<p>In my opinion, your solution is correct.</p>
<p>The contract curve <strong>must be such that <span class="math-container">$x_{1B} = x_{2B}$</span>,</strong> otherwise it could be possible to <strong>increase the utility of one consumer without decreasing the utility of the other.</strong></p>
<p>Consider the following graph, where I have represented a generic Leontief utility function of a consumer <span class="math-container">$B$</span>, <span class="math-container">$u_B = \min\{{ax,by}\}$</span>, and the indifference (convex)  curves of a differentiable utility function of a consumer <span class="math-container">$A$</span>.</p>
<p><a href="https://i.sstatic.net/9t5sz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9t5sz.png" alt="enter image description here" /></a></p>
<p>The optimum bundles for consumer <span class="math-container">$B$</span> with Leontief preferences must be on the line of slope <span class="math-container">$a/b$</span>, <span class="math-container">$y=\frac {a}{b} x$</span>, where <span class="math-container">$ax =by$</span>.</p>
<p>And the contract curve must be that line  <span class="math-container">$y=\frac {a}{b} x$</span>.<span class="math-container">$^1$</span></p>
<p>In fact, if we were at a point outside the line <span class="math-container">$y=\frac {a}{b} x$</span> as, for example, <span class="math-container">$A$</span>, where the other consumer <span class="math-container">$A$</span> has a utility <span class="math-container">$\overline U$</span>, we could go to point <span class="math-container">$C$</span> where the consumer with Leontief preferences increases their utility, and the utility of the other consumer is still <span class="math-container">$\overline U$</span>. Contradicting the assumption that consumer <span class="math-container">$B$</span> is maximizing their level of utility with</p>
<p><span class="math-container">$x_{1A}^{\frac{1}{2}} x_{2A}^{\frac{1}{2}} = \overline{U}\;\;\; (1)$</span>.</p>
<p>Vice versa,  seeing it from  the other side, so to speak, if you want to maximize the utility of consumer <span class="math-container">$A$</span> for a given indifference curve of the Leontief utility function, you must place the bundle  at the  vertex of that indifference curve.</p>
<p><strong>Actually, you are using equation <span class="math-container">$(1)$</span>:</strong>  the implicit assumption, from a mathematical point of view, is that there is an indifference curve of consumer <span class="math-container">$A$</span> (the blue lines) which passes through each vertex of the Leontief indifference curves, as points <span class="math-container">$B$</span> and <span class="math-container">$C$</span>, which is guaranteed by the form of the utility function</p>
<p><span class="math-container">$$u_A = x_{1A}^{\frac{1}{2}} x_{2A}^{\frac{1}{2}}$$</span>.</p>
<p>In other words, the red line <span class="math-container">$y=\frac {a}{b} x$</span> crosses, at every point,  an indifference curve of consumer <span class="math-container">$A$</span>.</p>
<p>And, in general, you are using the form of the indifference curves of consumer <span class="math-container">$A$</span>. I mean, you are not maximizing  utility for the Leontief function in the emptiness, from a mathematical point of view, you use the properties of the utility function of consumer <span class="math-container">$A$</span>, beginning with the very basic requirement that the functions are defined where needed.</p>
<p>Actually, nothing very different from maximization with the budget constraint.</p>
<p>As for your objection that you don't use, in this case, the equations <span class="math-container">$(1)-(3)$</span>, this is not completely true, because you then use them to find the bundle of goods <span class="math-container">$x_{1A}, x_{2A}$</span> of consumer <span class="math-container">$A$</span>, and the related level of utility.</p>
<hr />
<p><span class="math-container">$^1$</span> <sup> In this Edgeworth Box the contract curve doesn't end, as usual, at the top right point <span class="math-container">$0_A$</span>, but I think that it is correct when we have Leontief preferences: once consumer <span class="math-container">$B$</span>  is on the right vertical line, the utility of consumer <span class="math-container">$B$</span> cannot increase, it is 'stopped' by the total endowment of good <span class="math-container">$x$</span>.</sup></p>

CC BY-SA 4.0
2023-03-11T02:18:50.917Z
Thanks! I appreciate your complete and general answer! I’m finding things confusing for Leontief and linear preferences in general equilibrium. I’ll ask a question soon for two linear utilities.
Nicolas Torres
CC BY-SA 4.0
2023-03-11T09:09:23.790Z
You are welcome. I agree with you, things are more confusing when there isn't differentiability.
BakerStreet
CC BY-SA 4.0
2023-03-12T18:42:58.500Z
I just asked the question about linear utilities here: https://economics.stackexchange.com/questions/54702/how-to-find-the-contract-curve-when-both-agents-have-linear-utilities
Nicolas Torres
CC BY-SA 4.0
2023-03-10T20:43:37.787Z
2023-03-10T21:50:41.357Z
2023-03-10T21:50:41.357Z
BakerStreet
<p>I'm trying to solve the following excercise:</p>
<p>Find the contract curve for an economy where agents' (<span class="math-container">$A$</span> and <span class="math-container">$B$</span>) preferences and endowments are given by:</p>
<p><span class="math-container">$u_A = x_{1A}^{\frac{1}{2}} x_{2A}^{\frac{1}{2}}$</span></p>
<p><span class="math-container">$u_B = \min\{x_{1B},x_{2B}\}$</span></p>
<p><span class="math-container">$(\omega_{1A},\omega_{2A}) = (\alpha,\beta)$</span></p>
<p><span class="math-container">$(\omega_{1B},\omega_{2B}) = (\beta,\alpha)$</span></p>
<p>I know the contract curve in this case can be found by solving:</p>
<ul>
<li><span class="math-container">$\max x_{1A}^{\frac{1}{2}} x_{2A}^{\frac{1}{2}}$</span></li>
</ul>
<p>subject to</p>
<p><span class="math-container">$ \min \{x_{1B},x_{2B}\} = \overline{U}$</span></p>
<p><span class="math-container">$x_{1A} + x_{1B} = \alpha + \beta$</span></p>
<p><span class="math-container">$x_{2A} + x_{2B} = \alpha + \beta$</span></p>
<p>or by solving</p>
<ul>
<li><span class="math-container">$\max \min\{x_{1B},x_{2B}\}$</span> <span class="math-container">$(0)$</span></li>
</ul>
<p>subject to</p>
<p><span class="math-container">$x_{1A}^{\frac{1}{2}} x_{2A}^{\frac{1}{2}} = \overline{U}$</span> <span class="math-container">$(1)$</span></p>
<p><span class="math-container">$x_{1A} + x_{1B} = \alpha + \beta$</span> <span class="math-container">$(2)$</span></p>
<p><span class="math-container">$x_{2A} + x_{2B} = \alpha + \beta$</span> <span class="math-container">$(3)$</span></p>
<p>I know the usual ways to find contract curves are either by forming a Lagrangian or as a shortcut, start from solving <span class="math-container">$MRS_A = MRS_B$</span>.</p>
<p>However, the presence of a Leontief function makes either of these problems unsolvable through methods involving differentiability (Lagrangian or MRS shortcut).</p>
<p>I think the latter problem is the easier one, I'd start by setting <span class="math-container">$x_{1B} = x_{2B}$</span> as the standard procedure for solving a constrained Leontief optimization problem (equation <span class="math-container">$(0)$</span>).</p>
<p>I know equation <span class="math-container">$x_{1B} = x_{2B}$</span> is a valid equation for a contract curve. Is this already the contract curve?</p>
<p>What I found weird is I got a valid equation for a contract curve without even using the constraints <span class="math-container">$(1)-(3)$</span>.</p>
<p>If it isn't the contract curve, I wouldn't know how to use equation <span class="math-container">$(1)$</span> involving <span class="math-container">$A$</span>'s utility.</p>

<general-equilibrium><leontief>
How to find the contract curve when an agent has Leontief utility?
Nicolas Torres
# How to find the contract curve when an agent has Leontief utility?

(Asked by: username_0 on 2023-03-10)

I'm trying to solve the following excercise:

Find the contract curve for an economy where agents' ($A$ and $B$) preferences and endowments are given by:

$u\_A = x\_{1A}^{\frac{1}{2}} x\_{2A}^{\frac{1}{2}}$

$u\_B = \min\{x\_{1B},x\_{2B}\}$

$(\omega\_{1A},\omega\_{2A}) = (\alpha,\beta)$

$(\omega\_{1B},\omega\_{2B}) = (\beta,\alpha)$

I know the contract curve in this case can be found by solving:

* $\max x\_{1A}^{\frac{1}{2}} x\_{2A}^{\frac{1}{2}}$

subject to

$ \min \{x\_{1B},x\_{2B}\} = \overline{U}$

$x\_{1A} + x\_{1B} = \alpha + \beta$

$x\_{2A} + x\_{2B} = \alpha + \beta$

or by solving

* $\max \min\{x\_{1B},x\_{2B}\}$ $(0)$

subject to

$x\_{1A}^{\frac{1}{2}} x\_{2A}^{\frac{1}{2}} = \overline{U}$ $(1)$

$x\_{1A} + x\_{1B} = \alpha + \beta$ $(2)$

$x\_{2A} + x\_{2B} = \alpha + \beta$ $(3)$

I know the usual ways to find contract curves are either by forming a Lagrangian or as a shortcut, start from solving $MRS\_A = MRS\_B$.

However, the presence of a Leontief function makes either of these problems unsolvable through methods involving differentiability (Lagrangian or MRS shortcut).

I think the latter problem is the easier one, I'd start by setting $x\_{1B} = x\_{2B}$ as the standard procedure for solving a constrained Leontief optimization problem (equation $(0)$).

I know equation $x\_{1B} = x\_{2B}$ is a valid equation for a contract curve. Is this already the contract curve?

What I found weird is I got a valid equation for a contract curve without even using the constraints $(1)-(3)$.

If it isn't the contract curve, I wouldn't know how to use equation $(1)$ involving $A$'s utility.

**Tags:** general-equilibrium, leontief


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## 1 answers

### Answer by username_1 on 2023-03-10. Score: 3 | Accepted answer

In my opinion, your solution is correct.

The contract curve **must be such that $x\_{1B} = x\_{2B}$,** otherwise it could be possible to **increase the utility of one consumer without decreasing the utility of the other.**

Consider the following graph, where I have represented a generic Leontief utility function of a consumer $B$, $u\_B = \min\{{ax,by}\}$, and the indifference (convex) curves of a differentiable utility function of a consumer $A$.

[![enter image description here](https://i.sstatic.net/9t5sz.png)](https://i.sstatic.net/9t5sz.png)

The optimum bundles for consumer $B$ with Leontief preferences must be on the line of slope $a/b$, $y=\frac {a}{b} x$, where $ax =by$.

And the contract curve must be that line $y=\frac {a}{b} x$.$^1$

In fact, if we were at a point outside the line $y=\frac {a}{b} x$ as, for example, $A$, where the other consumer $A$ has a utility $\overline U$, we could go to point $C$ where the consumer with Leontief preferences increases their utility, and the utility of the other consumer is still $\overline U$. Contradicting the assumption that consumer $B$ is maximizing their level of utility with

$x\_{1A}^{\frac{1}{2}} x\_{2A}^{\frac{1}{2}} = \overline{U}\;\;\; (1)$.

Vice versa, seeing it from the other side, so to speak, if you want to maximize the utility of consumer $A$ for a given indifference curve of the Leontief utility function, you must place the bundle at the vertex of that indifference curve.

**Actually, you are using equation $(1)$:** the implicit assumption, from a mathematical point of view, is that there is an indifference curve of consumer $A$ (the blue lines) which passes through each vertex of the Leontief indifference curves, as points $B$ and $C$, which is guaranteed by the form of the utility function

$$u\_A = x\_{1A}^{\frac{1}{2}} x\_{2A}^{\frac{1}{2}}$$.

In other words, the red line $y=\frac {a}{b} x$ crosses, at every point, an indifference curve of consumer $A$.

And, in general, you are using the form of the indifference curves of consumer $A$. I mean, you are not maximizing utility for the Leontief function in the emptiness, from a mathematical point of view, you use the properties of the utility function of consumer $A$, beginning with the very basic requirement that the functions are defined where needed.

Actually, nothing very different from maximization with the budget constraint.

As for your objection that you don't use, in this case, the equations $(1)-(3)$, this is not completely true, because you then use them to find the bundle of goods $x\_{1A}, x\_{2A}$ of consumer $A$, and the related level of utility.

---

$^1$  In this Edgeworth Box the contract curve doesn't end, as usual, at the top right point $0\_A$, but I think that it is correct when we have Leontief preferences: once consumer $B$ is on the right vertical line, the utility of consumer $B$ cannot increase, it is 'stopped' by the total endowment of good $x$.

> **Comments:**
> **username_0**: Thanks! I appreciate your complete and general answer! I’m finding things confusing for Leontief and linear preferences in general equilibrium. I’ll ask a question soon for two linear utilities.  
> **username_1**: You are welcome. I agree with you, things are more confusing when there isn't differentiability.  
> **username_0**: I just asked the question about linear utilities here: https://economics.stackexchange.com/questions/54702/how-to-find-the-contract-curve-when-both-agents-have-linear-utilities  

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