<p>Common utility function means: Here utility functions of the two consumers are of the same form:
<span class="math-container">$u_i (x_i, y_i) = x_i^\alpha y_i^{1-\alpha}$</span>  where <span class="math-container">$0 &lt; \alpha&lt; 1$</span> and <span class="math-container">$i\in\{1,2\}$</span>
Society has <span class="math-container">$10$</span> units of X and <span class="math-container">$10$</span> 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 <span class="math-container">$((5,5),(5,5))$</span>.</p>
<p>One way to do this problem is to pick the target allocation as your endowment allocation: <span class="math-container">$(e_1,e_2)=((5,5),(5,5))$</span>, now check that the prices <span class="math-container">$(p_X,p_Y) = (\alpha,(1-\alpha))$</span> will support it as an equilibrium allocation.</p>

CC BY-SA 4.0
2023-03-18T04:20:43.927Z
+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.
BB King
CC BY-SA 4.0
2023-03-18T07:20:20.710Z
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
CC BY-SA 4.0
2023-03-18T01:57:28.103Z
2023-03-18T02:23:07.303Z
2023-03-18T02:23:07.303Z
Amit
<p>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.</p>

CC BY-SA 4.0
2023-03-15T19:06:11.247Z
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 $e_1+p*e_2$.
aliosha karamazov
CC BY-SA 4.0
2023-03-15T19:54:59.243Z
@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
CC BY-SA 4.0
2023-03-15T20:10:17.160Z
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)?
aliosha karamazov
CC BY-SA 4.0
2023-03-15T18:39:56.303Z
2023-03-15T18:39:56.303Z
1muflon1
<p>Let's consider an exchange economy with two identical consumers. The
common utility function is:
<span class="math-container">$$u^i (x_1, x_2) = x_1^α  x_2^{1-α} \;\;\; \text{for} \;\;\; 0 &lt; α &lt; 1.$$</span></p>
<p>Society has 10 units of <span class="math-container">$x_1$</span> and 10 units of <span class="math-container">$x_2$</span> in all. We have to find endowments <span class="math-container">$e_1$</span> and <span class="math-container">$e_2$</span> and Walrasian
equilibrium prices that will support as a Walrasian Equilibrium Allocation the
equal-division allocation giving both consumers the bundle <span class="math-container">$(5,5)$</span>.</p>
<p>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?</p>
<p>In the solution, we derive the first consumer's demand function which assumed the budget constraint:</p>
<p><span class="math-container">$$ \max x_1^α  x_2^{1-α} \;\;\; \text{s.t.} \;\;\; x_1 + p + x_2 = e_1 + pe_2.$$</span></p>
<p>Why do we assume the consumer's income is <span class="math-container">$e_1 + pe_2$</span> and why do multiply by prices only the second good? Did we normalize here?</p>

<microeconomics><utility><profit-maximization><walrasian>
What is the budget constraint when we assume a common utility function?
aliosha karamazov
# What is the budget constraint when we assume a common utility function?

(Asked by: username_0 on 2023-03-15)

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?

**Tags:** microeconomics, utility, profit-maximization, walrasian

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## 2 answers

### Answer by username_1 on 2023-03-18. Score: 3

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.

> **Comments:**
> **username_2**: +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.  
> **username_1**: 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.  

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### Answer by username_3 on 2023-03-15. Score: 1

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.

> **Comments:**
> **username_0**: 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 $e_1+p*e_2$.  
> **username_3**: @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  
> **username_0**: 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)?  

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