Price mechanism question (economics)

Question

There are two goods in the market $X1$ and $X2$ and both of their prices are positive. If an agent's utility function is given as: Q1 = amount of X1 agent buys, Q2 = amount of X2 agent buys, $u = Q1 + |Q2-5|$ Identify the demand of this agent.

Note: agent has to spend his initial endowment.

Answer 1

Clearly it depend on the prices, say $P1$ and $P2$, and on the agent's endowment, say $m$.

There is no point in demanding $Q2 \gt 5$: utility would be increased reducing $Q2$ to $5$ and spending the saved amount on $X1$.

For small endowments the agent will maximise utility by spending on the good with the lower price. Indeed if $P1 \lt P2$ it will always be better to buy more of $X1$ and none of $X2$. And similarly if $P1 \gt P2$ it will be better to buy more of $X2$ and none of $X1$ until $Q2=5$. So

• if $P1 \lt P2$, then demand is $Q1=m/P1$ of $X1$ and $Q2=0$ of $X2$
• if $P1 \gt P2$ and $m/P2 \le 5$, then demand is $Q1=0$ of $X1$ and $Q2=m/P2$ of $X2$
• if $P1 \gt P2$ and $m/P2 \ge 5$, then demand is $Q1=(m-5\times P2)/P1$ of $X1$ and $Q2=5$ of $X2$

If $P1=P2$ then there are several solutions which are convex combinations of those bullet points.