Leo's utility function is min{x, 3y + 2z}. The price of x is $1, the price of y is $9, and the price of z is $8. Leo's income is $8. How many units of x does Leo demand?
A) 2
B) 1.60
C) 5
D) 7
E) None of the above.

Question

Quizplus · Accepted Answer

Since $1x$ is cheaper than $3y+2z$, it is optimal for Leo to buy as much $x$ as possible without exceeding his budget. Leo's maximum affordable amount of $x$ is $8/1 = 8$, so he buys 2 units of $x$ at a cost of $2*1=$$2.00$. 
Then, he has an income of 8 - 2 = 6 dollars left. To find out whether he should buy any units of $y$ and/or $z$, we can compare the prices and see how much utility each would provide. One unit of $y$ costs $9, while one unit of $z$ costs $8. We can calculate the marginal utility of each as follows:

- For $y$: min($2,3(1)+2(0)) = 2$ (since Leo has already bought 2 units of $x$, he can't substitute them for $y$)
- For $z$: min($2,3(0)+2(1)) = 2$ (same reasoning as above)

Since the marginal utility per dollar is the same for both $y$ and $z$, Leo should buy a combination of the two that exhausts his remaining budget while keeping the ratios of the prices equal. That is:

- $6/$$17 is approximately 0.35
- $0.35 * 9 = 3.15$, rounded down to 3 units of $y$
- $0.35 * 8 = 2.8$, rounded down to 2 units of $z$

Therefore, Leo demands 2 units of $x$, 3 units of $y$, and 2 units of $z$, which exactly exhaust his income of 8 dollars.

Leo's Utility Function Is Min{x, 3y + 2z}