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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?

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