8.1 Introduction

As we’ve seen, the process for maximizing a utility function subject to a budget constraint is exactly the same as maximizing a utility function subject to a linear PPF. If we’re given specific values of $p_1$, $p_2$, and $m$, the process is in fact exactly the same as in Chapters 5 and 6. For example, suppose you had to maximize the Cobb-Douglas utility function \(u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}\) subject to a PPF with the equation \(2x_1 + x_2 = 12\) Since the MRS for this utility function is $x_2/x_1$ and the MRT of this PPF is 2, you asserted that the optimum would be characterized by the tangency condition \(\begin{aligned} MRS &= MRT\\ {x_2 \over x_1} &= 2\\ x_2 &= 2x_1 \end{aligned}\) And you plugged that into the PPF constraint to find the optimal choice: \(\begin{aligned} 2x_1 + 2x_1 &= 12\\ 4x_1 &= 12\\ x_1^\star &= 3\\ x_2^\star &= 2x_1^\star = 6 \end{aligned}\) This is exactly the same problem — and the same procedure you would follow — as if you had to maximize that utility function subject to the budget line defined by $p_1 = 2$, $p_2 = 1$, and $m = 12$.

However, in this section we’re now interested in characterizing the optimal bundle as a function of prices and income. Therefore we’ll be interested in solving this problem, keeping $p_1$, $p_2$, and $m$ as variables. Therefore our tangency condition will set the MRS equal to the generic slope of the budget line, $p_1/p_2$: \(\begin{aligned} MRS &= {p_1 \over p_2}\\ {x_2 \over x_1} &= {p_1 \over p_2}\\ x_2 &= {p_1 \over p_2}x_1 \end{aligned}\) We’ll then plug this into the generic budget line, $p_1x_1 + p_2x_2 = m$: \(\begin{aligned} p_1x_1 + p_2\left[{p_1 \over p_2}x_1\right] &= m\\ p_1x_1 + p_1x_1 &= m\\ 2p_1x_1 &= m\\ x_1^\star(p_1,p_2,m) &= {m \over 2p_1}\\ x_2^\star(p_1,p_2,m) &= {p_1 \over p_2}x_1^\star(p_1,p_2,m) = {m \over 2p_2} \end{aligned}\) Note that if you then plug in $p_1 = 2$, $p_2 = 1$, and $m = 12$, you get the same bundle as before; but now we have a more general demand function expressing our optimal bundle $(x_1,x_2)$ as a function of the parameters of the model: $p_1$, $p_2$, and $m$.

In the diagram below you can see how this works. The left-hand graph shows a 3D rending of the utility function; the “hill” represents the amount of utility at every bundle $(x_1,x_2)$. The budget line, in green, shows a path over the hill. The blue dot shows the highest point along the path.

The diagram on the right shows the projection of this onto plane: that is, the budget line and the indifference curve through the optimal bundle.

Try changing the prices and income. You can see that the hill does not change but the path over the hill does change, and therefore the highest point along the path changes:

This is one of the hardest concepts to grasp, especially because it seems like indifference curves “shift” as you change the budget line (and therefore the optimal point). Key to understanding this is that prices do not affect utility: changes in the prices do not affect the curvature of the hill, but only the location of the path over the hill.

In this chapter we’ll derive the optimal choice for a number of the standard utility functions.