the market for firearms: a comparative static analysis
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A paper I wrote that demonstrates the mathematical economics technique of comparative static analysis on the market for firearms. The document was created using LaTeX. If you download, please leave a comment. Thanks!TRANSCRIPT
The Market for Firearms:
A Comparative Static Analysis
Chris Vecchio
December 10, 2009
John Carroll University
1 Introduction
Citizens of a country demand firearms for a number of reasons. These include
but are not limited to, personal protection, hunting, gun collecting, and po-
tential criminal activities. The three year period of 2007-2009 saw a massive
increase in the demand for firearms of all types. A figure that is widely used
to estimate the demand for firearms is the number of background investigations
that are completed each month. This investigation is mandatory for any person
wishing to buy a firearm. According to USA Today, there was a 31% increase
between 2007 and 2008 in the number of background checks performed by the
Federal Bureau of Investigation for potential firearm owners. (Neary, 2009) In
addition, according to Ted Novin of the National Shooting Sports Foundation
(NSSF) the National Instant Criminal Background Check System (NICS) con-
ducted 973,003 background checks in September 2008. In 2009, that number
had increased to 1,093,230 checks, a 12.4 percent increase. (Kakkuri, 2010)
According to the NSSF, there has also been a “significant spike” in demand
for ammunition. (National Shooting Sports Foundation, 2009) This evidence
leads to the question as to what variables influence the supply and demand for
firearms and what is the relationship between them. This paper uses compar-
ative statics to study the effects of changes in exogenously determined income,
tastes, expected prices, price of complementary goods, and excise taxes on the
supply and demand for firearms.
2 Model
The demand and supply for firearms will be represented by the following system
of equations. Assuming a unique interior solution exists, quantity demanded
1
equals quantity supplied when the market is in equilibrium. Thus, we have
QD = QS
QD = D (P, Y0, A0, E0, C0)
QS = S (P,E0, T0)
(1)
Where P is the price of firearms, Y0 is income, A0 is consumer tastes, E0 rep-
resents the expectations by consumers and producers of future firearms prices,
C0 is the price of ammunition, and T0 is an excise tax on the supplier. If we let
QD = Q = QS in (1), the model can be expressed by the following two equation
system:
F 1 (P,Q;Y0, A0, E0, C0, T0) = D (P, Y0, A0, E0, C0)−Q = 0
F 2 (P,Q;Y0, A0, E0, C0, T0) = S (P,E0, T0)−Q = 0(2)
The required signed partial derivatives are as follows:
DP < 0 DY0> 0 DA0
> 0 DE0> 0 DC0
< 0
SP > 0 SE0 > 0 ST0 < 0(3)
The partial derivatives with respect to price DP and SP are obtained from the
laws of demand and supply. This model assumes that firearms are a normal good
and therefore DY0 > 0. In addition, DA0 > 0 implies that as consumers’ tastes
change in favor of owning more firearms, their demand for firearms will increase
and vice versa. The variable C0 represents the price of complementary goods and
therefore DC0 < 0. An increase in the price of complementary goods will cause
a decrease in the demand for firearms. The expected future price of firearms
E0 will affect the current demand and supply for firearms. As witnessed in the
past year, if consumers expect prices to increase in the future, their current
demand for firearms will increase. The effects of expected prices on current
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supply are slightly harder to determine. However, based on evidence from the
past year, suppliers in the United States have reacted to the expected price
increase by increasing their production immediately, causing current supply to
increase. (National Shooting Sports Foundation, 2009) Thus, SE0> 0. Finally,
ST0< 0 indicates that an increase in a tax on firearms will cause a decrease in
supply.
3 Implicit Function Theorem
The model assumes that all partial derivatives are continuous. To verify if the
implicit function rule applies we check:
|J| =
∣∣∣∣∣∣∣∂F 1
∂P∂F 1
∂Q
∂F 2
∂P∂F 2
∂Q
∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣DP −1
SP −1
∣∣∣∣∣∣∣ = SP −DP > 0
Because the Jacobian is non-zero, there exists a 5-dimensional neighborhood,
N , in which the implicit function theorem can be applied and we can write the
endogenous variables implicitly as functions of the exogenous variables:
P ∗ = P ∗ (Y0, A0, E0, C0, T0) Q∗ = Q∗ (Y0, A0, E0, C0, T0)
We can then write the equilibrium identities as:
D (P ∗, Y0, A0, E0, C0)−Q∗ ≡ 0
S (P ∗, E0, T0)−Q∗ ≡ 0(4)
3
4 Comparative Static Analysis
To conduct the comparative static analysis, we take total differentials of (4) to
obtain:
DP∗ · dP ∗ + DY0· dY0 + DA0
· dA0 + DE0· dE0 + DC0
· dC0 − dQ∗ = 0
SP∗ · dP ∗ + + SE0· dE0 + ST0
· dT0 − dQ∗ = 0(5)
Rearranging (5) to get endogenous variables on the left and exogenous variables
on the right gives:
DP∗ · dP ∗ − dQ∗ = −DY0· dY0 −DA0
· dA0 −DE0· dE0 −DC0
· dC0
SP∗ · dP ∗ − dQ∗ = − SE0 · dE0 − ST0 · dT0
This system can be rewritten in matrix form as:
DP∗ −1
SP∗ −1
dP ∗
dQ∗
=
−DY0· dY0 −DA0
· dA0 −DE0· dE0 −DC0
· dC0
−SE0 · dE0 − ST0 · dT0
To analyze the effects of a change in an exogenous variable, ceteris paribus,
we use Cramer’s Rule to solve for the various comparative static derivatives.
The first comparative static calculation will be demonstrated in its entirety. If
the only changing exogenous variable is income, all other exogenous variable
differentials will be equal to zero. We then divide through by dY0to obtain:
DP∗ −1
SP∗ −1
∂P∗
∂Y0
∂Q∗
∂Y0
=
−DY0
0
(6)
4
We now use Cramer’s Rule to solve for∂P∗
∂Y0:
∂P ∗
∂Y0=
∣∣∣∣∣∣∣−DY0 −1
0 −1
∣∣∣∣∣∣∣SP∗ −DP∗
=DY0
SP∗ −DP∗> 0 (7)
The remaining comparative static calculations are completed in a similar fash-
ion.
∂P ∗
∂A0=
∣∣∣∣∣∣∣−DA0 −1
0 −1
∣∣∣∣∣∣∣SP∗ −DP∗
=DA0
SP∗ −DP∗> 0 (8)
∂P ∗
∂E0=
∣∣∣∣∣∣∣−DE0 −1
−SE0−1
∣∣∣∣∣∣∣SP∗ −DP∗
=DE0
− SE0
SP∗ −DP∗= Indeterminate (9)
∂P ∗
∂C0=
∣∣∣∣∣∣∣−DC0 −1
0 −1
∣∣∣∣∣∣∣SP∗ −DP∗
=DC0
SP∗ −DP∗< 0 (10)
∂P ∗
∂T0=
∣∣∣∣∣∣∣0 −1
−ST0−1
∣∣∣∣∣∣∣SP∗ −DP∗
=−ST0
SP∗ −DP∗> 0 (11)
∂Q∗
∂Y0=
∣∣∣∣∣∣∣DP∗ −DY0
SP∗ 0
∣∣∣∣∣∣∣SP∗ −DP∗
=DY0
SP∗
SP∗ −DP∗> 0 (12)
∂Q∗
∂A0=
∣∣∣∣∣∣∣DP∗ −DA0
SP∗ 0
∣∣∣∣∣∣∣SP∗ −DP∗
=DA0
SP∗
SP∗ −DP∗> 0 (13)
5
∂Q∗
∂E0=
∣∣∣∣∣∣∣DP∗ −DE0
SP∗ −SE0
∣∣∣∣∣∣∣SP∗ −DP∗
=DE0
SP∗ − SE0DP∗
SP∗ −DP∗> 0 (14)
∂Q∗
∂C0=
∣∣∣∣∣∣∣DP∗ −DC0
SP∗ 0
∣∣∣∣∣∣∣SP∗ −DP∗
=DC0
SP∗
SP∗ −DP∗< 0 (15)
∂Q∗
∂T0=
∣∣∣∣∣∣∣DP∗ 0
SP∗ −ST0
∣∣∣∣∣∣∣SP∗ −DP∗
=−ST0
DP∗
SP∗ −DP∗< 0 (16)
These comparative statics results allow us to reach conclusions about how
changes in the exogenous variables affect the equilibrium values of the endoge-
nous variables. First, if income increases, both equilibrium price and quan-
tity will increase. Second, if consumer tastes increase in favor of owning more
firearms the equilibrium quantity and price will increase. Third, if the price
of the complementary good ammunition increases, the equilibrium price of
firearms will fall. If expectations about future prices increase, it is certain
that equilibrium quantity will increase. However, it is not possible within this
general form model to determine the effect of a change in future price expec-
tations on equilibrium price. The sign of this result will ultimately depend on
whetherDE0 > SE0 or DE0 < SE0 . That is, it depends on whether the changes
in the expectations have more of an impact on demand or supply. Finally, a
higher excise tax on firearms will lower equilibrium price and quantity.
5 Policy Implications
This section will use the comparative static results determined above to examine
the effects of two government policy changes on the market for firearms. The
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first policy is an increased waiting period for obtaining a firearm. In an effort to
reduce the number of violent crimes committed by those who purchase a firearm
in the ‘heat of the moment’, many states have mandated waiting periods to
allow more time to ‘cool off.’ Potential firearm consumers may be less inclined to
obtain the weapons because of the increased waiting period. This policy assumes
that the increased waiting period diminishes consumers’ tastes for firearms. The
comparative static results (8) and (13) tell us that this would cause a decrease
in the equilibrium quantity and price of firearms.
The second policy involves a change in the excise tax on firearms. The
current excise tax in place by the Alcohol and Tobacco Tax and Trade Bureau
is 10% of sale price for pistols and revolvers, and 11% of sale price for all other
firearms and ammunition. (Alcohol and Tobacco Tax and Trade Bureau, 2010)
As comparative static result (16) indicates, an increase in this tax would reduce
equilibrium purchases of these weapons. This would provide a means by which
proponents of gun control could achieve their objective of reducing the amount
of firearms purchased legally.
6 Discussion/Possible Extensions
An immediate extension to the model presents itself in the form of taking into
consideration substitute goods. Specifically, if the price of firearms increases
sufficiently, consumers will begin to substitute away from firearms to other types
of weapons. This claim is supported by research conducted by Cook (1981) who
found that a decrease in the availability of firearms caused a substitution away
from firearms to other weapons. A more comprehensive model could also include
the effects of changes in resource prices on manufacturers’ output decisions. As
resource prices increase, the costs of production will increase and therefore lead
to a decrease in the supply of firearms.
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A more interesting extension would be to examine the formal market for
firearms directly alongside alternative markets for them. Cook and Leitzel
(1996) concluded that an increase in the excise tax on firearms in the formal
market leads to greater demand for firearms in the informal market. A model
in which both markets are considered simultaneously would allow for a more
detailed examination of possible effects.
In addition, it would be informative to further investigate the indeterminate
impact of the future price expectations on equilibrium price. One potential
approach to this is discussed by Chiang and Wainwright (2005, pp. 527-532).
This method involves the use of differential equations.
A final consideration is that the exogenous variable E0 involves an additional
aspect of expectations. It is possible that the demand for firearms has increased
because of a ‘fear effect’ not stemming from any fear of an increase in price, but
rather, from the fear (or expectation) of a decline in the availability firearms.
Bice and Hemley (2002, pp. 251-65) found “some evidence that the demand
for handguns rises in anticipation of restrictive gun legislation.” This assertion
is articulated by Ted Novin, director of public affairs for NSSF who said that
the demand for firearms “is largely being driven by the political concerns of
gun owners. . . ” (Kakkuri, 2010) One source of these fears are rumors that the
Obama administration plans to increase the federal tax on gun ammunition by
500 percent. The University of Pennsylvania’s website factcheck.org discusses
the validity of this claim and finds no evidence to support it. (Gore, 2009) A
more detailed survey of firearm consumers would be needed to determine the
source of the ‘fear effect.’
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References
[1] Alcohol and Tobacco Tax and Trade Bureau, (2010, February 8). TTB Taxand Fee Rate. Retrieved February 8, 2010, from Alcohol and Tobacco Taxand Trade Bureau: http://www.ttb.gov/tax audit/atftaxes.shtml
[2] Bice, D. C., Hemley, D. D. (2002). The Market for New Handguns: AnEmpirical Investigation. The Journal of Law Economics , 251-65.
[3] Chiang, A. C., & Wainwright, K. (2005). Fundamental Methods of Mathe-matical Economics. New York: McGraw-Hill/Irwin.
[4] Cook, P. J. (1981). The Effect of Gun Availability on Violent Crime Pat-terns. Annals of the American Academy of Political and Social Science ,63-79.
[5] Cook, P. J., & Leitzel, J. A. (1996). ”Perversity, Futility, Jeop-ardy”: An Economic Analysis Of The Attack On Gun Control. Re-trieved February 2, 2010, from Law and Contemporary Problems:http://www.saf.org/lawreviews/cookandleitzel.htm
[6] Gore, D. (2009, June 23). 500 Percent Ammo Tax? Retrieved December 2,2009, from FactCheck.org: http://www.factcheck.org/2009/06/500-percent-ammo-tax/
[7] Kakkuri, M. (2010, January 20). Is the Frenzy Over? Retrieved February2, 2010, from Gun Digest: http://gundigest.com/article/is-the-frenzy-over/
[8] National Shooting Sports Foundation. (2009). NSSF. Retrieved Decem-ber 5, 2009, from Answers To Ammunition Availability Questions:http://www.nssf.org/media/FactSheets/Ammunition.cfm
[9] Neary, B. (2009). USA Today. Retrieved December 8,2009, from Fear of regulation drives gun, ammo shortage:http://www.usatoday.com/news/nation/ 2009-03-29-ammo-shortage N.htm
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