Finding the consumer optimum
A household splits its budget between two goods to get the most utility. At the optimum the highest reachable indifference curve just touches the budget line: the marginal rate of substitution equals the price ratio.
Formulas and rules
- Budget line: p₁·x₁ + p₂·x₂ = B, slope −p₁/p₂
- Optimum: MRS = (∂U/∂x₁) / (∂U/∂x₂) = p₁/p₂
- Cobb-Douglas U = x₁^a · x₂^b: x₁* = a/(a+b) · B/p₁, x₂* = b/(a+b) · B/p₂
- Budget share of good 1 = a/(a+b), whatever the prices and budget
- Check: the optimum uses up the budget exactly
Your turn
Three problems on this topic. Type numbers like 12.5 or 1,250 and elasticities with their sign.
Task
A household with U(x₁, x₂) = x₁ · x₂ and a budget of €256.00 buys at prices p₁ = €4.00 and p₂ = €4.00. Now the price of good 1 falls to €2.00. Find the new quantity of good 1, the change and the effect on demand for good 2.
New quantity x₁
Change in x₁
Demand for good 2
Numbers like 12.5 or 1,250; elasticities with their sign (−0.75); percentages as a number (12.5). Round to 2 decimal places.
Worked example
A household has a budget of €1,683.00 for two goods. Good 1 costs €11.00, good 2 costs €9.00. Find the most of each good it can buy, the slope of the budget line (x₂ against x₁) and how much of good 2 it can buy if it buys 45 units of good 1.
Worked solution
- 1.Budget line: 11·x₁ + 9·x₂ = 1,683
- 2.Only good 1: x₁ = 1,683 / 11 = 153; only good 2: x₂ = 1,683 / 9 = 187
- 3.Solved for x₂: x₂ = 187 − (11/9)·x₁, slope −p₁/p₂ = −1.22
- 4.x₂ = (1,683 − 11 × 45) / 9 = 132
Typical mistake: Giving the slope as −p₂/p₁. The budget line falls by p₁/p₂ units of good 2 per unit of good 1: one more unit of good 1 costs p₁, which buys p₁/p₂ units of good 2.
A household’s utility function is U(x₁, x₂) = x₁² · x₂². Prices are p₁ = €8.00 and p₂ = €8.00, the budget is €2,304.00. Find the optimal bundle and the share of the budget spent on good 1.
Worked solution
- 1.Marginal rate of substitution: MRS = (∂U/∂x₁) / (∂U/∂x₂) = 2x₂ / (2x₁)
- 2.Optimum: MRS = p₁/p₂ → 2x₂ / (2x₁) = 8/8 → 8·x₂ = (2/2) · 8·x₁
- 3.Into the budget line: p₁x₁ · (1 + 2/2) = 2,304 → x₁* = 2/4 × 2,304 / 8 = 144
- 4.x₂* = 2/4 × 2,304 / 8 = 144
- 5.Budget share of good 1 = exponent / sum of exponents = 2/4 = 50.00%
Typical mistake: Building the MRS the wrong way round (marginal utility of good 2 over good 1) and then setting it equal to p₁/p₂. Check: the optimum must use up the budget exactly.
Common questions
Why does demand for good 2 not change when good 1 gets more expensive (Cobb-Douglas)?
The household always spends the same share of its budget on each good. If good 1 gets dearer it buys less of it but spends the same money on it, so the same amount is left for good 2.
How do I build the marginal rate of substitution?
Marginal utility of good 1 divided by marginal utility of good 2. With U = x₁^a · x₂^b most terms cancel and you get a·x₂ / (b·x₁).