Example: For \(U = x_1 \cdot x_2\) at bundle \((2, 6)\):

\[MRS = \frac{MU_1}{MU_2} = \frac{x_2}{x_1} = \frac{6}{2} = 3\]

👉 The consumer would give up 3 units of good 2 for 1 more unit of good 1.

Derivation (along an indifference curve, utility stays constant):

\[dU = 0\]

\[\frac{\partial U}{\partial x_1} dx_1 + \frac{\partial U}{\partial x_2} dx_2 = 0\]

\[MU_1 \cdot dx_1 + MU_2 \cdot dx_2 = 0\]

Rearranging:

\[-\frac{dx_2}{dx_1} = \frac{MU_1}{MU_2}\]

\[\boxed{MRS = \frac{MU_1}{MU_2}}\]

👉 This formula lets us compute the MRS directly from the utility function — no graph needed!

Now we can put everything together:

In words: Choose the bundle on the highest possible indifference curve that is still affordable (on or below the budget line).

Since more is better (non-satiation), the consumer will always spend all income ➡️ the optimal bundle is on the budget line.

At the optimum, the slope of the IC equals the slope of the budget line:

Interpretation of \(\frac{MU_1}{p_1} = \frac{MU_2}{p_2}\): The marginal utility per euro spent must be equal for both goods. If it is not, the consumer can improve by reallocating spending.

Only when MRS = \(p_1/p_2\) is there no room for improvement — the consumer is at the optimum!

Analogy from Lecture 3: This is exactly the cost-benefit principle applied to marginal reallocation of spending.

The method (2 equations, 2 unknowns):

Equation 1 — Tangency condition:

\[MRS = \frac{p_1}{p_2} \implies \frac{MU_1}{MU_2} = \frac{p_1}{p_2}\]

Equation 2 — Budget constraint:

\[p_1 x_1 + p_2 x_2 = M\]

Steps:

1️⃣ Compute \(MU_1\) and \(MU_2\) from the utility function

2️⃣ Set \(MRS = p_1 / p_2\) and solve for \(x_2\) in terms of \(x_1\) (or vice versa)

3️⃣ Substitute into the budget constraint

4️⃣ Solve for \(x_1^*\) and \(x_2^*\)

Problem: \(U(x_1, x_2) = x_1 \cdot x_2\), with \(p_1 = 20\), \(p_2 = 10\), \(M = 100\).

Step 1: Marginal utilities

\[MU_1 = \frac{\partial(x_1 x_2)}{\partial x_1} = x_2 \qquad MU_2 = \frac{\partial(x_1 x_2)}{\partial x_2} = x_1\]

Step 2: Tangency condition

\[\frac{MU_1}{MU_2} = \frac{p_1}{p_2} \implies \frac{x_2}{x_1} = \frac{20}{10} = 2 \implies x_2 = 2x_1\]

Step 3: Substitute into budget constraint

\[20 x_1 + 10(2x_1) = 100 \implies 20x_1 + 20x_1 = 100 \implies 40x_1 = 100\]

Step 4: Solve

\[\boxed{x_1^* = 2.5 \text{ meals}} \qquad x_2^* = 2(2.5) = \boxed{5 \text{ museums}}\]

\[U^* = 2.5 \times 5 = 12.5\]

At the optimum \((x_1^*, x_2^*) = (2.5, 5)\):

\[\frac{MU_1}{p_1} = \frac{x_2}{p_1} = \frac{5}{20} = 0.25 \text{ utils per euro}\]

\[\frac{MU_2}{p_2} = \frac{x_1}{p_2} = \frac{2.5}{10} = 0.25 \text{ utils per euro}\]

✅ Equal! The last euro spent on meals gives the same additional satisfaction as the last euro spent on museums.

👉 If \(MU_1/p_1 > MU_2/p_2\), the consumer should shift spending toward good 1 (meals give more “bang for the buck”).

👉 If \(MU_1/p_1 < MU_2/p_2\), shift toward good 2.

Problem: \(p_1 = 5\), \(p_2 = 10\), \(M = 200\).

Step 1: \(MU_1 = 0.4 \cdot x_1^{-0.6} \cdot x_2^{0.6} \qquad MU_2 = 0.6 \cdot x_1^{0.4} \cdot x_2^{-0.4}\)

Step 2: Tangency

\[\frac{MU_1}{MU_2} = \frac{0.4 \, x_2^{0.6} \, x_1^{-0.6}}{0.6 \, x_1^{0.4} \, x_2^{-0.4}} = \frac{0.4}{0.6} \cdot \frac{x_2}{x_1} = \frac{2}{3} \cdot \frac{x_2}{x_1}\]

Setting equal to \(p_1/p_2 = 5/10 = 1/2\):

\[\frac{2}{3} \cdot \frac{x_2}{x_1} = \frac{1}{2} \implies x_2 = \frac{3}{4} x_1\]

Step 3: Budget constraint: \(5x_1 + 10 \left(\frac{3}{4}x_1\right) = 200 \implies 5x_1 + 7.5x_1 = 200 \implies 12.5x_1 = 200\)

Step 4: \(\boxed{x_1^* = 16} \qquad x_2^* = \frac{3}{4}(16) = \boxed{12}\)

💡 Shortcut for Cobb-Douglas \(U = x_1^a x_2^b\): spend fraction \(\frac{a}{a+b}\) of income on good 1, and \(\frac{b}{a+b}\) on good 2!

Verify with our previous example: \(a = 0.4\), \(b = 0.6\), \(M = 200\), \(p_1 = 5\), \(p_2 = 10\)

\[x_1^* = \frac{0.4}{1} \cdot \frac{200}{5} = 0.4 \times 40 = 16 \quad \checkmark\]

\[x_2^* = \frac{0.6}{1} \cdot \frac{200}{10} = 0.6 \times 20 = 12 \quad \checkmark\]

👉 This shortcut works for any Cobb-Douglas utility. The exponents determine budget shares.

1️⃣ MRS = rate at which the consumer trades good 2 for good 1 along an IC

2️⃣ Diminishing MRS ➡️ indifference curves are convex ➡️ consumers prefer variety

3️⃣ Utility functions assign numbers to bundles; \(MRS = MU_1/MU_2\)

4️⃣ Optimal choice: highest IC touching the budget line ➡️ tangency condition \(MRS = p_1/p_2\)

5️⃣ Equivalent condition: \(MU_1/p_1 = MU_2/p_2\) (equal marginal utility per euro)

6️⃣ Cobb-Douglas shortcut: spend fraction \(a/(a+b)\) on good 1

Next lecture: Lecture 8 — From individual demand to market demand and linear demand curves.

Question: A consumer has utility \(U = x_1 \cdot x_2\) and currently consumes the bundle \((4, 8)\). The MRS at this point is:

A. 32

B. 4

C. 2

D. 0.5

Question: At the optimal bundle, if \(MU_1/p_1 > MU_2/p_2\), the consumer should:

A. Buy more of good 1 and less of good 2

B. Stay at the current bundle — it is already optimal

C. Buy more of good 2 and less of good 1

D. Increase total spending

A tourist in the Algarve has a daily budget of €120 to split between boat tours (\(x_1\), price €30 each) and restaurant meals (\(x_2\), price €20 each). Their utility function is \(U(x_1, x_2) = x_1^{0.5} \cdot x_2^{0.5}\).

1. Write the budget constraint equation and find the intercepts.

2. Compute \(MU_1\) and \(MU_2\). Derive the MRS.

3. Using the tangency condition (\(MRS = p_1/p_2\)) and the budget constraint, find the optimal bundle \((x_1^*, x_2^*)\).

4. Verify your answer using the Cobb-Douglas shortcut (\(a = b = 0.5\)).

5. Compute the utility at the optimum. Now suppose the tourist’s budget increases to €180 (everything else unchanged). Find the new optimal bundle and new utility. By what percentage did utility increase?