The Cafe Owner's Math

Open a cafe and work out how many cups to sell to come out ahead.

The cost of one more cup of ☕ coffee is the marginal cost. It climbs as the cups pile up. While the price beats that cost, each sale adds profit. Stop where the two become equal — that rule is the formula below.

For example, a cup costs 2,000 won to make and sells for 4,500, so each cup adds 2,500. By evening, ingredients run low and one more cup costs 5,000. From then on, every extra cup loses 500 won.

MC=ΔCΔqMC = \frac{\Delta C}{\Delta q}
P=MC (profit max)P = MC \ \text{(profit max)}

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Examples & Exercises

The cafe's fixed cost is 2020, marginal cost is MC=ΔC/Δq=4+2qMC = \Delta C / \Delta q = 4 + 2q, and a cup sells for P=20P = 20.

How many cups maximize profit?

Fixed cost 2020, price 2020 — the default setup.
Show proof

  1. While the cost of one more cup (MCMC) is below the price (PP), each extra cup adds profit. Stop exactly where the two meet.

  2. At P=MCP = MC, 20=4+2q20 = 4 + 2q gives q=8q = 8 cups. Total cost is 20+4⋅8+82=11620 + 4 \cdot 8 + 8^2 = 116 and revenue is 160160, so profit is 4444.

Rent (the fixed cost) jumps from 2020 to 6060.

What happens to the best quantity, and what is the new profit?

Only the fixed cost is 6060 — everything else default.
Show proof

  1. A fixed cost is not part of the cost of "one more cup". The MCMC curve does not move, so the P=MCP = MC point — q=8q = 8 cups — does not move either.

  2. Profit, though, is shaved by exactly the rent hike: 160−(60+32+64)=4160 - (60 + 32 + 64) = 4. Same quantity, thinner pocket — that is what a fixed cost does.

1. Intro to Microeconomics