This paper examines the effects of an excise tax imposed on a monopolist's product. Then simple but quite general polynomial demand and cost curves are introduced and discussed, as is the Microsoft Excel workbook that embeds the functions. Finally, exercises based on selected special cases illustrating the use of the workbook are sketched.

Bradley N. Hopkins

J. Wilson Mixon, Jr.

English

Research Papers in Economics

Analysing the Effects of
Excise Taxes Using
Microsoft Excel
J. Wilson Mixon, Jr. and Bradley N. Hopkins
Campbell School of Business, Berry College
Introduction
Robert L. Bishop (1968) provides a general treatment of
speciﬁc and ad valorem sales taxes in perfectly competitive
industries (those in which ﬁrms are price takers) and in
simple monopolies (those in which a single seller sells a
product for a single price). For competitive industries,
Bishop’s analysis conﬁrms the standard analysis that
appears in textbooks, in which the incidence of a tax
imposed on a product is distributed between buyers and
sellers according to the shapes of the industry demand
curve and the industry supply curve. 
Bishop emphatically points out, however, that this analysis
does not easily extend to the monopolistic case. Indeed,
he calls into question the applicability of the concept of
shared incidence in this setting (1968: 215): ‘The concept
of the “incidence” of a tax is … anomalous…. . In one
sense, … the monopolist … pays the whole tax and more;
and the burden on consumers must be added to that.’ In
addition, Bishop shows that it is quite possible that the
price that a monopolist charges can rise by more than the
speciﬁc tax. (Bishop extends the analysis to ad valorem
taxes, but the analysis here is limited to speciﬁc taxes. The
generalisation is straightforward.) 
The central statement of Bishop’s analysis is this (1968:
201): ‘As an antidote to excessive preoccupation with the
linear case, it is important to notice that the monopolist’s
price rises either more or less sharply according as [the]
demand [for its product] is concave from above or below.’
Thus, second derivatives come into play in the case of
monopoly, but not in the case of competition. Despite this
warning, issued almost four decades ago, textbooks still
routinely represent demand curves with straight lines.
Following Bishop, this paper examines the effects of an
excise tax imposed on a monopolist’s product. The
remainder of the paper is organised as follows. The next
section reviews salient aspects of Bishop’s development.
Then simple but quite general polynomial demand and
cost curves are introduced and discussed, as is the
Microsoft Excel workbook that embeds the functions.
Finally, exercises based on selected special cases
illustrating the use of the workbook are sketched.1
Demand, supply and tax incidence in
competitive markets 
The most pertinent of Bishop’s derivations appear below.
The ﬁrm is assumed to maximise proﬁts, which consist of
revenue (R) less cost (C). The following relationships
pertain:
• R = pq, where p is the height of the (inverse) demand
curve at each quantity (q)
• RN = p + qpN, where pN is the slope of the inverse
demand curve (RN = dR/dq is marginal revenue)
• RO = 2pN + qpO, where pO is the rate at which the
slope of the demand curve changes as q changes and
RO is the rate at which marginal revenue changes as q
changes
• C = cq, where c is the height of the average cost curve
at each quantity (q)
• CN = c + cpN, where cN is the slope of the average cost
curve, dc/dq (CN is marginal cost)
• CO = 2cN + qcO, where cO is the rate at which the
slope of the average cost curve changes as q changes
and CO is the rate at which marginal cost changes as q
changes. 
Bishop directly demonstrates the result that every
principles textbook reports for a competitive industry:
• dp/dt = pN/(pN – sN), = –pN/(sN +( –pN)) 
where sN is the slope of the competitive supply curve and
dp/dt is the price change per one-unit change in the tax
CHEER  Volume 18 Page 25rate. From this result one can quickly determine that some
commonly-reported results hold:
• dp/dt = t/2 if the elasticity of demand equals that of
supply at the equilibrium price in the absence of taxes
(Bishop’s analysis is in terms of slope, but p and q are
the same for both demand and supply so equal slopes
imply equal elasticities and vice versa)
• if dp/dt = t/2, then the tax (t) must be equally shared
between buyers and sellers in this case
• if –pN > sN (the absolute value of the inverse demand
curve’s slope exceeds that of the supply curve), dp/dt >
1⁄2 (that is, if the demand curve is less elastic than the
supply curve, buyers pay more than 
1⁄2 of the tax)
• if –pN < sN (the absolute value of the inverse demand
curve’s slope is less than that of the supply curve),
dp/dt < 
1⁄2 (that is, if the demand curve is more elastic
than the supply curve, buyers pay more than 
1⁄2 of the
tax),
• dp/dt = 0 if the demand curve is horizontal (all of the
tax is absorbed by sellers), and
• dp/dt = 1 if the supply curve is horizontal (all of the tax
is paid by the buyers). 
Demand, cost and tax incidence in
monopoly markets
Bishop’s treatment of the competitive market reveals no
surprises. Results typically shown in textbooks are
demonstrated in a concise fashion. The main point of
Bishop’s analysis, and the focus of all that follows here, is
that the results are much more problematic when the seller
faces a downward-sloping demand for its product. In these
cases, the curvature of both the demand curve and the
average cost curve can affect the way that a tax increase is
shared between buyers and the seller.
Now the effect of the tax on price is as follows:
• dp/dt = pN/(RO – CO) 
which can be shown to be positive because pN is negative
and RN (marginal revenue) must cut CN (marginal cost)
from above (so RO < CO at the quantity for which RN =
CN. (Otherwise, RN would come to exceed CN as quantity
increases, so the ﬁrm cannot be at its proﬁt-maximizing
quantity.)
Bishop shows that for the special case of linear demand
and cost curves, a result very much like that of the
competitive case occurs:
• dp/dt = pN/(2(pN – cN)). 
Here pN < 0, and cN < 0, so dp/dt is indeed positive.
Furthermore, dp/dt is less than 1.0. If cN = 0, then dp/dt =
1⁄2. If cN > 0 (a more likely case), dp/dt < 
1⁄2. The reason is
that, in this case the reduction in quantity causes the level
of the ﬁrm’s marginal cost to fall (movement along the MC
curve). If the average cost curve slopes downward over
the relevant range, dp/dt can exceed 1.0 (the price can rise
by more than the excise tax). This happens when the
absolute value of the average cost curve’s slope is between
one-half the demand curve’s slope and the value of its
slope. Such a steep slope for the average cost curve seems
unlikely, but cannot be ruled out theoretically.
The main point of Bishop’s development is that dp/dt can
exceed 1.0 even if the marginal cost curve is not
downward sloping, and that outcome is a function of the
curvature of the demand curve. Bishop says:
‘As an antidote to an excessive preoccupation with the
linear case, … notice that the monopolist rises either
more or less sharply according as [the demand curve] is
concave from above or below. In general, … the effect
depends on the slope of the AR [demand curve] relative
to the difference in the slopes of the MR and MC. …
This the fundamental difference between the
monopolistic and competitive cases: the effect under
competition depends solely on the ﬁrst derivatives
[slopes] of the demand and supply functions, but under
monopoly it depends not only on the ﬁrst derivatives of
AR and MC but also on the second derivative of AR
[pO]. Even with constant MC, it is … possible for a
speciﬁc tax to increase the monopolist’s price by more
than the tax. This will be so whenever AR is more
sharply downward sloping than MR… . In other words,
dp/dt is greater than unity when the demand curve’s
upward concavity is strong enough… (1968: 200).
The model
Exploring the conﬁgurations suggested by Bishop requires
a speciﬁc model of demand and cost. The model outlined
below is sufficiently ﬂexible to address the questions at
hand. The general forms for the demand and cost
functions are these: 
p = α + βq–ρ – γqµ (1)
and
c = λ + δq–ε + κqν (2)
Equation (1) is the inverse demand curve, showing
willingness to pay. Equation (2) describes the average cost
curve. These functional forms are general enough to
generate curves with varying curvature characteristics.
They include the following important special cases: 
• linear demand curve:2 ρ = 0 and µ =1 (p = α + γq)
• constant-price-elasticity demand curve: α = g = 0; in this
case, the elasticity of demand is 1/ ρ (p = βq–ρ) 
• constant unit cost (horizontal marginal cost = average
cost): δ = κ = 0 (c = λ). 
The model is put to work in an Excel workbook. The
workbook shows the equilibrium value in the absence of a
tax. It also shows, in a separate sheet, the optimal value
and calculates the deadweight loss that results from the
monopolists’ production of a quantity below the one for
which CN = p. This second sheet becomes important in
considering the losses due to the imposition of a tax. The
results of imposing a tax on a monopolist are shown in a
Page 26 CHEER  Volume 18third sheet. Each sheet requires the discovery of a value
that optimises some function. In the ﬁrst sheet, the
optimising condition is that RN = CN, the proﬁt-
maximizing rule for the monopolist in the absence of a
tax. In the second sheet the optimising condition is that p
= CN, the condition for efficient resource allocation.
Finally, in the third sheet the optimising condition is that
RN = CN + t, where t is the per-unit tax rate.
Equilibrium, no tax
Figure 1 shows the results of implementing Solver given
one set of parameter values. We selected these values
intentionally to provide a ‘reasonable-looking’ demand
curve (downward-sloping and without obvious curvature
anomalies) that, nonetheless, generates a price rise in
response to a speciﬁc tax that exceeds the tax rate. Along
with the graph, Figure 1 shows the critical values for
evaluating the effect of an excise tax on the price. Of
immediate interest is the last entry, dp/dt = 2.543469252.
This implies that a tax of $1 will result in a price increase
of about $2.54.
Excel’s Solver is required to ﬁnd the quantity that
minimises the absolute difference between marginal
revenue (MR) and marginal cost (MC). The table reports
the average value of MR and MC as well as each
separately. In fact, the two are so close that the difference
(0.0000001865) does not appear in the tabled values of MR
and MC (RN and CN). To see why the problem must be
solved numerically rather than analytically, consider the
determination of q*, the proﬁt-maximizing quantity. The
marginal revenue and marginal cost functions are as
follows:
MR = α + (1– ρ)βq–ρ – (µ + 1)γqµ (3)
and
MC = λ + (1– ε)δq–ε +(ν + 1)κqν (4)
One cannot simply set MR = MC and solve for q because
the polynomial resulting from the equality may be of any
order.
Efficiency
As does Bishop, this paper focuses mainly on price but
also addresses the question of efficiency. Accordingly,
once the demand and cost curves are speciﬁed and the
equilibrium values are determined, the user is invited to
examine a second sheet to see the implications of
monopoly for efficiency. In this case, the efficient quantity
is 8.24 units rather than the equilibrium value of 3.68 units.
The deadweight loss associated with monopoly is
$6,497.72. To get a sense of how large the deadweight loss
is, consider that the equilibrium expenditure on this good
is about $3861.67*3.6768 or $14,198.59. That is, the
deadweight loss is almost one-half as large as spending.
We emphasise that the demand curve is not necessarily
representative.
Equilibrium, with tax
We now turn to the effect of imposing an excise tax on
this good. The output from the ﬁrst sheet predicts that the
price will rise by about $2.54 per $1.00 tax. In fact, the rise
is a bit larger, $3.065 per $1.00 when a $200 per-unit tax is
imposed. Also of note is the size of the added deadweight
loss (DWL) when the tax is imposed. The addition to DWL
is $2,109.81, over three times the amount of tax revenue
raised, $605.89. Thus it costs the private sector $605.89 +
$2,109.81 or $2,715.70 to deliver $605.89 to the
government. This is a striking example of what Bishop
means when he says (1968: 105):
‘The concept of the “incidence” of the tax as between
the consumers and the monopolistic producer is even
more anomalous than in the competitive case, because
of the intensiﬁed deadweight loss. In one case, … the
monopolist … pays the whole tax and more; and the
burden on consumers must then be added to that. With
so much deadweight loss, there does not seem to be
any meaningful way of saying what fractions of the tax
are paid by consumers and producers.’ 
Exercises
Once the model has been reviewed, the workbook allows
the user to specify values and examine their implications
for the impact of a speciﬁc tax. We provide values for ﬁve
cases:
• Linear demand curve. The user is invited to conﬁrm the
textbook ﬁndings regarding the sharing of the tax
between seller and buyers. Two exercises are provided.
The ﬁrst retains the U-shaped average cost curve shown
in Figure 1. In the second, the case of AC = MC =
constant is examined. The user conﬁrms that the
monopoly quantity is one-half the efficient quantity and
that the price rises by one-half as much as the tax. 
• Constant-elasticity demand curve. The user sees that in
this case, if the marginal cost curve is horizontal the
ﬁrm engages in a type of mark-up pricing and, as
shown by Mixon (1986) the price rises by more than the
tax.
• Upward-sloping MR curve. Economists offer both
theoretical and empirical reasons to expect that virtually
all demand curves are downward sloping. Textbook
representations of demand curves and marginal revenue
curves typically show the latter below the former and
downward sloping as well. A downward-sloping
demand curve need not, however, imply a downward-
sloping marginal revenue curve. Formby, Layson and
CHEER  Volume 18 Page 27
Figure 1. Illustrative exampleSmith (1982) show that many of the demand curves that
authors of classic textbooks have drawn to illustrate the
demand/marginal revenue relationship, in fact imply
marginal revenue curves that have some upward-
sloping region (though the authors mistakenly drew
their corresponding marginal revenue curves as
downward sloping). 
• Downward-sloping MC curve. Bishop (1968: 101) points
out that a tax can raise price by more than the tax rate
if the marginal cost curve is sufficiently downward
sloping. Such might be the case with a ‘natural
monopolist’. 
• Concave demand curve. This case in included mainly
for the sake of completeness. Like the linear case, the
concave demand curve yields the result that dp/dt < 1.
Summary
This paper develops the analysis of the effect of an excise
tax on the price charged by a proﬁt-maximising
monopolist. Following Bishop, it shows that the impact of
an excise tax imposed on a monopolist differs from that of
a competitive industry in two important ways. First and
more striking is the result that the price can rise by more
than the tax and that this can happen under conditions
that cannot be ruled out a priori. The second difference is
the size of the deadweight loss imposed on a monopolist:
this loss can be a multiple of the amount of tax revenue
raised. These points are driven home by demonstrating
their occurrence in a set of downloadable exercises based
on a simple model and executed in a Microsoft Excel
workbook, which is also available for downloading.
Notes
1 An overview that contains a link to the workbook along with
a set of exercises is available at: http://csob.berry.edu/faculty/
economics/taxincidence/taxincidence.html. 
2 In this familiar case, RO = 2pN (the MR curve has twice the
slope of the demand curve). Bishop (1968: 200) shows that
dp/dt = pN/(RO – CO), so that dp/dt > 1 only if –pN > –CO.
That is, the price rises by more than the tax only if the
marginal cost curve is negatively sloped and it is more steeply
sloped than the demand curve. But the latter result violates
the second-order condition: the ﬁrm would be minimising
proﬁts and not maximising them.
References 
Bishop, R. L. (1968) ‘The Effects of Speciﬁc and Ad Valorem
Taxes’, Quarterly Journal of Economics, 82, 198–218.
Formby, J. P., Layson, S. and Smith, W. J. (1982) 
‘The Law of Demand, Positive Sloping Marginal Revenue, and
Multiple Proﬁt Equilibria’, 
Economic Inquiry, 20, 303–311.
Mixon, J. W. Jr. (1986) ‘On the Incidence of Taxes on a
Monopolist’s Price: A Pedagogical Note’, 
Journal of Economic Education, 17, 201–203.
Contact details
J. Wilson Mixon, Jr. (Corresponding author)
Dana Professor of Economics
Campbell School of Business
Berry College
E-mail: wmixon@berry.edu
Bradley N. Hopkins
Economics Major
Berry College
Page 28 CHEER  Volume 18

On the Incidence of Taxes on a Monopolist’s Price: A Pedagogical Note’,

The Effects of Speciﬁc and Ad Valorem Taxes’,

The Law of Demand, Positive Sloping Marginal Revenue, and Multiple Proﬁt Equilibria’,

Analysing the Effects of Excise Taxes Using Microsoft Excel

http://www.economicsnetwork.ac.uk/cheer/ch18/mixon.pdf

Analysing the Effects of Excise Taxes Using Microsoft Excel

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