# MRS in Walrasian equilibrium price expression?

I have the utility function $U^A=\alpha ln(x^A)+\beta ln(y^A)$ for consumer $A$ and $U^B=\gamma ln(x^B)+\phi ln(y^B)$ for consumer $B$. They are endowed with $\omega^h_x$ and $\omega^h_y$ of $x$ and $y$, where $h=A,B$. $p_y = 1$.

They are allowed to trade and I have solved for the Walrasian equilibrium price $p^*_x$ and found that this is

$$p^*_x=\frac{\alpha \omega^A_y+\gamma \omega^B_y}{\beta \omega^A_x+\phi \omega^B_x}$$

I believe that I'm missing out on some basic intuition here. What is the significance of $\alpha$ being swapped from good $x$ (in the utility function) to $y$ (in the price), for instance? There also seems to be a link here to the MRS? It seems like the first term in the nominator divided by the first term in the denominator yields the MRS for $A$ and the same for the second terms for $B$?

• How did you solve for Walrasian equilibrium? Given that you could solve the equations your MRS related questions are very unclear to me. – Giskard Mar 7 '18 at 10:20
• Perhaps showing your derivation steps explicitly will help both you and the people who try to help better understand where your confusion lies. – Herr K. Mar 7 '18 at 23:29