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?
