8.7 Price and Income Offer Curves

Demand functions describe optimal behavior as a function of prices and income. One way of visualizing this behavior is to see how the optimal bundle moves in good 1-good 2 space due to a change in the price of a good or the consumer’s income. These kind of visualizations are called offer curves: they indicate the various “offers” the consumer would make when faced with different scenarios.

One way of thinking about an offer curve is as a parametric function. You might remember from a previous math course that the graph of a parametric function illustrates a case in which both $x$ and $y$ are functions of some third variable, $t$: that is, it plots the point $(x(t), y(t))$ as $t$ varies over a set range. For example, the two functions \(\begin{aligned}x(t) = t \cos(2\pi t)\\y(t) = t \sin(2 \pi t)\end{aligned}\) generate a spiral as $t$ increases from 0. Use the slider in the graph below to generate this spiral:

This graph isn’t a function in the usual sense of plotting $y$ as a function of $x$, or vice versa; rather, it illustrates the case in which the coordinates of the point $(x,y)$ are themselves functions of some third thing — in this case, $t$.

Similarly, offer curves plot the path an optimal bundle takes as price or incomes change. For example, the graph below shows the optimization problem for the Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$. We’ve established that the demand functions for this utility function are \(\begin{aligned}x_1^*(p_1,p_2,m) = {m \over 2p_1}\\x_2^*(p_1,p_2,m) = {m \over 2p_2}\end{aligned}\)

Price Offer Curves

The price offer curve for this function will illustrate how the optimal bundle changes as the price of one of the goods changes, holding the price of the other good and income constant. Let’s plot the price offer curve for good 1. Note that when $p_1$ changes, the coordiante of $x_1$ changes, but the coordinate of $x_2$ does not; therefore the optimal bundle moves to the left and right as $p_1$ changes. The following graph illustrates this:

Check the “show POC” box to see the price offer curve defined by this movement; note that as the price of good 1 changes, the optimal point moves along the POC. Doing so will also show a series of budget lines corresponding to $p_1 = 2, 4, 6,$ and $8$, and the corresponding optimal bundles.

The price offer curve for good 2, by a similar logic, would be a vertical line: as $p_2$ changes, the value of $x_2^\star$ would change, but the value of $x_1^\star$ would not.

Not all utility functions have POC’s which are vertical or horizontal lines. For example, with a perfect complements utility function, the POC will just connect all the points corresponding to the base of the “L” of the indifference curves. We’ll talk more about the price offer curves for complements and substitutes in the next section.

By and large, calculating the equation for price offer curves is beyond the scope of this course.

Income Offer Curves

The income offer curve illustrates how the optimal bundle changes as income changes, holding all prices constant. Note that in our Cobb-Douglas example, the consumer is spending half their income on good 1, and the other half on good 2, so as income increases or decreases, the consumer just scales their purchase up or down:

Check the “show IOC” box to see the income offer curve defined by this movement; note that as income changes, the optimal point moves along the IOC. As above, a few budget lines corresponding to $m = 40, 80, 120$, and $160$ will be shown as well.

Unlike price offer curves, there is a way to calculate the equation of the IOC:

• If the optimum is found at a tangency point, then the equation of the IOC is the tangency condition $MRS = p_1/p_2$. This makes sense, because $p_1/p_2$ is constant at every point along an IOC; so the IOC corresponds to all the points at which the MRS is equal to that particular price ratio. For example, in the Cobb-Douglas example above, the IOC is shown for $p_1/p_2 = 2$. Since the MRS of the utility function $u(x_1,x_2) = x_1x_2$ is $MRS(x_1,x_2) = x_2/x_1$, it follows that the tangency condition is \(MRS = {p_1 \over p_2} \Rightarrow {x_2 \over x_1} = 2 \Rightarrow x_2 = 2x_1\)You can see that the IOC does, indeed, have a slope of 2 in this example.
• If the optimum is found according to some other rule, such as “always buy at the base of the L” (perfect complements) or “spend all your money on good 1” (perfect substitutes if $MRS > p_1/p_2$), then the IOC just follows that rule. So the IOC of a perfect complements function will connect all the bases of the L-shaped indifference curves, and the IOC for a situation in which you only buy good 1 corresponds to the horizontal axis.

More complicated behavior can also be captured by the IOC. For example, consider the quasilinear utility function $u(x_1,x_2) = 80 \ln x_1 + x_2$. As we derived here, when $p_2 = 1$, this utility function corresponds to the demand functions \(\begin{aligned} x_1^\star(p_1,m) &= \begin{cases} {80 \over p_1} & \text{ if }m \ge 80\\ \\ {m \over p_1} & \text{ if }m \le 80 \end{cases}\\ \\ x_2^\star(p_1,m) &= \begin{cases} m - 80 & \text{ if }m \ge 80\\ \\ 0 & \text{ if }m \le 80 \end{cases} \end{aligned}\) With $p_1 = 2$, this means that the consumer will buy 40 units of good 1 as long as they have at least $m = 80$, with their optimal bundle being characterized by a tangency condition; and if they have $m < 80$, they won’t be able to afford 40 units of good 1, so their optimal bundle is a corner solution in which they spend all their money on good 1. As above, this graph shows their optimal bundle at different incomes:

Note that here the IOC has two segments, corresponding to the two “rules” for the optimal bundle: a lower horizontal section that corresponds to the rule “buy only good 1 when $m < 80$,” and a vertical section that corresponds to the rule “buy at the point of tangency if $m > 80$.”