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: