# Price setting for a pure exchange economy, with two goods and two agents, with possible different utility functions that depends on total consumption

Consider a pure exchange economy comprised of two agents with differentiable utility functions that are strictly increasing and strictly concave. There are only two goods in this economy (x,y) and each agent has a positive endowment of both goods. The utility functions are generic e possibly different, but both depend on total individual consumption x+y. Find the equilibrium prices.

My doubt, in this case, is if I can address the question by specifying different utility functions to both agents that depend on total consumption and satisfy the assumption (e.g. two different Cobb-Douglas)?

From that I could use the budgt set of both agents, from the initial endowmenst, to determine the quantities relative to the prices and the, from de $$MRS(x_1, x_2)$$ determine the prices suach as $$\frac{p_1}{p_2}=MRS(x_1, x_2)$$ and $$MRS(x_1, x_2)=\frac{\frac{du(x_1,x_2}{dx_1}}{\frac{du(x_1,x_2}{dx_2}}$$.

From that, I would get the relative prices as a function of the quantities, substitute one of these quantities on the budget set and get the prices relative to the initial endowments.

Is that sufficient to address the question posed?

• Cobb-Douglas does not depend on "total consumption x+y", it is a bivariate function $U(x,y)$. What exactly are you asking? Mar 23, 2021 at 10:34
• "My doubt, in this case, is if I can address the question by specifying different utility functions to both agents that depend on total consumption and satisfy the assumption (e.g. two different Cobb-Douglas)?" --- No, most certainly not. Mar 25, 2021 at 2:30