I have a two-good, two-consumer exchange economy. $(\omega_{i1}, \omega_{i2})\in \mathbb{R}_{+}^2$ denote consumer $i$'s endowments of commodities 1 and 2 for $i\in\{1,2\}$. Utility functions are given by $x_{11}^{\alpha}x_{12}^{1-\alpha}$ (consumer 1), $x_{21}^{\beta}x_{22}^{1-\beta}$ (consumer 2).
Normalizing $p_1=1$, I find that, in any Walrasian equilibrium, $p_2=\frac{\omega_{11}(1-\alpha)+(\omega_{21})(1-\beta)}{\alpha(\omega_{12})+\beta(\omega_{22})}$. I am asked to consider the effect of changing endowments to $(\omega_{11}^*, \omega_{12}^*)=(\omega_{11}-\epsilon, \omega_{12}-\epsilon)$ and $(\omega_{21}^*, \omega_{22}^*)=(\omega_{21}+\epsilon, \omega_{22}+\epsilon)$ on equilibrium prices (in this case, $p_2$), as a function of $\alpha$ and $\beta$.
What is the intuition behind the result that $p_2$ is decreasing in the endowment of good 2, but increasing in the endowment of good 1?
Is it that, as the endowment of good 2 increases, the price of good 2 must fall in order to induce higher demand for good 2 to ensure the market for good 2 clears?
On the other hand, as the endowment of good 1 increases, the price of good 2 must increase, in order to ensure that consumers buy less of good 2 and more of good 1, to ensure that the market for good 1 clears.
Concerning $\alpha$ and $\beta$. The higher is $\alpha$, the greater is the share of wealth consumer 1 spends on good 1 (and so the smaller the share spent on good 2). Thus, upon an increase in $\alpha$, the amount by which $p_2$ must increase in response to an increase in consumer 1's endowment of good 1 (in order to ensure that the market for good 1 clears, by inducing consumer 1 to spend less on good 2 and more on good 1) decreases (as consumer 1 becomes progressively ‘more inclined’ to consume good 1).
Similar analysis can be conducted for $\beta$.
Is this reasoning correct?
