<p>Consider a pure exchange economy with two consumers - A and B, and two goods - X and Y with
<span class="math-container">$u_A(x_A, y_A) = x_Ay_A$</span>, and <span class="math-container">$u_B(x_B, y_B) = x_By_B$</span>.
Suppose endowments of A and B are
<span class="math-container">$\omega_A = (1,0)$</span> and <span class="math-container">$\omega_B = (0,1)$</span>.
In this case, set of all Pareto efficient allocations <span class="math-container">$\{((x_A, y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|y_A=x_A \wedge x_A+x_B = 1 \wedge y_A+y_B=1\}$</span> is also the Core of this economy (This is the 45 degree line connecting the origins of A and B in the Edgeworth box). However there is only one Competitive Equilibrium Allocation which is <span class="math-container">$\left(\left(x_A, y_A\right),\left(x_B,y_B\right)\right)=\left(\left(\frac{1}{2}, \frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)\right)$</span> with supporting prices <span class="math-container">$(p_X, p_Y) = (1,1)$</span>.</p>

CC BY-SA 4.0
2023-01-03T06:26:36.537Z
2023-01-03T06:26:36.537Z
Amit
<p>Suppose we have two agents say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> with endowments <span class="math-container">$\omega^A = (\omega_1^A, \omega_2^A)$</span> and <span class="math-container">$\omega^B =(\omega_1^B, \omega_2^B)$</span> both in <span class="math-container">$\mathbb{R}^2$</span>. Agents have Cobb-Douglas utility functions:</p>
<p><span class="math-container">$$u_A(x^A) = (x_1^A)^\alpha(x_2^A)^{1-\alpha},\\
u_B(x^B) = (x_1^B)^\beta(x_2^B)^{1-\beta}.$$</span></p>
<p>where <span class="math-container">$\alpha,\beta \in (0,1).$</span></p>
<p>Now, because Cobb-Douglas functions are nice concave utility functions, we know that the contract curve (the set of Pareto efficient allocations) is precisely the set of points with <span class="math-container">$MRS_A = MRS_B,$</span> that is, where
<span class="math-container">$$
\frac{\alpha x_2^A}{(1-\alpha)x_1^A} = \frac{\beta x_2^B}{(1-\beta)x_1^B}
$$</span>
Combining this with the feasibility conditions:
<span class="math-container">$$x_1^A + x_1^B = \omega_1^A + \omega_1^B \equiv \omega_1,\\
x_2^A + x_2^B = \omega_2^A + \omega_2^B \equiv \omega_2.$$</span>
which we can write as
<span class="math-container">$$x_1^B = \omega_1-x_1^A,\\
x_2^B = \omega_2-x_2^A.$$</span>
Substituting these into the contract curve condition, we have
<span class="math-container">$$
\frac{\alpha x_2^A}{(1-\alpha)x_1^A} = \frac{\beta (\omega_2-x_2^A)}{(1-\beta)(\omega_1-x_1^A)}
$$</span>
which, after some rearranging, gives us
<span class="math-container">$$\alpha x_2^A(1-\beta)(\omega_1-x_1^A) = \beta (\omega_2-x_2^A)(1-\alpha)x_1^A$$</span>
Solving this for <span class="math-container">$x_2^A$</span> will give you the contract curve. Now, the core of the economy is all the points on the contract curve which deliver weakly greater utility to BOTH agents than the initial endowment. If you plot the contract curve in the Edgeworth box, it's going to look something like a line from one corner to the other, and if you plot the indifference curves corresponding to the initial endowments (i.e. the curve <span class="math-container">$u_A(x_1^A, x_2^A) = u_A(\omega_1^A, \omega_2^A))$</span>, the core is all points on the contract curve between <span class="math-container">$A$</span> and <span class="math-container">$B$</span>'s indifference curves (essentially because these are the points both agents would be weakly better off at, and there are no Pareto improvements at these points).</p>
<p>Finally we get to the question: What is the Walrasian equilibrium? I'm only going to sketch an answer to that here. To get the WE, we would need to solve each agent's constrained maximisation problem where the objective function is the utility and the constraint is their budget:
<span class="math-container">$$p_1x_1^i + p_2x_2^i = p_1 \omega_1^i + p_2\omega_2^i$$</span>
for <span class="math-container">$i\in\{A,B\}$</span>. This turns out not to be too tricky because the utility functions are so nice. Once you have the demand function for each agent, you can use the fact that markets clear in a WE to find a unique price ratio <span class="math-container">$p_1/p_2$</span> that clears both markets. This price ratio will lead to a unique allocation in the core, so to answer your question you can choose any allocation in the core that's not this one!</p>

CC BY-SA 4.0
2022-04-07T16:47:04.373Z
I know how to get the WE, and already derived both contract curves. And I know the core of the economy is the contract curve bounded by the level curves of the initial utility. But I can't seem to understand how an allocation can be in the core of the economy but not be a WE.
Abraham Villalobos Jaimes
CC BY-SA 4.0
2022-04-14T00:45:03.167Z
The WE gives you a specific allocation in the core, but in general the core is a continuum. If the two indifference curves are already tangent to each other at the endowment point, then the core is precisely the WE (it's a single point, the point of intersection between the two indifference curves), otherwise the core is ALL PE allocations that make players weakly better off than their endowments, and the WE is a specific one of these PE allocations, namely the one corresponding to the price that clears both markets.
deej
CC BY-SA 4.0
2022-04-07T13:05:59.603Z
2022-04-07T13:05:59.603Z
deej
<p>So I was given this problem where both agents have Cobb-Douglas utility functions and I'm asked to find an allocation that's in the core but not a Walrasian equilibrium. Isn't the core of an economy the segment of the contract curve that lies between the original indifference curves?</p>

CC BY-SA 4.0
2022-04-06T21:36:12.780Z
Yes, it is. At least in a two-agent Edgeworth-box economy.
Michael Greinecker
CC BY-SA 4.0
2022-04-07T19:34:51.503Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Giskard
<microeconomics><general-equilibrium><walrasian>
When is an allocation in the core of an economy but it's not a Walrasian equilibrium?
Abraham Villalobos Jaimes
# When is an allocation in the core of an economy but it's not a Walrasian equilibrium?

(Asked by: username_0 on 2022-04-06)

So I was given this problem where both agents have Cobb-Douglas utility functions and I'm asked to find an allocation that's in the core but not a Walrasian equilibrium. Isn't the core of an economy the segment of the contract curve that lies between the original indifference curves?

**Tags:** microeconomics, general-equilibrium, walrasian
> **Comments:**
> **username_1**: Yes, it is. At least in a two-agent Edgeworth-box economy.  
> **username_2**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  

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## 2 answers

### Answer by username_3 on 2023-01-03. Score: 1

Consider a pure exchange economy with two consumers - A and B, and two goods - X and Y with
$u\_A(x\_A, y\_A) = x\_Ay\_A$, and $u\_B(x\_B, y\_B) = x\_By\_B$.
Suppose endowments of A and B are
$\omega\_A = (1,0)$ and $\omega\_B = (0,1)$.
In this case, set of all Pareto efficient allocations $\{((x\_A, y\_A),(x\_B,y\_B))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|y\_A=x\_A \wedge x\_A+x\_B = 1 \wedge y\_A+y\_B=1\}$ is also the Core of this economy (This is the 45 degree line connecting the origins of A and B in the Edgeworth box). However there is only one Competitive Equilibrium Allocation which is $\left(\left(x\_A, y\_A\right),\left(x\_B,y\_B\right)\right)=\left(\left(\frac{1}{2}, \frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)\right)$ with supporting prices $(p\_X, p\_Y) = (1,1)$.


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### Answer by username_4 on 2022-04-07. Score: 0

Suppose we have two agents say $A$ and $B$ with endowments $\omega^A = (\omega\_1^A, \omega\_2^A)$ and $\omega^B =(\omega\_1^B, \omega\_2^B)$ both in $\mathbb{R}^2$. Agents have Cobb-Douglas utility functions:

$$u\_A(x^A) = (x\_1^A)^\alpha(x\_2^A)^{1-\alpha},\\
u\_B(x^B) = (x\_1^B)^\beta(x\_2^B)^{1-\beta}.$$

where $\alpha,\beta \in (0,1).$

Now, because Cobb-Douglas functions are nice concave utility functions, we know that the contract curve (the set of Pareto efficient allocations) is precisely the set of points with $MRS\_A = MRS\_B,$ that is, where
$$
\frac{\alpha x\_2^A}{(1-\alpha)x\_1^A} = \frac{\beta x\_2^B}{(1-\beta)x\_1^B}
$$
Combining this with the feasibility conditions:
$$x\_1^A + x\_1^B = \omega\_1^A + \omega\_1^B \equiv \omega\_1,\\
x\_2^A + x\_2^B = \omega\_2^A + \omega\_2^B \equiv \omega\_2.$$
which we can write as
$$x\_1^B = \omega\_1-x\_1^A,\\
x\_2^B = \omega\_2-x\_2^A.$$
Substituting these into the contract curve condition, we have
$$
\frac{\alpha x\_2^A}{(1-\alpha)x\_1^A} = \frac{\beta (\omega\_2-x\_2^A)}{(1-\beta)(\omega\_1-x\_1^A)}
$$
which, after some rearranging, gives us
$$\alpha x\_2^A(1-\beta)(\omega\_1-x\_1^A) = \beta (\omega\_2-x\_2^A)(1-\alpha)x\_1^A$$
Solving this for $x\_2^A$ will give you the contract curve. Now, the core of the economy is all the points on the contract curve which deliver weakly greater utility to BOTH agents than the initial endowment. If you plot the contract curve in the Edgeworth box, it's going to look something like a line from one corner to the other, and if you plot the indifference curves corresponding to the initial endowments (i.e. the curve $u\_A(x\_1^A, x\_2^A) = u\_A(\omega\_1^A, \omega\_2^A))$, the core is all points on the contract curve between $A$ and $B$'s indifference curves (essentially because these are the points both agents would be weakly better off at, and there are no Pareto improvements at these points).

Finally we get to the question: What is the Walrasian equilibrium? I'm only going to sketch an answer to that here. To get the WE, we would need to solve each agent's constrained maximisation problem where the objective function is the utility and the constraint is their budget:
$$p\_1x\_1^i + p\_2x\_2^i = p\_1 \omega\_1^i + p\_2\omega\_2^i$$
for $i\in\{A,B\}$. This turns out not to be too tricky because the utility functions are so nice. Once you have the demand function for each agent, you can use the fact that markets clear in a WE to find a unique price ratio $p\_1/p\_2$ that clears both markets. This price ratio will lead to a unique allocation in the core, so to answer your question you can choose any allocation in the core that's not this one!

> **Comments:**
> **username_0**: I know how to get the WE, and already derived both contract curves. And I know the core of the economy is the contract curve bounded by the level curves of the initial utility. But I can't seem to understand how an allocation can be in the core of the economy but not be a WE.  
> **username_4**: The WE gives you a specific allocation in the core, but in general the core is a continuum. If the two indifference curves are already tangent to each other at the endowment point, then the core is precisely the WE (it's a single point, the point of intersection between the two indifference curves), otherwise the core is ALL PE allocations that make players weakly better off than their endowments, and the WE is a specific one of these PE allocations, namely the one corresponding to the price that clears both markets.  

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