# What is the budget constraint when we assume a common utility function?

Let's consider an exchange economy with two identical consumers. The common utility function is: $$u^i (x_1, x_2) = x_1^α x_2^{1-α} \;\;\; \text{for} \;\;\; 0 < α < 1.$$

Society has 10 units of $$x_1$$ and 10 units of $$x_2$$ in all. We have to find endowments $$e_1$$ and $$e_2$$ and Walrasian equilibrium prices that will support as a Walrasian Equilibrium Allocation the equal-division allocation giving both consumers the bundle $$(5,5)$$.

Now, I do not understand why we are requested to find endowments when they are supposed to be given initially in this exchange economy. What does the common utility function imply here?

In the solution, we derive the first consumer's demand function which assumed the budget constraint:

$$\max x_1^α x_2^{1-α} \;\;\; \text{s.t.} \;\;\; x_1 + p + x_2 = e_1 + pe_2.$$

Why do we assume the consumer's income is $$e_1 + pe_2$$ and why do multiply by prices only the second good? Did we normalize here?

Common utility function means: Here utility functions of the two consumers are of the same form: $$u_i (x_i, y_i) = x_i^\alpha y_i^{1-\alpha}$$  where $$0 < \alpha< 1$$ and $$i\in\{1,2\}$$ Society has $$10$$ units of X and $$10$$ units of Y. and you need to find an endowment allocation and the Walrasian equilibrium prices that will support as a Walrasian Equilibrium Allocation the equal-division allocation i.e. the bundle $$((5,5),(5,5))$$.

One way to do this problem is to pick the target allocation as your endowment allocation: $$(e_1,e_2)=((5,5),(5,5))$$, now check that the prices $$(p_X,p_Y) = (\alpha,(1-\alpha))$$ will support it as an equilibrium allocation.

• +1. But, while this is a way to find a solution for endowments etc, it doesn’t per se necessarily find all solutions, i.e rule out other solutions for the initial endowment. Of course, there (likely) aren’t any other solutions, but your method cannot guarantee it. Mar 18, 2023 at 4:20
• There are other possibilities for the initial endowment. In fact, any allocation from the set of all feasible allocations $((x_1,y_1),(x_2,y_2))$ that satisfy $\alpha x_1 + (1-\alpha)y_1 = 5$, can serve as a choice for the initial endowment.
– Amit
Mar 18, 2023 at 7:20

The budget constraint is not caused by the utility function. Indeed we normalize one of the prices. Prices are relative so you can always select one good as a numeraire and normalize its price to 1.

• How can we derive the budget constraint in such a case? I do not understand why we assume that the income of the first consumer is $e_1+p*e_2$. Mar 15, 2023 at 19:06
• @alioshakaramazov the consumer has endowment e1 and e2 that can be sold at price 1 and p respectively. Thus the income of consumer is e1+p*e2
– 1muflon1
Mar 15, 2023 at 19:54
• I think I got confused when the exercise started to derive the first consumer's demand function, since we have two consumers. I assumed e1 is the endowment of consumer 1 for goods 1 and 2 respectively, thus it would not need to include e2. This is the definition of endowment as far as I know. Are we assuming here that this is a common budget constraint (including endowment of consumer 1 and 2)? Mar 15, 2023 at 20:10