# Given a Walrasian equilibrium, compute all intitial endowments that lead to given Walrasian equilibrium

If you would be so kind to take a look at the following problem:*

Consider an economy with two agents Stan ($$S$$) and Laurel ($$L$$) and two goods: Cans of coke ($$C$$) and Pretzels ($$P$$). The total quantities of these two goods are fixed at 6 cans of coke and 10 pretzels. The utility function of Stan is

$$U_L = C_s^{1/3}*P_s^{2/3}$$and the utility function of Laurel is $$U_L = C_L^\beta*P_L^{1-\beta}$$ Stan is endowed with 3 cans of coke and 3 pretzels $$((C_S, P_S) = (3, 3))$$ and Laurel with 3 cans of coke and 7 pretzels $$((C_L, P_L) = (3, 7))$$.

Calculate all initial endowments from which trades lead to $$(C_S, C_L, P_S, P_L) = (3, 3, 3, 7)$$ as the Walrasian equilibrium.

I am stuck solving the problem.

So far I have:

• $$MRS_S=MRS_L$$ needs to hold for a Pareto efficient allocation

• $$\frac{1}{2}\frac{P_S}{C_S}=\frac{\beta}{1-\beta}\frac{P_L}{C_L}$$

• $$P=P_L+P_S=10$$, so $$P_L=10-P_S$$

• $$C=C_L+C_S=6$$, so $$C_L=6-C_L$$

• $$\frac{1}{2}\frac{P_S}{C_S}=\frac{\beta}{1-\beta}\frac{10-P_S}{6-C_S}$$

Am I in the right direction; How should I solve this problem?