library(tidyverse)
ggplot() +
stat_function(fun = function(x1) 5 - x1, color = "black") +
xlim(0, 5)
6 Choice
For more information on these topics, see Varian Chapter 5: Choice.
6.1 Objective
In this chapter, you will combine the budget set and the theory of preferences to examine the optimal choice of consumers. Earlier, I stated that the economic model of consumer choice is that people choose the best bundle they can afford. Now I can rephrase that more professionally as, “consumers choose the most preferred bundle from their budget sets.”
6.2 Optimal Choice
I’ll plot the budget line \(p_1 x_1 + p_2 x_2 = 5\) for the case where the price of both goods 1 and 2 are 1 and the consumer has an income of 5 to spend.
Then I’ll add indifference curves of the utility function \(u(x_1, x_2) = x_1 x_2\).
p <- ggplot() +
stat_function(fun = function(x1) 5 - x1, color = "black") +
stat_function(fun = function(x1) 2/x1, color = "red") +
stat_function(fun = function(x1) 4/x1, color = "orange") +
stat_function(fun = function(x1) 6.25/x1, color = "gold1") +
stat_function(fun = function(x1) 8/x1, color = "green") +
stat_function(fun = function(x1) 10/x1, color = "blue") +
xlim(0, 5) +
ylim(0, 5) +
geom_point(data = point_data, aes(x = x, y = y), color = "black", size = 3) +
transition_reveal(along = -x)
animate(p)
Let your eye follow the black dot along the budget line: along that line, the consumer exhausts their income on goods 1 and 2. The budget line crosses the red indifference curve (\(u = 2\)), so you know that the consumer can get more than a utility of 2. The budget line also crosses the orange indifference curve (\(u = 4\)), so you know that the consumer can get more than a utility of 4. Something interesting happens at the gold indifference curve (\(u = 6.25\)): the budget line doesn’t cross the gold indifference curve, the budget line is tangent to the gold indifference curve. The point of tangency represents the bundle that will earn the consumer the most utility (\(u = 6.25\)), while still being affordable.
As long as preferences are convex, indifference curves are smooth, and there is an interior solution, then the optimal choice \((x_1^*, x_2^*)\) is where the indifference curve is tangent to the budget line. At that point, the slope of the indifference curve (the MRS) is equal to the slope of the budget line (\(-p_1/p_2\)).
6.3 Perfect Substitutes
Consider this budget line and these indifference curves for butter and margarine:
6.4 Classwork 6
Consider the utility function: \(u(x_1, x_2) = x_1^{0.5} x_2^{0.5}\). This is a monotonic transformation of the \(u(x_1, x_2) = x_1 x_2\) utility function that we’ve explored a couple of times, so indifference curves are the well behaved ones we’re used to seeing. Take another monotonic transformation (natural log) to see that \(u(x_1, x_2) = 0.5 \ln x_1 + 0.5 \ln x_2\): I’ll do this to make \(u\) easier to work with. If the optimal bundle is an interior point, then we have 2 equations:
\(MRS = \frac{-p_1}{p_2}\), where \(MRS\) is the slope of the indifference curve. We learned last chapter that \(MRS = -\frac{\ \partial u(x_1, x_2)/\partial x_1}{\partial u(x_1, x_2) / \partial x_2}\).
Budget constraint must hold at an equality: \(p_1 x_1 + p_2 x_2 = m\). Suppose \(p_1 = 1\) and \(p_2 = 1\), and income \(m = 5\).
And we have 2 unknowns: \(x_1\) and \(x_2\). Solve for the 2 equations and 2 unknowns: you should get \(x_1 = 2.5\) and \(x_2 = 2.5\).
A utility function from part a) that’s of the form \(u(x_1, x_2) = x_1^a x_2^b\), where \(a\) and \(b\) are non-negative and \(a + b = 1\), is called a Cobb-Douglas utility function. It has a useful property where the exponents \(a\) and \(b\) tell you the share of the income that will be spent on goods 1 and 2 at the optimal bundle. For instance, in part a), the exponents were (0.5, 0.5) and you showed that when prices were equal, we bought equal amounts of good 1 and good 2. Write a proof of the fact that if utility is Cobb-Douglas, then you’ll spend \(a\) of your income on good 1 and \(b\) of your income on good 2.
Suppose that we observed a consumer’s choices at several prices and income levels (see the table below). \(s_1\) and \(s_2\) are the shares of the consumer’s income spent on goods 1 and 2.
This data shows that the consumer’s share of spending on goods 1 and 2 stays very stable at \(0.25\) and \(0.75\), respectively, so the Cobb-Douglas utility function would be an excellent choice to model this consumer’s preferences: \(u(x_1, x_2) = x_1^{0.25} x_2^{0.75}\). Add another column \(u\) that calculates the consumer’s utility during each year.
We can use this model of the consumer’s preferences to evaluate the impact of proposed policy changes (to do counterfactual analysis). Suppose for example that the government was contemplating imposing a system of taxes that would result in prices (2, 3) and the consumer having income 200. How much of goods 1 and 2 would the consumer choose to purchase, and what would their utility be? Would they be better or worse off compared to the years in the data?
Economists often analyze how a proposed policy might affect different groups of individuals, such as teenagers, middle-class households, and the elderly. Each group has unique preferences for various goods, which are evident in their consumption expenditure patterns. By estimating a utility function that describes these consumption patterns, economists can forecast demand and evaluate the potential impact of policy proposals.
| year | \(p_1\) | \(p_2\) | m | \(x_1\) | \(x_2\) | \(s_1\) | \(s_2\) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 100 | 25 | 75 | .25 | .75 |
| 2 | 1 | 2 | 100 | 24 | 38 | .24 | .76 |
| 3 | 2 | 1 | 100 | 13 | 74 | .26 | .74 |
| 4 | 1 | 2 | 200 | 48 | 76 | .24 | .76 |