# Perfect Complements - Walrasian Equilibrium

For a homework , I struggled to solve the following question but couldn't go further:

endowment of person 1 = (30,0)

endowment of person 2 = (0,20)


utility functions are such that: \begin{eqnarray*} U (a_1,b_1) & = & \min(a_1,b_1) \\ \\ U (a_2,b_2) & = & \min(4a_2,b_2). \end{eqnarray*}
What I am doing is setting a1 equal to b1 and 4a2 equal to b2. After that I'm writing these; $$p_1 a_1 + p_2 b_1 = 30p_1 \mbox{ and } p_1a_2 + p_24a_2 = 20p_2$$ Finally, by looking at feasibility condition, I'm writing down: \begin{eqnarray*} \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\ \\ \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20. \end{eqnarray*}

Here, if I do some calculations, they result in $p_1 = p_2$ and accordingly $a_1=b_1=15, a_2=4$ and $b_2=16.$ But then the problem is that there is excess demand for good2 and there isn't a WE.

Alternatively, I am considering to have $p_2 = 0$ so that there exist a Walrasian Equilibrium.

BUT, I'm stuck at this point and lack the correct intuition for solving next steps. Or, I might be in a completely incorrect way. Please, explain me what will be done with this excess demand?

• Why are you considering $p_2 = 0$? Notice that everyone likes good 2 'more' than good 1. And the total supply of good 1 exceeds that of good 2. This means that good 2 will always become scarce before good 1 does. Hence you should consider $p_1 = 0$. – Giskard Mar 19 '16 at 18:21
If the prices $p_1$ and $p_2$ are positive than as you pointed out the equations \begin{eqnarray*} \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\ \\ \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20. \end{eqnarray*} hold. This is troublesome, because subtracting the first equation from the second yields \begin{eqnarray*} \frac{3 \cdot 20p_2}{p_1+4p_2} & = & -10 \end{eqnarray*} which has no solution for positive prices. However if a good has zero price, than the demand equations are different as no one minds buying superflous goods as long as they are free. For $p_2 = 0$ you would have \begin{eqnarray*} \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\ \\ \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & \leq & 20. \end{eqnarray*} This is because $\frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2}$ is a minimum that the buyers need but they do not mind having more as good 2 is free. There is also a formulation of equilibrium, the 'value-equilibrium' which is that for good $i$ \begin{eqnarray*} p_i \cdot D_i(p) = p_i \cdot S_i(p) \mbox{ and } D_i(p) \leq S_i(p) \end{eqnarray*} meaning that there can be oversupply of zero price goods.
Unfortunately this equation system would still imply \begin{eqnarray*} \frac{3 \cdot 20p_2}{p_1+4p_2} & \leq & -10 \end{eqnarray*} which has no solution in non-negative prices. On the other hand, for $p_1 = 0$ you would have \begin{eqnarray*} \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & \leq & 30 \\ \\ \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20. \end{eqnarray*} which implies \begin{eqnarray*} \frac{3 \cdot 20p_2}{p_1+4p_2} & \geq & -10. \end{eqnarray*} As $p_1 = 0$ any positive $p_2$ is a solution for this.
• I suggest you draw the indifference curves and the budget set with $p_1 = 0$. Perhaps it will help you see the optimal consumptions better. – Giskard Mar 20 '16 at 8:28