The consumer optimum: equal bang per buck

Consumer choice
How does a customer split a budget between my product and the alternatives?

Learning goals

After this page, you can:

  • Explain “equal bang per buck”: at the best bundle, the last dollar spent on each good adds the same satisfaction.
  • Find the best bundle from a short marginal-utility table and two prices.
  • Explain why a price cut shifts spending toward the cheaper good.

The idea

You stand at the food truck with $30. The first smoothie is worth a lot to you. The third one is worth much less. The same is true for burritos. So you keep buying whichever gives you more satisfaction per dollar — until the money is gone. That is the whole theory on this page.

Three plain facts about preferences.

  1. More is better. Given the choice, people take more of a good thing.
  2. Variety is valued. The fifth burrito adds less satisfaction than the first. Economists call the extra satisfaction from one more unit its marginal utility (MU). It falls as you buy more: this is diminishing marginal utility.
  3. People trade off. To get more of one good inside a budget, you give up some of another.

Bang per buck. Divide the marginal utility of a good by its price: MU ÷ P. This is the satisfaction you get from the next dollar spent on that good — its “bang per buck.”

The rule. Spend each next dollar on the good with the higher bang per buck. As you buy more of a good, its MU falls, so its bang per buck falls too. You stop moving money when the two are equal. At the best bundle:

  • the whole budget is spent, and
  • the last dollar on each good adds the same satisfaction.

If one good gives 5 per dollar and the other gives 3, you can do better: move a dollar from the 3 to the 5. Keep moving until the numbers meet.

What a price cut does. When the price of burritos falls, burrito MU ÷ P rises. Burritos now win more of the “next dollar” contests. The customer buys more burritos. This is the demand curve, seen from inside the customer’s head.

Substitutes and complements from the inside. For substitutes (two coffee brands), a cheaper rival pulls dollars away from you. For complements (burger and fries), a cheaper burger raises the value of buying fries too.

What this means for a seller. To win more of the customer’s budget, you can lower the price, raise the perceived MU (quality, brand, convenience), or do both.

Key rule

\[ \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} \quad \text{(with the whole budget spent)} \] If MUX/PX > MUY/PY: move dollars toward X.

Try it

The table shows the satisfaction from each burrito and each smoothie. Income is $45. Burritos cost $9 and smoothies $6.

  1. Click “Spend on the next unit” a few times. Which good does each dollar go to, and why?
  2. Click “Spend the whole budget.” What is the final bundle?
  3. Click the price-cut button. How does the bundle change?

Worked example

Problem. Income $45. Burritos $9, smoothies $6. Marginal utilities:

Unit 1 2 3 4 5
MU of burritos 36 30 24 18 12
MU of smoothies 24 20 16 12 8

Step 1. Bang per buck. Burritos (÷ 9): 4.0, 3.3, 2.7, 2.0, 1.3. Smoothies (÷ 6): 4.0, 3.3, 2.7, 2.0, 1.3.

Step 2. Spend dollar by dollar. The two lists are equal at each step, so the student buys in pairs: burrito 1 and smoothie 1 (both 4.0), then burrito 2 and smoothie 2 (both 3.3), then burrito 3 and smoothie 3 (both 2.7). Cost: 3 × $9 + 3 × $6 = $45. The budget is spent.

Step 3. Best bundle: 3 burritos and 3 smoothies. The last dollar on each good adds 2.7.

Step 4. Burritos drop to $6. Burrito bang per buck becomes 6, 5, 4, 3, 2. Now burritos win more contests. The new best bundle is 4 burritos and 3 smoothies ($42). The remaining $3 cannot buy another unit. The price cut pulled spending toward burritos.

Check yourself

Check 1.

At a customer’s current bundle, the last dollar spent on coffee adds 5 units of satisfaction and the last dollar spent on pastries adds 3. To do better, the customer should:

Optimum: MU per dollar equal across goods.

Check 2.

‘Variety is valued’ in consumer choice means:

Diminishing marginal utility is why people spread spending across goods.

Check 3.

The price of good X falls and nothing else changes. The customer’s best bundle:

A price cut raises that good’s MU per dollar → more of it.

Check 4.

Marginal utilities per unit — burritos: 36, 30, 24; smoothies: 24, 20, 16. Burritos cost $9, smoothies $6, and the customer has $30. Which bundle follows the equal-bang-per-buck rule and spends the whole $30?

Compare MU/P across goods unit by unit; spend where it is higher until the money is gone.

Check 5.

A smoothie shop wants customers to buy more smoothies without cutting the price. According to the consumer-optimum logic, what else works?

Two levers: lower P or raise MU — both raise MU per dollar.

Video

Video coming soon.