Remove Linear Good From Quasi-linear Utility Function

Question

Given a quasi-linear utility function: $u(x_1, x_2) = f(x_1) + \beta x_2$, $\beta > 0 $

What would happen if good 2 ($x_2$) is removed from the market? Would the new utility function be: $u(x_1) = f(x_1)$? If so, would the demand function simply be $x_1 = m/p_1$? I want to find the new demand function as well as the new indifference curves and draw these in a diagram, but I am unsure how to proceed. Surely I cannot graph them in $x_1, x_2$ space if there is no more $x_2$. Or would I still graph them in $x_1, x_2$ space as vertical lines?

I hope my question is understandable, thx for any help.

Answer 1

This is one possible interpretation. Good 2 being removed from the market can simply be interpreted as $x_2 = 0$. In an economic interpretation the good does not simply disappear from the utility function in the sense that preferences do not change, it is just the availability of the good that changes. This is an external condition, so you can simply think of this as a market constraint $x_2 = 0$.

Now, looking at indifference curves as the different bundles for which the consumer obtains the same level of utility, and defining this level as $k$. It is clear that for any $k$ when there is only one good, each "indifference curve" will consist of only one point (in particular $x_{1}|u(x_1,0) = k$). In a 2-D graph this will simply correspond some point ($x_1$,0) for each $k$ level.

The demand function should be quite straightforward.

Answer 2

If you just have one good in your utility function, then you really dont need an indifference curve. The reason we use IC is because we cant draw 3-D graphs properly without a computer (Remember $U(x,y)$ requires 3 axes to plot: X-axis, Y-axis, and a Z-axis to plot the values of U). So indifference curve (which basically draws the contours of the utility function) provides an easy illustration.

Given this caveat, lets plot the IC of $U(x,y) = f(x)$. The right way to think about this is that good $Y$ is available, but the consumer does not derive any utility from it.

Recall the definition of an indifference curve: An indifference curve curve with utility level $k$ is the set of all bundles $(x,y)$ such that $U(x,y)=k$.

In our case, suppose $(x^*,0)$ provides the utility level $k$: $U(x^*,0)=f(x^*)=k$. What other bundles give the same utility? Note that you can change $y$ and still receive utility $k$: $U(x^*,0)=U(x^*,1)=.....=k$ and so on. Thus the IC that yields utility $k$ is a vertical line (parallel to the Y-axis).

Assuming that $f(x)$ is increasing, vertical lines towards the right correspond to higher utility levels (obvious, since the value of $x$ increases as we move right).

You can now easily verify that the optimal bundle should be $x^*=\frac{m}{p}$. Draw the vertical IC. Draw the downward sloping budget line. Since higher utility is achieved by moving right, keep moving till you reach the extreme right point of your budget set - this is exactly the point $(\frac{m}{p},0)$.