2
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.