# How to calculate the intercept of an indifference curve in an Edgeworth box?

Consider an exchange economy with two goods (Good 1 and 2) and two individuals ($$A$$ and $$B$$):

• $$A$$'s utility function is $$u_A(x_{A_1}, x_{A_2}) = 2x_{A_1} + 5x_{A_2}$$

• $$B$$'s utility function is $$u_B(x_{B_1}, x_{B_2}) = 3x_{B_1} + 7x_{B_2}$$

• $$A$$'s initial endowment is $$(\omega_{A_1}, \omega_{A_2}) = (4, 6)$$

• $$B$$'s initial endowment is $$(\omega_{B_1}, \omega_{B_2}) = (10, 2)$$

The core condition requires:

$$2x_{A_1} + 5x_{A_2} \geq 2\omega_{A_1} + 5\omega_{A_2} = 2 \times 4 + 5 \times 6 = 38$$

and

$$3x_{B_1} + 7x_{B_2} \geq 3\omega_{B_1} + 7\omega_{B_2} = 3 \times 10 + 7 \times 2 = 44$$

The line with equation $$2x_{A_1} + 5x_{A_2} = 38$$ hits the left boundary at:

$$(x_{A_1}, x_{A_2}) = (0,38/5), ~~~~(x_{B_1}, x_{B_2}) = (14, 2/5)$$

The line with equation $$3x_{B_1} + 7x_{B_2} = 44$$ hits the left boundary at:

$$(x_{A_1}, x_{A_2}) = (0,54/7) ~~~ (x_{B_1}, x_{B_2}) = (14, 2/7)$$

I don't understand how these intercepts have been calculated.

I've used the MRS and endowment point to calculate the $$(x_{A_1}, x_{A_2})$$ intercepts correctly:

$$x_{A_2} = -\frac{2}{5}x_{A_1} + \frac{38}{5}$$

But get an incorrect $$(x_{B_1}, x_{B_2})$$ intercept:

$$x_{A_1} = 14, ~~ x_{A_2} = 2$$

Please can someone enlighten me? No worries if you can't.

Alex

In an edgeworth box model the total amount of goods must be constant. In your problem this translates to

$$\begin{eqnarray} x_{A_1} + x_{B_1} &=& \omega_{A_1} + \omega_{B_1} = 10 + 4 = 14 \tag{1}\\ x_{A_2} + x_{B_2} &=& \omega_{A_2} + \omega_{B_2} = 6 + 2 = 8 \tag{2}\\ \end{eqnarray}$$

So let's consider the first intercept

$$(x_{A_1}, x_{A_2}) = (0, 38/5) \tag{3}$$

which you properly derived. If you replace these values in Eq. (1) you get

$$0 + x_{B_1} = 14 ~~~\Rightarrow~~~ x_{B_1} = 14 \tag{4}$$

and if you replace in Eq. (2) you get

$$\frac{38}{5} + x_{B_2} = 8 ~~~\Rightarrow~~~ x_{B_2} = \frac{2}{5} \tag{5}$$

So there you have it

$$(x_{A_1}, x_{A_2}) = (0,38/5), ~~~(x_{B_1}, x_{B_2}) = (14,2/5)$$

You can do the same for the other intercept, I will leave that for you to complete