We examine the marginal impact on revenue of increasing the quantity of capital. If we
begin at point KA and increase capital by ΔK, the extra output is MPKA· ΔK represented
by the area of the small rectangle.
ΔK
MPKA
MPK
ΔK
KA
If the firm faces a rental rate of R
P
it will optimally choose to rent K*.
R/P
MPK
K*
We can divide up K* into increments of ΔK.. The output that we produce would be the
sum of those increments multiplied by the extra output that those increments produce.
That is, we can write output, as the sum of a bunch of rectangles underneath MPK out to
K*.
R/P
MPK
K*
If we divide the increments very finely, so each ΔK was infinitely small, output would be
the area under the trapezoid.
R/P
MPK
K*
.
The amount paid to the owners of capital is R ⋅ K * which is represented by the rectangle
P
under the price line
R/P
MPK
K*
The area above the price line is the output retained by the firm (including wages). We
refer to this triangle as surplus.
Surplus
R/P
MPK
K*
Now consider if the government were to impose a tax wedge so that we could say that
R
P
= R + tw . This reduces the optimal capital stock to K** and reduces
P
1 − taxwedge
output. The new surplus is also smaller left for the firm. The lost surplus is divided into
two parts. The rectangle given by tw·K** is the tax bill collected by the government. This
is a loss for the firm but a gain for the government and at the level of society overall it is
a wash. But there is one small triangle which is a loss of surplus for the firm but no gain
for the government. We call this the deadweight loss because it is a net loss for society.
This loss represents the goods not produced because the distortionary taxation reduces the
optimal level of output.
Surplus’
Deadweight Loss
R/P+tw
Tax Bill
R/P
MPK
K**
K*
Now consider what happens if the government raises the taxwedge so much that tw
doubles.
Deadweight Loss 2
R/P+2tw
Tax Bill 2
R/P+tw
Deadweight Loss 1
Tax Bill 1
R/P
MPK
K***
K**
K*
The tax wedge has doubled. We see two things. First, the tax bill has less than doubled.
This is because raising taxes increases revenue on any unit of capital used, but also
simultaneously reduces the actual quantity of goods used. Is Tax Bill 2, in fact, larger,
than Tax Bill 1. In theory, it could be larger or smaller. Consider the extreme case.
Compare the tax bill when the tax wedge is twA than when it is twB
R/P+twB
R/P+twA
Tax
Bill A
R/P
MPK
K*A
K*
We see that the firm pays a tax when the tax wedge implies twA but when the tax wedge
is raised so high that we have twB , the firm finds it unprofitable to hire any capital,
produces no goods and pays no taxes. Therefore by definition, taxes fall when tax rates
rise. On the other hand if tw=0, then the government would also collect zero tax bill and
if they raised the tax wedge to twA . Therefore, the relationship between the tax bill and
tax revenue is non-linear. This relationship is referred to as the Laffer curve. Most
empirical studies suggest that for most types of taxes, the top of the Laffer curve is above
current tax rates in almost all developed and developing
countries.
Tax Bill
tw
The second finding is that the deadweight loss increases and in fact more than doubles. In
fact, in this case, the deadweight loss increases 4 fold when the tax wedge doubles. When
the tax wedge triples, the deadweight loss increases 9 fold.
Deadweight Loss 3
R/P+3tw
R/P+2tw
Deadweight Loss 2
R/P+tw
Deadweight Loss 1
R/P
MPK
K****
K***K*** K**
K*
When the original tax rate is tripled, the deadweight lost increases 9 fold. In general, we
might think of the deadweight loss increasing quadratically with the tax wedge. Why
does the deadweight loss rise so sharply? When the tax wedge is zero, the marginal
product of the final increment of capital barely creates any surplus. When the tax wedge
is raised, the firm cuts back on this capital, but it barely had any impact on surplus
anyway. But as the tax wedge rises, the capital that goes unrented and rented has larger
and larger impacts on output and surplus.
Bigger surplus loss
No big loss
R/P
MPK
K*
Another implication is tax smoothing. Say the government has some funds that they want
to collect.over the course of two consecutive periods. They could 1) raise taxes a lot
today and go back to normal tomorrow; 2) wait until tomorrow to raise taxes and raise
them a lot then; or 3) raise taxes evenly in both periods. The best choice is the middle. As
the deadweight loss is quadratic in the tax wedge, the total deadweight loss is minimized
by not raising taxes too much in any period.
Deadweight
Loss
tw
twHI/2
twHI
The average of the deadweight losses when the tax wedge is zero in one period and high
in the next is greater than the deadweight loss at any of the intermediate points.