How to find the contract curve when an agent has Leontief utility?

Question

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.

Answer 1

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 $A$.

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.

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$^1$ ^{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 $x$.}