Notes 7: Consumption and Taxes
In these notes, we will discuss, in a mathematical framework, how households optimize
their decisions about labor, leisure, and consumption.
Let’s start with a simple base case-- the optimal consumption decision of a
household who has preferences over two goods: x and y:
Suppose the household has the following utility function: U(x,y).
The household also faces the following constraint:
Px x + Py y = I
(1)
where: Px = price of good x
Py = price of good y
I = household income.
We will assume that the household takes Px, Py and I as given. If the household
maximizes its utility (chooses the optimal allocations of x and y), they would set:
MUx/ Px = MUy/ Py
(2)
where MUx is the marginal utility the household receives from consuming the last unit of
x.
This relationship comes straight from micro: Households want to set the additional
utility per dollar of consuming x to the additional utility per dollar of consuming y. If
MUx/Px > MUy/Py, the household could be made happier by taking the last dollar spent
on y and using it to buy a dollar’s worth of x. The dollar’s worth of x would yield higher
additional (marginal) utility to the household.
To solve for the optimal x and y, we have two equations ((1) and (2)) and two unknowns
((x) and (y)). This is exactly what we did in our micro class.
But, this is a macro class - why do we care about this?
Let’s place this problem in a macro context—the consumption and leisure choices of
a household:
We will define:
x = consumption
y = leisure
I = PVLR
Px = price of consumption (something like the CPI)
1
Py = price of leisure (this is what we defined as the after-tax real wage - see labor
market notes for further explanation).
Households take the CPI, their after tax real wage and their PVLR as given (all of those
are equilibrium values set in markets in which each household is a price taker). The
household then chooses, given its budget constraint, the optimal allocations of leisure and
consumption.
Let’s review the labor market in the context of the above framework. This above
framework is NOT new - the labor supply discussion in Notes 4 assumed households
maximize utility - this is the same assumption we are making now. Now, I will prove
this mathematically by using the result from household utility optimization. Y
Note: The intuition approach and this mathematical approach are EXACTLY the same.
As taxes increase, after tax wages will fall (for every given before tax wage). In other
words, the price of leisure (Py) will fall. Suppose the household can only adjust the
amount of leisure they can consume (i.e., they cannot adjust consumption). This is an
unrealistic assumption, but it provides the intuition. In reality, the household will adjust
both.
For now, let’s focus on how the household will adjust its leisure allocation:
Let’s focus on equation (2) to start. This says that as Py falls, in order for (2) to hold,
MUy must fall (again, we assumed that households cannot adjust consumption, so MUx is
constant - as is Px by assumption). If households are maximizing utility, then a fall in P y
will result in MUy falling, which means that households will increase the amount of
leisure they consume (recall from micro that an increase in y leads MUy to fall, according
to diminishing marginal utility). This is the substitution effect of changing labor
taxes on leisure. As after tax wages fall, the price of leisure falls. This will cause the
household to rebalance its consumption bundle towards leisure because it is relatively
cheaper. The substitution effect of changing labor taxes says that leisure will increase
and work will fall (work is the opposite of leisure).
There is also an income effect (that is the effect through equation (1) - the budget
constraint). As after tax wages fall, households become permanently poorer (PVLR
falls). As a result, they can afford less x and y. So, they consume less of both. This is
the income effect. The income effect says that as households become poorer, they cannot
afford as much leisure so they will consume less leisure (work more).
For labor supply, the income and substitution effects go in opposite directions. We
knew that already.
2
How will consumption respond to an increase in labor income taxes?
The income effect from an income tax increase with consumption is easy. The increase
in income taxes causes PVLR to fall - we are permanently poorer and as a result going to
buy less of the goods we care about ---- i.e., consumption and leisure. The income effect
from an income tax increase says households will purchase less consumption.
Let’s look at equation (2) again: In that equation, households equate the marginal
utility per dollar for both x and y. If y gets cheaper (leisure gets less expensive)
households will switch towards leisure. But, they can also equate (2) by changing the
MUx. After the price of leisure fell, (2) will hold with equality if MUx increased (holding
MUy) constant.
How does the MUx increase?
By decreasing the amount of x consumed. When labor income taxes increase, the
substitution effect says that households will consume less consumption (substitute
consumption for leisure! - hence the term ‘substitution effect’.)
Why do this analysis?
Notice the differences - with labor supply - a change in labor income taxes could increase
or decrease our desire to work depending on the size of the income and substitution
effects. This is NOT the case with consumption. Both the income effect and the
substitution effect for changes in labor income taxes go in the same direction!
Increasing (decreasing) labor income taxes will cause consumption to unequivocally fall
(increase). The only thing we do not know is how much consumption will change - this
depends on whether the tax cut is permanent or temporary, whether households are PIH
households or Keynesian or whether households are liquidity constrained (see the Notes
6).
Additional Comments on Consumption and Interest Rates
The same analysis as above could be used to discuss consumption and interest rate
changes. Instead of x and y being consumption today and leisure today, make x and y,
consumption today and consumption tomorrow.
In the real world, utility functions might be over w, x, y and z - consumption today,
leisure today, consumption tomorrow and leisure tomorrow. The marginal utility per
dollar of all four goods would have to be set equal to each other - that gives us three
equations, plus one equation for the budget constraint - we would have four equations
and four unknowns. For now, we will only focus on the consumption today and
consumption tomorrow component.
3
The interest rate is the price of consumption today. In other words, the interest rate is
what you would have to give up in terms of consumption tomorrow in order to consume
today. If you consume today, you cannot save. If you save, you can consume tomorrow.
For a given amount of saving, the higher the interest rate, the more you can consume
tomorrow. So, by consuming today, you are giving up consumption tomorrow (all else
equal). The interest rate tells you how much consumption tomorrow you are giving
up to consume today. So, basically, the interest rate is the price of consumption today
(in terms of consumption tomorrow). The higher the interest rate, the higher the price of
consumption today.
Let’s define:
x = consumption today
y = consumption tomorrow.
From equation (2) at the beginning of this chapter, if interest rates go up, the Px will go
up. The substitution effect says that you will consume less x (consumption today) and
more y (consumption tomorrow) as interest rates go up.
What about the income effect?
Well, that depends on whether you are a net borrower or a net lender.

Net lenders: In Notes 6, I assumed that everyone was a net lender—that is they
have positive savings. If you have positive savings - as interest rates go up, you
become permanently richer. This makes sense because your interest INCOME
will increase. As a result, you will consume more today and more tomorrow.

Net borrowers: This is not the case if you are a net borrower—that is—you have
negative savings. If you have negative savings, as interest rates go up, you
become permanently poorer. This makes sense because your interest
PAYMENTS go up, reducing your income. In this case, you want less
consumption today and less consumption tomorrow.
To summarize:
Net Savers (positive saving)
Income Effect
Substitution Effect
Consumption Today
Increase
Decrease
Consumption Tomorrow
Increase
Increase
If Y is held fixed, the income effect says that C will increase today. That implies that
S(private) today will fall (S(private) = Y - C). The substitution effect says that S(private)
will increase. Y is fixed and C today falls. So, the effect on household savings today is
also ambiguous - if households are net savers.
4
Net Borrowers (negative saving)
Income Effect
Substitution Effect
Consumption Today
Decrease
Decrease
Consumption Tomorrow
Decrease
Increase
Given that Y is fixed (all else equal), both the income and substitution effect says that
household savings today will increase if interest rates today increase - if households are
net borrowers.
5