# 18.3 Endowment optimization

Once we have an endowment budget line, the optimization problem is no different than with an exogenous income: we’re trying to get to the highest possible indifference curve along the budget constraint. However, while our optimal bundle was previously a function of prices and income, now the optimal bundle will be a function of prices and the endowment.

To illustrate this point, let’s think of the simple Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$. The optimal bundle is found by the at the point along the budget line where $MRS = x_2/x_1 = p_1/p_2$; solving for $x_2$, this gives us the **tangency condition**
\(x_2 = {p_1 \over p_2}x_1\)
If we plug this into the **income budget constraint** with exogenous income, we get
\(\begin{aligned}
p_1x_1 + p_2\left[{p_1 \over p_2}x_1 \right] &= m\\
2p_1x_1 &= m\\
x_1^\star &= {1 \over 2}{m \over p_1}
\end{aligned}\)
and therefore
\(x_2^\star = {p_1 \over p_2}x_1^\star = {1 \over 2}{m \over p_2}\)
If we plug the tangency condition into the **endowment budget constraint** for endowment $(e_1,e_2)$, we get
\(\begin{aligned}
p_1x_1 + p_2\left[{p_1 \over p_2}x_1 \right] &= p_1e_1 + p_2e_2\\
2p_1x_1 &= p_1e_1 + p_2e_2\\
x_1^\star &= {1 \over 2}{p_1e_1 + p_2e_2 \over p_1} = {1 \over 2}\left(e_1 + {p_2 \over p_1}e_2\right)
\end{aligned}\)
and therefore
\(x_2^\star = {p_1 \over p_2}x_1^\star = {1 \over 2}\left(e_2 + {p_1 \over p_2}e_1\right)\)

Note that these are really equivalent expressions, if we use monetary value of the endowment $\hat m = p_1e_1 + p_2e_2$ as the “income.” That’s a perfectly fine shortcut to use in solving these problems; just be sure to substitute the endowments back in at the end!

Finally, note that the expression for the optimal bundle given an endowment budget constraint depends *only on the ratio of the prices, not on the individual prices.* In other words, doubling all prices doesn’t change the location of the budget constraint (it still goes through the endowment with a slope of the price ratio), so it can’t affect the optimal bundle.

The graph below illustrates the optimal bundles for this utility function, given the two kinds of budget constraints: