Notes 6: Consumption
There are two predominant theories of consumption:
1)The Keynesian Consumption Function
2)The Permanent Income Consumption Function (Life Cycle Consumption Function).
Let us start with the second of the theories (the more difficult one) and then we will compare/contrast the two theories:
The Permanent Income Consumption Function (from now on, we will call it the PIH - consumption function, where the PIH stands for the Permanent Income Hypothesis) results from household optimizing behavior. Households do not care solely about their current income when they are making consumption decisions, they also care about their permanent (or life time) incomes.
Let us take the simple optimizing example that we covered in the Topic #3 slides. Suppose households have the following utility function U(.) over current (C) and future consumption (Cf):
U(C, Cf) = ln (C) + ln (Cf)
Does this utility function make sense? A little bit. It does possess the feature that households face diminishing marginal utility in both C and Cf. What is the marginal utility of current consumption? The more that a household consumes, the less additional benefit it receives from an additional unit of consumption. The marginal utility of current consumption is the derivative of household utility with respect to current consumption:
U(.)/C = 1/C ;where U(.) is just short hand for U(C, Cf).
Notation: the subscript f denotes a future period. Variables lacking a subscript are the variable today. In previous chapters, I used the notational convention of the subscript t representing today and the subscript t+1 representing tomorrow.
Note:The marginal utility of current consumption is decreasing in C. As C increases, the marginal utility tends toward zero. This feature of our utility function tends to match the real world. The more we consume, the more we become satiated. In this simple utility function, we will abstract from the fact that individuals also like leisure - we did the consumption-leisure trade off when discussion our model of labor supply. We could include leisure in the utility function, but that would just make the analysis a little messier.
is the household Time Discount Factor. Basically, it measures how patient people are. With very close to 1, the consumer is extremely patient. That is, he/she is indifferent between consumption today and consumption tomorrow. With less than 1, the consumer is impatient. That is, he/she prefers consuming today - all else equal -as opposed to consuming tomorrow. Data shows that most people are slightly impatient - they prefer to consume a little bit extra today - all else equal. In other words, they tend to ‘discount’ future utility (i.e., future consumption). People do, however, differ on how impatient they are. There is much work trying to estimate what people’s s look like. Some households look like they have s = 0.98 and others look like they have s close to 0.75 or lower.
We will take the above utility function as given. There are many other utility functions out there - some of them better represent the real world. You should know that economists working on consumption research spend lots of time using different utility functions - the one we are using is simple to use and captures some of the realistic features of household consumption behavior.
Along with the utility function, households also face a constraint to how much they can consume. We will assume that a household - over its life time - can only consume as much as the income it earns over its lifetime.
Let us build this budget constraint. We will assume that households start with some initial wealth (Wealth0). This could represent gifts from parents, past savings, whatever.
By the nature of our decision above, we assume two periods (today and tomorrow). We also assume that households can borrow funds anytime they want (we will relax this condition later).
Household savings between today and tomorrow can be defined as:
S1 = (Wealth0 + Y) - C,
where Y is the income the household earns today (in period 1). Households start with some initial wealth (Wealth0), earn some money today (Y) and make some consumption decisions today (C). Whatever is left after the household consumes is household savings. If C > Wealth + Y, the household is borrowing today (S1is negative). If C < Wealth + Y, the household would be saving today (S1 is positive). This is analogous to our definition of disposable income from the beginning of the class (setting taxes equal to zero - we will abstract from taxes for now and but we now add in some initial level of wealth).
What is the maximum amount I can consume tomorrow (period 2)? Well, my maximum consumption in period 2 would be if I consumed all my resources in period 2. What are my resources in period 2? The amount of money I saved in period 1 (which earned some return), plus the income I earned in period 2 (Yf). Formally,
Cf = S1(1 + r) + Yf
Cf = (Wealtho + Y - C) (1 + r) + Yf <substituting in above definition of S1
The above equation says that if you know Wealth0, Y, Yf and C, you uniquely know Cf (assuming households want to consume all their resources). To illustrate that this budget constraint makes sense, rewrite the definition of Cf as:
C + Cf/ (1 + r) = Wealth0 + Y + Yf / (1+ r) <moving C from the right to the left side of the equation and dividing through by (1+r)>
Rewriting the equation like this says that present value of lifetime consumption (C (consumption today) and Cf / (1+ r) (consumption tomorrow discounted back to today at the rate of interest)) is equal to the present value of lifetime resources.
So, households face the following problem:
maximize: ln (C) + ln (Cf) <household utility>
subject to: Cf = (Wealtho + Y - C) (1 + r) + Yf<budget constraint>
What is the solution to this problem? There are many ways to solve this but, here is my approach: The utility function has two arguments (C and Cf) - it is harder to maximize over two variables. So, I substitute one out. I combine the utility function and the budget constraint. In other words, households maximize:
ln (C) + ln [(Wealtho + Y - C) (1 + r) + Yf] <substitute budget constraint into utility function>
I then maximize over C (i.e., take derivative of utility function with respect to C). Notice that increasing C today will increase utility directly - but, by having higher C today, you have fewer saving today and lower Cf tomorrow. This second effect comes through the budget constraint (the second term in which C enters in the above ‘new’ utility function).
What is the derivative?
U(.)/C = 1/C - (1+r) {1/[(Wealtho + Y - C) (1 + r) + Yf]} <This is simple calculus>
We can rewrite the above as:
U(.)/C = 1/C - (1 + r){1/Cf} <Substitute out Cf = (Wealtho + Y - C) (1 + r) + Yf
What is the maximum for utility? Set the derivative equal to zero:
1/C - (1 + r) (1/Cf) = 0
Or, rewriting, we get:
Cf = (1 + r) C
Does this make sense intuitively? Yes - if is higher (more patient households), they should want to consume more in the future - relative to today’s consumption. That is exactly what this equation says. If interest rates are higher, the price of current consumption is high (you could save a dollar today and get (1+r) dollars tomorrow). If r increases, it pays to save and households should save more - this will increase consumption tomorrow. This is the substitution effect of changing interest rates on consumption. There is also an income effect on consumption from changing interest rates that works through the budget constraint (if r increases and we are net savers, we are richer - so we will want more of the things we like - C today and C tomorrow).
What is the optimal consumption for a household? We have two equations and two unknowns:
Equation (1) is the optimal consumption plan:
Cf = (1 + r) C
Equation (2) is the budget constraint:
Cf = (Wealtho + Y - C) (1 + r) + Yf
Solving these equations for C and Cf, we get:
C = [(Wealth0 + Y) (1 + r) + Yf]/ [(1+)(1+r)] or simplifying
C = (Wealth0 + Y)/(1+) + Yf / ((1+)(1+r))
Note:With this consumption function, an increase in r will actually cause C to fall. This is not a generic feature of all utility functions and it is only occurs when using log utility. While log utility is easy to use, it has the undesirable feature of suppressing income effects resulting from interest rate changes on consumption. In general, we will always assume that there is no effect of changes in r on current consumption (i.e., the income and substitution effects will cancel). This does not occur with log utility, but will occur with more realistic utility functions.
As in class, let us set = 1 and r = 0 (for illustrative purposes). Doing this gives us:
C = Cf = [(Wealth0 + Y) + Yf]/2
Basically, households smooth consumption over each period. What is the consumption level they choose? The level of consumption (in both periods) is their level of lifetime resources divided equally over both periods. If the household lived 3 periods, they would take their lifetime resources and divide them over three periods (and so on).
Summary #1:
What have we learned so far from the PIH - consumption function?
a)Households smooth consumption by taking lifetime resources and dividing them across their remaining lifetime.
b)An increase in will cause future consumption to be higher than current consumption (all else equal).
c)If households are net savers (see Notes #9), an increase in interest rates will cause future consumption to definitely increase (income and substitution effects go in same directions), but the effect on current consumption is ambiguous (income and substitution effects go in opposite directions). We will go over the case when households are net debtors soon.
d)Current consumption depends on both current income (Y) and future income (Yf) - this is different than the Keynesian consumption function!
The Keynesian consumption function says: C = a + b (Y) <formally it is C = a + b Yd - but, if T = 0, then Y = Yd>.
I am not going to go into depth with the examples we did in class - but, I do want to summarize the findings that we had. For those who want more details, see the actual notes from the Topic #3 slides.
Summary #2
What does the PIH tell us?
a)The marginal propensity to consume out of an expected income increase should be zero! If we knew that our income was going to increase, we should have already planned that into our optimal consumption path.
Let me make this statement more formally. Define the growth rate in consumption as:
(Cf - C)/ C
For PIH consumers (using the above formulas for optimal C and Cf), we get:
(Cf - C)/ C = (1+r) - 1 < just substitute in the above solutions>
If =1 and r = 0, then the growth rate in consumption is equal to zero (consumption is constant).
Notice, neither Y nor Yf enters into the growth rate formula at all!
For Keynesian consumers, we get:
(Cf - C)/ C = b(Yf - Y)/ (a + b Y)
If a = 0, then for Keynesian consumers:
(Cf - C)/ C = (Yf - Y)/ Y
The growth rate in consumption will equal the growth rate in income.
For PIH consumers, the growth rate in consumption has nothing to do with expected changes in income - if I knew my income was going to be high tomorrow, I would simply adjust my consumption today!
For Keynesian consumers, the growth rate in consumption is exactly equal to the growth rate in income (if a = 0).
b)For PIH consumers, the MPC (marginal propensity to consume) out of permanent unexpected shocks to income should be 1. The above discussion referred to predictable changes in income. The results differ if we are uncertain about our future income. Suppose we unexpectedly get a raise in our salary that will persist well into the future. As we saw with the examples in class, consumption will adjust (under the PIH), one for one with the unexpected income change if the unexpected income change in permanent.
c)The MPC out of temporary unexpected shocks to income should be equal to 1/LL where, LL is the expected length of life that the household faces. Households want to smooth that shock over all the periods of their life (so as to keep consumption relatively constant - or, more appropriately - the expected consumption growth rate equal to the above formula - which is not a function of current or future income).
Why are we studying these theories of consumption?
We may care about how households respond to things like tax cuts. If the tax cut is temporary - like most tax cuts are - the consumption response depends upon whether households follow PIH rules or Keynesian consumption rules. If the households are PIH consumers, the consumption effect of the tax cut would be expected to be small. If households are Keynesian, the consumption effect of the tax cut could be quite large (as households consume an additional b% of the tax cut). Knowing how consumers behave is key to evaluating the effectiveness of government policy. Empirically, it looks like both theories seem to describe some consumers. So, what researchers, myself included, are working on now is trying to figure out the percent of households who behave as PIH consumers and the percent that behave as Keynesians.
As I said in class, there are lots of bells and whistles being put on the PIH model to describe household behavior. The major one we talked about was liquidity constraints. Liquidity constraints prevent a household from borrowing.
You should know:
a)What liquidity constraints are
b)Why the presence of liquidity constraints make PIH consumers look more like Keynesian Consumers.
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