# Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$

I know how to solve the two-good case with $$u(x) = \min\{x1, x2\}$$, but the addition of $$x3$$ confuses me.

Problem

Derive the demand function $$x(p,w)$$ from $$u(x) = \min\{x1, x2\} + x3$$

What I did so far

We assume that in optimum $$x1 = x2$$.

Set up the budget constraint $$p1x1 + p2x2 + p3x3 = w$$

Rewrite budget constraint as $$(p1+p2)x1 + p3x3 = w$$ or $$(p1+p2)x2 + p3x3 = w$$

We can write $$x1*$$ or $$x2* = \frac{w-p3x3}{p1+p2}$$ and $$x3* = \frac{w-(p1+p2)a}{p3}$$

Confusion

How to proceed? Can I still use a Lagrangian to solve this?

Hint: Imagine that there are two coffee bars, $$A$$ and $$B$$. There is only one type of coffee in the world. My preferences are such that I always want 1 unit of sugar with 1 unit of coffee; if I consume units of coffee and sugar in the ratio $$1:1$$, additional units of only one of the two don't give me extra utility.

At coffee bar $$A$$ they sell coffee and sugar separately. That is: there is one counter for coffee, where coffee is sold for a price of $$p_1$$ per unit. There is another counter for sugar, where I can buy one unit of sugar for a price of $$p_2$$.

At coffee bar $$B$$ however, they only sell one unit of coffee with one unit of sugar in a bundle, the price of which is equal to $$p_3$$. They do not sell anything separately

At what prices do I prefer to buy my coffee with sugar at $$A$$, at what prices do I prefer to buy my coffee with sugar at $$B$$?

• If I understand you correctly, you buy at bar $A$ for $p1+p2$, and at bar $B$ for $p3$. – Anne1005 Oct 9 '19 at 18:55
• That is correct, but what I meant with the last sentence was: At what values of $p_1$, $p_2$, $p_3$ is it cheaper to buy my coffee with sugar at $A$? – Milton Keynes Oct 9 '19 at 19:36
• I edited the answer to make it more clear, try to relate the little story above to the problem you have to solve – Milton Keynes Oct 9 '19 at 19:39
• Thank you. In that case it means that you will buy at $A$ if $p1 + p2$ < $p3$. If $p1 + p2$ > $p3$, you will buy at $B$. If they are equal, you are indifferent. – Anne1005 Oct 9 '19 at 20:09
• Exactly! And if you now combine this with the fact that your budget is equal to $w$ you have your demand function. – Milton Keynes Oct 9 '19 at 21:09

for this problem you must conciser two possible branches of the utility function:

$$u(\text{x})=x_1+x_3\ \ \text{if} \ \ x_1 $$u(\text{x})=x_2+x_3\ \ \text{if} \ \ x_1>x_2$$

The demands of these then proceed how you would for any case of perfect substitutes. However you must list them for each case.

Hope this helps