Download On the Government Spending Multiplier

On the Government Spending Multiplier
Intermediate Macroeconomics – Fall 2010
The University of Notre Dame
Professor Eric Sims
There has been much recent attention given to the concept of the government spending multiplier. In this document I discuss what the multiplier is,
how it is derived, and when it might be large.
In the model with which we have worked so far, output is exogenously
given. That is, y s = y fixed. The time path of government spending is specified exogenously, and the government finances its spending with lump sum
taxes. The demand side of the economy is simply the aggregate accounting
identity, assuming that consumers behave optimally:
yd = c + g
In our model, Ricardian equivalence holds, which means that people behave as though the government balances its budget period by period. Subbing in the generic form for the consumption function, we therefore get:
y d = c(y d − g, y 0 − g 0 , r) + g
I put y d as the argument in the consumption function so as to make
explicit that consumption depends on an endogenous variable, output. The
function as written above implicitly defines aggregate demand as a function
of exogenous variables (g and g 0 ) and the real interest rate, r. It is implicit
because the left hand side variable also appears on the right hand side. The
equilibrium of the economy can be expressed graphically as follows:
1
ys
1+r
1+r*
yd
y
To determine how much the output demand curve shifts when government
spending increases, we need to implicitly differentiate (also called totally
differentiate) the demand curve (since it is an implicit function). This says
that the total change in the left hand side variable is (approximately) equal
to the marginal derivatives times the changes in the variables on the right
hand side. Applying this to above, we get:
dy d =
∂c
∂c
∂c
(dy d − dg) + 0 (dy 0 − dg 0 ) + dr + dg
∂y
∂y
∂r
∂c
∂y
is the derivative of consumption with respect to its first argument
∂c
(which is “net income”, y − g); ∂y
0 is the derivative with respect to the
∂c
second argument; and ∂r
is the derivative with respect to third argument.
d
Now “solve” for dy , which appears on both sides of the equal sign:
!
∂c
∂c
∂c
∂c
dy d = − dg + 0 (dy 0 − dg 0 ) + dr + dg
1−
∂y
∂y
∂y
∂r
!
∂c
∂c
∂c
= 1−
dg + 0 (dy 0 − dg 0 ) + dr
∂y
∂y
∂r
Now if we assume that dy 0 = 0 and dg 0 = 0 (which is fine . . . we want
to isolate the effect of the increase in current g only). Furthermore, assume
that dr = 0. This is fine if we are trying to figure out what the “shift” to
the right in the output demand curve is. Then we are left with:
2
!
!
∂c
∂c
1−
dy d = 1 −
dg
∂y
∂y
Simplifying, we can get an expression for the effect of government spending on demand:
dy d
=1
dg
In other words, an increase in government spending generates a one for
one increase in total output demand at a given interest rate, assuming that
the increase in government spending has no implications for future government spending or output.
The above is a partial equilibrium statement, as it holds the real interest
rate fixed. In equilibrium, however, the real interest rate will have to move
so as to equate demand with supply. Graphically, we can see the equilibrium
effects of an increase in government spending below:
ys
1+r
1+r1*
Horizontal shift =
change in g
1+r*
yd
y
y + dg
y
Here, we see that, if output is exogenously supplied, that the government
spending multiplier is zero in equilibrium. Intuitively, the increase in government spending must increase the real interest rate, which “crowds out”
consumption.
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So how do traditional Keynesian models generate bigger government
spending multipliers? For one, they don’t feature Ricardian equivalence.
In a traditional Keynesian model, consumption is a function of current output and the real interest rate only (i.e. consumption is not forward-looking).
This would only obtain in our model if there were liquidity constraints or
something similar. Then the implicit function characterizing aggregate demand becomes:
y d = c(y d , r) + g
Totally differentiating this object yields:
dy d =
∂c d ∂c
dy + dr + dg
∂y
∂r
Set dr = 0 and solve:
!
∂c
1−
dy d = dg
∂y
⇒
d
dy
1
=
∂c > 1
dg
1 − ∂y
∂c
Here the multiplier is greater than one because the MPC (i.e. ∂y
) must
be less than one. The bigger is the MPC the bigger is the multiplier. In
words, the demand curve shifts out more than one for one with an increase
in government spending in the absence of Ricardian equivalence. The above
expression is the traditional Keynesian government spending multiplier that
you see thrown around, and probably remember from your principles courses.
Of course, the above is still a partial equilibrium statement, because it
holds the real interest rate fixed. In equilibrium demand must still equal
supply. This means that, even without Ricardian equivalence, the multiplier
in equilibrium must still be zero. Because demand shifts out by more, this
just means that the real interest rate will have to increase by more to keep
equilibrium output unchanged. See the graph below.
4
ys
1+r
1+r2*
Horizontal shift =
change in g/(1-mpc)
1+r*
yd
y
y + dg/(1-mpc)
y
Basically, for the government spending multiplier to be large, we (i) must
have an elastic supply curve (i.e. y s is not vertical) and (ii) it helps if there is
no Ricardian equivalence. We will see that once we introduce variable labor
supply, the output supply curve is no longer vertical, and the government
spending multiplier is no longer zero in equilibrium. Nevertheless, it will
not be very large and will not “work” according to the traditional Keynesian
understanding. To get the traditional Keynesian multiplier, you need the
output supply curve to be basically flat. This could happen if the real
interest rate gets “stuck” at a lower bound, although it’s not clear why this
would happen, given that the zero lower bound only applies to the nominal
interest rate.
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