Consumer Choice

Perloff 4; WB 5 and 6.
We examine the foundation of the Demand curve: how each consumer chooses which goods to buy, and how much of them.

In applying consumer theory, we will see how Demand can be derived from these choices, and revisit a few key Demand shifters: the good's price, other goods' prices and income.

The graphing techniques for the supply-and-demand model are again useful here.

In our model the consumer chooses her “best affordable bundle.” This is explained in three steps:

  • A bundle is a combination of goods.
  • The affordable bundles are determined by the budget.
  • The best affordable bundle is determined by preferences.

Bundles

Consumers choose what to buy and how much. Mathematically, this choice is a bundle — an amount (possibly none) of each good.

We restrict attention to two goods, X and Y, but this is not as restrictive as it sounds.1 X could be the good we are interested in, while Y is a composite good representing spending on everything else.

We also generally restrict attention to “goods.” As discussed in the "Preferences" section below, for every bad there is an equivalent good. Products that are good up to a point and then bad (like how much water you drink within a day) can also be dealt with if we allow the consumer to throw away, or freely dispose of, any unwanted goods.

Budget

A consumer's budget describes the bundles of goods she can afford. A full description can be made using how much money she has (called “income” below) and the prices of each good.

Given income I and prices PX and PY,

  • Graph the consumer's budget set.
  • Find the consumer's marginal rate of transformation (MRT).
  • Identify how much a given bundle costs and whether or not it is affordable.
  • Identify a bundle, X and Y, based on expenditures on each good.
  • Find an affordable bundle, given some additional constraints (e.g., a minimum/maximum amount of X, a particular amount of X).

Budget changes

Given income I and prices PX and PY, show on a graph how the budget is affected by…

  • A decrease in income, or an increase in the income tax rate.
  • An increase in the price of one good, or a new tax on that good.
  • A change in both prices by a factor $\alpha$ (equivalent to when income changes by the factor $1/\alpha$).

Non-linear budgets

Given income I and prices PX and PY for food (X) and a composite good (Y), graph the budget set when…

  • The consumer has a given number of food stamps.
  • The consumer has a given number of food stamps that can be resold on the black market at a given price.
  • Graph the budget set when there is a maximum quota of Y that can be consumed (for given numbers).
  • Graph the budget set when there is a minimum quota of Y that must be consumed (for given numbers).
  • Graph the budget set when there is a quantity discount on X (i.e., the price is lower when you buy more; for given numbers).
  • Graph the budget set when there is an optional “membership fee” that reduces the price of X (for given numbers, see Perloff Ch 4, q21).

Preferences

A consumer's preferences describe how the consumer would decide between a pair of bundles. We can represent these preferences with a utility function, provided some assumptions hold.

Key assumptions

We assume that the consumer is

  • Insatiable — no matter how much of X and Y are being consumed, a little more of either is always better.
  • Decisive — she can compare any two bundles. For any bundles a and b, one of the following comparisons holds
    • a is preferred to b
    • b is preferred to a
    • the consumer is indifferent between a and b
  • Consistent — no comparisons form a loop.

These are usually called (strict) monotonicity, completeness and transitivity, respectively.

If monotonicity is reversed for good X so that it is a “bad,”2, we can still look at this case:

  • Choose some $\bar X$ that is so high that it will never be consumed.
  • Define a new good Z = $\bar X$-X.

To clarify how this works, try graphing indifference curves for XY and ZY.

Continuity and ICs

Under one additional assumption — that the preferences are “continuous”3 — we can, given a bundle, draw the set of other bundles that leave the consumer just as well off.

  • How do we know that this indifference set is a curve, not a band?
  • How do we know that the indifference curve (IC) is downward sloping?
  • How do we know that no two ICs cross?

The slope of the IC is the marginal rate of substitution (MRS). It represents the trade-off between X and Y that the consumer is willing to make.

Convexity

Our final assumption is that the consumer's indifference curves are strictly convex, or equivalently that the MRS is decreasing. This means that the curves “bow” towards the origin.

Three important types of preferences (almost) satisfy all our assumptions: perfect substitutes, perfect complements and Cobb-Douglas, or "imperfect substitutes". See Perloff, Figure 4.4. With perfect complements, strict monotonicity does not hold; and with perfect substitutes, strict convexity does not hold.

From a graph of ICs…

  • Recognize which type of preferences are being graphed.
  • With perfect substitutes, identify the ratio at which the consumer is willing to trade Y for X (it's the slope of the ICs or the MRS).
  • With perfect complements, identify the ratio in which the consumer likes to consume X and Y (it's the slope of a line through the “corners” of the ICs).

(Reversing the analysis above…)

  • For a given ratio at which the consumer is willing to trade Y for X, graph the ICs.
  • For a given ratio in which the consumer likes to consume X and Y, graph the ICs.

Utility representation

With the five assumptions above, we can represent the preferences using a utility function. This will be useful when we solve the consumer's problem.

Find the utility function…

  • For a given ratio at which the consumer is willing to trade Y for X.
  • For a given ratio in which the consumer likes to consume X and Y.
  • From a graph of perfect complements ICs.
  • From a graph of perfect substitutes ICs.

(Reversing…) From the utility function

  • Graph the ICs (but not for Cobb-Douglas utility).
  • With perfect substitutes, identify the ratio at which the consumer is willing to trade Y for X.
  • With perfect complements, identify the ratio in which the consumer likes to consume X and Y .

The slopes of the graphs UX and UY are the marginal utilities MUX and MUY. It is shown in class that the slope of the consumer's indifference curve (the MRS) at any point is $-MU_X/MU_Y$.

Utility maximization

The consumer's problem is to find the best bundle, X and Y, that is within her budget.

  • Find the best bundle, X* and Y*, given the budget, I, PX, and PY, and a utility function.
  • Find the best bundle, X* and Y*, given the budget, I, PX, and PY, and a description of preferences.

We need to solve for two numbers, so we will need two equations. The first condition is common to the three problems: all income is spent.

(1)
\begin{equation} I=XP_X+YP_Y \end{equation}

Spending more is not affordable, and spending less can never be optimal (by monotonicity). The second condition differs in each case.

Perfect substitutes

Because the goods are perfectly substitutable, the consumer will buy only one good. Compare the utility from spending all income on one good with the utility from spending all income on the other. Whichever is higher is the one the consumer buys, and gives the second equation needed to solve the consumer's problem.

Perfect complements

The consumer always wants to consume the two goods together in a particular ratio. The second equation says that they will do so. For example, if the consumer likes to to have 3 units of good X for every 4 units of good Y, then the second equation is 4X=3Y (this is not a typo!).

Cobb-Douglas

At the consumer's optimal bundle, her indifference curve will be just tangent to the budget line. So the second condition in this case is that the slopes match MRS = MRT. Intuitively, this is because the following two trade-offs must be the same:

  • The trade-off between X and Y you are willing to make.
  • The trade-off between X and Y you can make at market prices.

An alternative way of solving the Cobb-Douglas case is noting that if U = XAYB and A+B=1, then A is the fraction of income spent on X, while B is the fraction of income spent on Y. If A+B$\neq$1, then A/(A+B) is the fraction spent on X, while B/(A+B) is the fraction spent on Y.

Non-linear budgets

For budgets that are piecewise linear (made up of several linear budgets) and convex, solve the consumer's problem on each budget and compare utilities at each solution.

For budgets that are piecewise linear and concave (bow away from the origin), (i) solve the consumers problem on each budget; (ii) eliminate solutions that are outside the budget; (iii) if two or more solutions are left, compare utilities at each remaining solution; (iv) if no solutions are left, compare utilities at "corners" of the budget line.

With three goods

Consider the model in which the consumer has fixed income I, to spend on goods X, Y and Z. For each of the three types of preferences described above, we can solve the problem similarly.

  • For Cobb-Douglas preferences, use the income-share result above — if U = XAYBZC and A+B+C=1, then…
  • For perfect substitutes, only one good is bought, so compare the utility from spending all consumption on each.
  • For perfect complements, there is a ratio in which the three goods are consumed; this gives us two more equations.

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