# Utility Maximization

Business I · Microeconomics · https://tryals.app/learn/business-i/utility-maximization

## The Optimal Bundle

The consumer reaches the highest indifference curve on their **budget constraint**: 

$$p_x x + p_y y = M$$

where $p_x, p_y$ are prices and $M$ is income. The **budget line** slope is $-p_x/p_y$.

**The optimal bundle** occurs where the highest indifference curve is **tangent** to the budget line, so $MRS = p_x/p_y$, or:

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$

**Marginal utility per euro is equalised**. If $MU_x/p_x > MU_y/p_y$, buy more $x$.

| Object | Slope | Meaning |
|---|---|---|
| Budget line | $-p_x/p_y$ | Market exchange rate |
| Indiff. curve | $-MRS$ | Consumer exchange rate |
| Optimum | Equal | No trade remains |

**Corner solutions**: if MRS always exceeds the price ratio, buy only one good (perfect substitutes).

**Cobb-Douglas utility** ($U = x^a y^{1-a}$): spend fraction $a$ of income on $x$ and $(1-a)$ on $y$.

> **Pitfall:** Equalising $MU_x = MU_y$ instead of $MU/p$. A good with double the $MU$ is only worth buying if it costs less than twice as much.

## Budget Line Shifting

The **budget set** contains all affordable bundles; rational consumers choose a point on the line itself.

| Change | Budget Line | Reveals |
|---|---|---|
| Income rises | Parallel shift out | Engel curve |
| $p_x$ falls | Pivots out from $y$-intercept | Demand for $x$ |

For a price decrease, the new $x$-intercept is $M / p_x'$. 

**Taxes and subsidies**: A **per-unit tax** on $x$ steepens the line. A **lump-sum tax** shifts it inward (reducing $M$, slope unchanged).

**Insight**: A lump-sum tax leaving the same revenue makes the consumer better off than a per-unit tax because it avoids relative price distortion.

> **Tip:** An income change moves **both** intercepts proportionally (slope fixed); a price change moves **one** intercept only. This decodes any diagram.

## Practice questions

5 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. At the optimum the indifference curve is tangent to the budget line. What does tangency mean in words a shopper would recognise?

A. Both goods are being bought in equal quantities
B. She has spent her entire budget, which is what the optimum requires
C. Her willingness to swap equals the market's rate
D. The two goods give equal marginal utility per unit

**Answer:** C. Her willingness to swap equals the market's rate

**Why:** Slope of the indifference curve is willingness to trade, slope of the budget line is the market's rate, and equality means no profitable rearrangement is left. Spending the whole budget is necessary and not sufficient, every point on the line does that, and only one is optimal.

Page: https://tryals.app/practice/business-i/utility-maximization/at-the-optimum-the-indifference-curve-is-tangent-to-the-budget-line

### 2. If income doubles from 100 to 200 while prices stay fixed, the budget line shifts outward. **Estimate the new $x$-intercept** if $p_x = 5$.

**Answer:** 40 (within ±3)

**Why:** New $x$-intercept $= M/p_x = 200/5 = 40$. The old intercept was 20, it doubles, just like income.

Page: https://tryals.app/practice/business-i/utility-maximization/if-income-doubles-from-100-to-200-while-prices-stay-fixed-the-budget

### 3. Which are true at the optimal consumption bundle?

A. The indifference curve is tangent to the budget line
B. $MU_x / p_x = MU_y / p_y$
C. $MRS = p_x / p_y$
D. The consumer can afford a higher indifference curve

**Answer:** A. The indifference curve is tangent to the budget line; B. $MU_x / p_x = MU_y / p_y$; C. $MRS = p_x / p_y$

**Why:** All three conditions are equivalent descriptions of the optimum. The consumer cannot reach a higher indifference curve, the tangent one is the highest affordable.

Page: https://tryals.app/practice/business-i/utility-maximization/which-are-true-at-the-optimal-consumption-bundle

### 4. Order the steps of utility maximisation.

**Answer:**

1. Write the budget constraint: $p_x x + p_y y = M$
2. Set up the tangency condition: $MRS = p_x / p_y$
3. Solve the two equations (tangency + budget) simultaneously
4. Find the optimal bundle $(x^*, y^*)$

**Why:** Constraint, tangency condition, solve, interpret, the recipe for consumer optimisation.

Page: https://tryals.app/practice/business-i/utility-maximization/order-the-steps-of-utility-maximisation

### 5. At the optimum, $MU_x/p_x = MU_y/p_y$. Why must the ratios be equal rather than the marginal utilities themselves?

A. Because what matters is utility per euro spent
B. Because the budget constraint forces the ratios to be equal automatically
C. Because the goods have different prices, so their utilities cannot be compared
D. Because marginal utility is not measurable in absolute terms

**Answer:** A. Because what matters is utility per euro spent

**Why:** Equalising bang-per-buck is the condition for no profitable reallocation remaining. Comparing raw marginal utilities ignores what each costs to obtain.

Page: https://tryals.app/practice/business-i/utility-maximization/at-the-optimum-mux-px-muy-py-why-must-the-ratios-be-equal-rather