1.22 Summary

These lecture notes have been a pretty comprehensive review of the consumer theory from Econ 50 that you should be familiar with. Let’s summarize the most important things you should know, and what you should be able to do.

Things you should know

• Good 1 - Good 2 space: a diagram showing units of some “good 1” on the horizontal axis, and some “good 2” on the vertical axis. All slopes in good 1 - good 2 space are measured in units of good 2 per unit of good 1.
• Budget constraint: the set of bundles/choices available to a consumer. If a consumer has income/wealth $m$ and faces constant prices $p_1$ and $p_2$, the consumer faces the linear budget constraint $p_1x_1 + p_2x_2 = m$. The slope of this budget line is $-p_1/p_2$ (the “price ratio”)
• Utility function: a functional form relating a bundle $(x_1,x_2)$ to a certain number of “utils”.
• Indifference curve: a contour line (level set) of a utility function
• Marginal rate of substitution: the amount of good 2 a consumer would be willing to give up to get another unit of good 1; equal to the ratio of the partial derivatives of the utility function: \(MRS(x_1,x_2) = \frac{\partial u(x_1,x_2)/\partial x_1}{\partial u(x_1,x_2)/\partial x_2} = \frac{MU_1}{MU_2}\)
• Demand Function: A function describing the optimal bundle a function of prices and income
• (Gross) Demand Curve: A plot in a diagram showing the quantity of a good demanded at different prices of that good, holding the price of other goods and income constant.

Specific utility functions and their behaviors

In this course we’ll be focused on utility functions of the form $u(x_1,x_2) = a v(x_1) + b v(x_2)$ for some function $v(x)$:

Cobb-Douglas utility function: $v(x) = \ln x$ \(u(x_1,x_2) = a \ln x_1 + b \ln x_2 \text{ or } u(x_1,x_2) = x_1^ax_2^b\)

• Can normalize using $\alpha = a/(a+b)$ to write as $\alpha \ln x_1 + (1-\alpha) \ln x_2$
• Intuitive description: the consumer views the two goods as independent goods (neither complements nor substitutes). The greater $a/b$ (or the greater $\alpha$), the more the consumer likes good 1 relative to good 2.
• Marginal rate of substitution: \(MRS = \frac{a}{b}\times\frac{x_2}{x_1}\)
• Optimal behavior given a linear budget constraint of the form $p_1x_1 + p_2x_2 = m$: spend fraction $a/(a+b)$ of $m$ on good 1, and fraction $b/(a+b)$ on good 2: \(x_1^\star(p_1,p_2,m) = \frac{a}{a+b} \times \frac{m}{p_1}\) \(x_2^\star(p_1,p_2,m) = \frac{b}{a+b} \times \frac{m}{p_2}\) This is the “Cobb-Douglas trick” and we will use it a lot!

“Weak substitutes” utility function: e.g., $v(x) = \sqrt{x}$ \(u(x_1,x_2) = a\sqrt{x_1} + b\sqrt{x_2}\)

• Intuitive description: the consumer views the two goods as substitutes, but not perfect substitutes; an increase in the price of one causes them to consume less of that good and more of the other. \textit{Note: this is true of any function of the form $v(x) = x^r$ with $0 < r < 1$; but the math is easier with $r = \frac{1}{2}$, as in this case.}
• Marginal rate of substitution: \(MRS = \frac{a}{b}\sqrt{\frac{x_2}{x_1}}\)
• Optimal bundle given a linear budget constraint of the form $p_1x_1 + p_2x_2 = m$: \(x_1^\star(p_1,p_2,m) = \frac{p_2}{p_1} \times \frac{a^2m}{b^2p_1 + a^2p_2}\) \(x_2^\star(p_1,p_2,m) = \frac{p_1}{p_2} \times \frac{b^2m}{b^2p_1 + a^2p_2}\)

Perfect substitutes utility function: $v(x) = x$ \(u(x_1,x_2) = ax_1 + bx_2\)

• Intuitive description: the consumer gets $a$ units of utility from each unit of good 1, $b$ units of utility from each unit of good 2, regardless of how many of each bundle they have. Therefor, this utility function represents preferences that are neither convex nor concave.
• Marginal rate of substitution: \(MRS = \frac{a}{b}\)
• Optimal behavior given a linear budget constraint of the form $p_1x_1 + p_2x_2 = m$: buy only good 1 if $a/b > p_1/p_2$; only good 2 if $a/b < p_1/p_2$. If $a/b = p_1/p_2$, indifferent between all possible points along the constraint.

Utility function for concave preferences: e.g., $v(x) = x^2$ \(u(x_1,x_2) = ax_1^2 + bx_2^2\)

• Intuitive description: the consumer has \textit{increasing} MRS as they get more of good 1; they would rather have lots of either good than a mix. \textit{Note: this is true of any function of the form $v(x) = x^r$ with $r > 1$; but the math is easier with $r = 2$, as in this case.}
• Marginal rate of substitution: \(MRS = \frac{a}{b} \times \frac{x_1}{x_2}\)
• Optimal behavior given a linear budget constraint of the form $p_1x_1 + p_2x_2 = m$: buy only good 1 if $a/b > p_1/p_2$; only good 2 if $a/b < p_1/p_2$. If $a/b = p_1/p_2$, indifferent between buying only good 1 or only good 2 (but NOT a combination).

Quasilinear utility function: $u(x_1,x_2) = v(x_1) + x_2$ (e.g., $a\sqrt{x_1} + x_2$ or $a \ln x_1 + x_2$)

• Intuitive description: The consumer’s marginal rate of substitution only depends on good 1. Often this is used to represent situations in which “good 2” is “money spent on other goods,” in which case the “price ratio” is just the price of good 1, so the budget constraint is $px_1 + x_2 = m$; and since each unit of good 2 brings a utility of 1, this means that utility is measured in dollars.
• Marginal rate of substitution: \(MRS = v'(x)\)
• Optimal behavior given a linear budget constraint: set $v’(x_1) = p$ and solve for $x_1^\star$; plug this into the budget constraint and solve for $x_2^\star$.
  • If $x_2^\star > 0$ and $x_2^\star > 0$, buy the bundle $(x_1^\star,x_2^\star)$.
  • If $x_2^\star \le 0$, buy only good 1.=
  • If $x_1^\star \le 0$, buy only good 2.

Things you should be able to do

• Transform utility functions using a monotonic transformation
• Given one of the utility functions above:
  • Draw an indifference curve
  • Find and plot the optimal bundle subject to a linear budget constraint
  • Find and plot the optimal bundle subject to a kinked budget constraint
  • Find and plot the demand curve for good 1 or good 2
• Be able to handle the special case in which $a + b = 1$ – e.g., $u(x_1,x_2) = \alpha \ln x_1 + (1-\alpha) \ln x_2$
• Be able to handle the special case in which $a = 1$ – e.g., $u(x_1, x_2) = \ln x_1 + \beta \ln x_2$