# Effect of change in endowments on prices in Walrasian equilibria

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?