Consider an exchange economy with 2 goods $$x,y$$ and $$n+1$$ consumers $$0,1,\cdots,n$$. Consumer 0 has utility function $$u_0(x,y)=x^\alpha y^{1-\alpha}$$ and endowment $$w_0=(1,0)$$. Other consumers $$1,\cdots, n$$ are identical with utility functions $$u_i(x,y)=x^\beta y^{1-\beta}$$ and endowments $$w_i=(0,1)$$. Suppose $$0<\alpha<\beta<1$$. If we denote the relative price of $$x$$ to $$y$$ is $$\bar{p}$$, then (if I'm correct) a Walrasian equilibrium should be $$x_1^*=\alpha, y_1^*=n\beta, x_2^*=\frac{1-\alpha}{n}, y_2^*=1-\beta$$, and $$\bar{p}=\frac{n\beta}{1-\alpha}$$.

But now suppose that the economy opens up to free trade and as a result, the relative price of $$x$$ to $$y$$ is doubled: $$\hat{p}=2\bar{p}$$. I need to answer 2 questions:

1. How much each good would the economy export or import in the new equilibrium? Who gains and who loses?
2. Find a system of lump-sum transfers so that after free trade, every consumer in the economy is strictly better off.

How should I proceed with the Walrasian equilibrium that I found?

First, let us observe the autarky equilibrium:

Consumer $$0$$'s utility maximisation problem is

$$\displaystyle\max_{x_0\geq 0,y_0\geq 0}x_0^\alpha y_0^{1-\alpha}$$ subject to $$px_0+y_0\leq p$$

Solving it we get:

$$(x_0^d,y_0^d)\left(p\right)=\left(\alpha, (1-\alpha)p\right)$$

Consumer $$i$$'s utility maximisation problem, where $$i\in\{1,\ldots,n\}$$, is

$$\displaystyle\max_{x_i\geq 0,y_i\geq 0}x_i^\beta y_i^{1-\beta}$$ subject to $$px_i+y_i\leq 1$$

Solving it we get:

$$\displaystyle(x_i^d,y_i^d)\left(p\right)=\left(\frac{\beta}{p}, 1-\beta\right)$$

Now we can consider market for, say $$y$$, and equate demand and supply and solve for $$p$$:

$$(1-\alpha)p+(1-\beta)n=n$$

and we get the equilibrium $$p$$ i.e. $$\overline{p}$$ as:

$$\overline{p}=\dfrac{\beta n}{1-\alpha}$$.

Corresponding consumption of the consumers are:

$$\overline{x}_0=\alpha, \ \overline{y}_0=\beta n, \ \overline{x}_i=\dfrac{1-\alpha}{n}, \ \overline{y}_i=1-\beta$$.

With trade, $$\hat{p}=2\overline{p}=\dfrac{2\beta n}{1-\alpha}$$.

Corresponding consumption of the consumers now are:

$$\hat{x}_0=\alpha, \ \hat{y}_0=2\beta n, \ \hat{x}_i=\dfrac{1-\alpha}{2n}, \ \hat{y}_i=1-\beta$$.

Clearly, the economy will export $$x$$, and the magnitude of export equals:

$$1-\alpha-\dfrac{(1-\alpha)}{2}=\dfrac{1-\alpha}{2}$$

and it will import $$y$$, and the magnitude of import equals:

$$2\beta n+(1-\beta)n-n=\beta n$$

As we can see by comparing the consumption bundles of $$0$$ and $$i$$s that consumer $$0$$ is better off from the trade, whereas consumers - $$i$$s are worseoff.

One easy way to find the transfers so that every consumer is strictly better off with trade is to make sure that transfers are such that they can all afford the autarky equilibrium bundle at the free trade prices i.e. Transfer from $$0$$ to others equal $$\hat{p}-(\hat{p}\alpha+\beta n)=\dfrac{2\beta n}{1-\alpha}-\left(\dfrac{2\alpha\beta n}{1-\alpha}+\beta n\right)=\dfrac{2\beta n}{1-\alpha}-\left(\dfrac{\alpha\beta n+\beta n}{1-\alpha}\right)=\beta n$$

Therefore, Transfer to each $$i$$ from $$0$$ equals $$\beta$$.

So, we have the post-transfer income of $$0$$ as $$\hat{p}-\beta n=\dfrac{\beta n(1+\alpha)}{(1-\alpha)}$$ and of $$i$$s as $$1+\beta$$. Corresponding choices of the consumers are:

$$\tilde{x}_0=\dfrac{\alpha(1+\alpha)}{2}, \ \tilde{y}_0=\beta n(1+\alpha), \ \tilde{x}_i=\dfrac{(1+\beta)(1-\alpha)}{2n}, \ \tilde{y}_i=(1-\beta)(1+\beta)$$.

• thank you! what about the system of lump-sum transfers?
– user45416
Oct 29, 2023 at 20:19
• Updated the answer with Lump-sum transfers from $0$ to others that will ensure that everyone is better off with trade.
– Amit
Oct 30, 2023 at 1:55
• may I ask why the way of finding such a scheme is to make sure that individuals can afford the autarky equilibrium bundle at the free trade price?
– user45416
Oct 30, 2023 at 19:21
• Choosing it this way will ensure that autarky equilibrium bundle is affordable to all the agents at the free trade prices. So whatever they will choose at the free trade prices will be at least as good as the autarky bundle. And if their choice is unique and different from the autarky bundle, then the consumers will strictly prefer that choice to the autarky bundle which is the case here.
– Amit
Oct 31, 2023 at 0:14
• Dear Amit, do you mind taking a look at this question? economics.stackexchange.com/questions/57123/…
– user45416
Dec 7, 2023 at 21:52