About marshallian demands

Question

Let an agent with utility function $U(x_1,x_2,x_3)=min\{ x_1, x_2 \} +x_3$. We want to maximize $U$ subject to $p_1 x_1+p_2 x_2 +p_3 x_3= I$

First intuitively $x_1^*=x_2^*$ substituting en the budget constraint I have $$x_1^*=x_2^*=\frac{I-p_3 x_3}{p_1+p_2}$$ I don't know how to solve for $x_3$. Then i thought in Kuhn-Tucker but I can't see how to solve it.

Answer 1

I will give a hint. To increase/decrease utility by 1 unit, you can do two things. Increase $x_1$ and $x_2$ by $\frac{1}{2}$ units and this would cost you $\frac{p_1+p_2}{2}$. The second option is changing $x_3$ by one unit that would cost $p_3$. One other option is of course using a linear combination of the above changes. If you remember from well behaved utility functions, you would like to buy things for which utility per price is higher. Now you would go for a similar reasoning here and would try to buy extra utility at a cheapest price.

Answer 2

We solve the problem

\begin{equation} \max U(x_1,x_2,x_3) = \min\{x_1,x_2\} + x_3 \end{equation}

subject to

\begin{equation} x_1 p_1 + x_2 p_2 + x_3 p_3 = I \end{equation}

From the min term we get that $x_1 = x_2$. Therefore the budget constraint becomes

\begin{equation} x_1 (p_1 + p_2) + x_3 p_3 = I \end{equation}

Solving for $x_1$ we get

\begin{equation} x_1 = \frac{I - x_3 p_3}{p_1 + p_2} \end{equation}

Since $x_1 = x_2$,

\begin{equation} x_2 = \frac{I - x_3 p_3}{p_1 + p_2} \end{equation}

Therefore, the min term becomes

\begin{equation} \min\{x_1,x_2\} = \frac{I - x_3 p_3}{p_1 + p_2} \end{equation}

Substituting this expression for the min term into the utility function we get

\begin{equation} W(x_3) := U(x_1(x_3),x_2(x_3),x_3) = \frac{I - x_3 p_3}{p_1 + p_2} + x_3 \end{equation}

Separating the fraction and rearranging we get

\begin{equation} W(x_3) = (1 - \frac{p_3}{p_1 + p_2}) x_3 + \frac{I}{p_1 + p_2} \end{equation}

Note that this is a straight line, with slope $1 - \frac{p_3}{p_1 + p_2}$.

Notice $1 - \frac{p_3}{p_1 + p_2} > 0 \iff 1 > \frac{p_3}{p_1 + p_2} \iff p_1 + p_2 > p_3$.

From here we get 3 cases:

• $p_1 + p_2 > p_3 \rightarrow$ spend everything on $x_3 \rightarrow x_1 = 0, x_2 = 0, x_3 = \frac{I}{p_3}$.
• $p_1 + p_2 < p_3 \rightarrow$ don't consume $x_3 \rightarrow x_1 = \frac{I}{p_1 + p_2}, x_2 = \frac{I}{p_1 + p_2}, x_3 = 0$
• $p_1 + p_2 = p_3 \rightarrow x_3$ value indifferent $\rightarrow x_1 = \frac{I - x_3 p_3}{p_1 + p_2}, x_2 = \frac{I - x_3 p_3}{p_1 + p_2}, 0 \leq x_3 \leq \frac{I}{p_3}$, i.e. the optimal bundles form a line in 3-D space that looks like this: