TaxationIncidence

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 (p. 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." [Note] 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 generalization is straightforward.) The central statement of Bishop's analysis is this (p. 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 age, textbooks still routinely represent demand curves with straight lines. This primer addresses one of the cases that Bishop analyzes, that of an excise tax imposed on a monopolist. The remainder of the paper is organized 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. This section is followed by a discussion of a Microsoft Excel workbook workbook that embeds the functions. Finally, a set of exercises based on selected special cases closes the primer. (The workbook might not open for users of Internet Explorer. If this is the case, right-click on the link above and select the "Save Target As..." option.) [Click here or on the highlighted "Excel workbook" above to download the workbook.]
Demand, Supply, and Tax Incidence in Competitive Markets We summarize the most pertinent of Bishop's derivations below. The firm is assumed to maximize 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), R' = p + qp', where p' is the slope of the inverse demand curve, (R' = dR/dq is marginal revenue), R" = 2p' + qp", where p" is the rate at which the slope of the demand curve changes as q changes and R" 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), C' = c + cp', where c' is the slope of the average cost curve, dc/dq (C' is marginal cost), C" = 2c' + qc", where c" is the rate at which the slope of the average cost curve changes as q changes and C" is the rate at which marginal cost changes as q changes. Bishop direct demonstrates the result that every principles textbook reports for a competitive industry: ^dp/_dt = ^p'/_{(p' - s')}, = ^-p'/_{(s' +( - p'))} where s' is the slope of the competitive supply curve and dp/dt is the price change per one-unit change in the tax rate. 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 -p' > s' (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 -p' < s' (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 this 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 = ^p'/_{(R" - C")}, which can be shown to be positive because p' is negative and R' (marginal revenue) must cut C' (marginal cost) from above (so R" < C" at the quantity for which R' = C'. (Otherwise, R' would come to exceed C' as quantity increase, 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 = ^p'/_{(2(p' - c'))}. Here p' < 0, and c' < 0, so dp/dt is indeed positive. Furthermore, dp/dt is less than 1.0. If c' = 0, then dp/dt = 1/2. If c' > 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). In the unlikely case that the average cost curve slopes downward, 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 [p"]. 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... (p. 200).

The Model and An Application

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 model. The general forms for the demand and cost functions are these:

	p = α + βq^-ρ- γq^μ	1
and
	c = λ + δq^-ε + κq^ν	2

The height of the demand curve, the willingness to pay, is p. Thus, (1) is the inverse demand curve. Equation (2) describes the unit (average) cost curve. These functional forms are general enough to generate curves with varying curvature characteristics. They include the following important special cases: Table 1. Initial Coefficient Values

Linear demand curve: ρ = 0 and μ =1 (p = α + γq),
Constant-price-elasticity demand curve: α = γ = 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).

Table 1 shows the initial set of values for the coefficients of equations (1) and (2). Figure 2 and Figure 3 are based on this set of coefficients.

Before analyzing the implications of this set of parameters, it is useful to replicate Bishop's notation and the relationships which are central to the analysis. Table 2 provides the necessary review. In Table 2, the following definitions apply: Central Relationships

R is revenue, p is price, q is quantity, C is total cost, c is per-unit cost.

R' is the marginal revenue: MR = ^dR/_dq

R" is the slope of MR: R" = ^d²R/_dq²

C' is the marginal cost: MC = ^dC/_dq

C" is the slope of MC: C" = ^d²C/_dq²

p' is the slope of the inverse demand curve: p' = ^dp/_dq. p' < 0.

p" is the rate at which p' changes as q changes: p" = d²p/dq²

c' is the slope of the average cost curve: c' = dc/dq.

c" is the rate at which c' changes as q changes: c" = d²c/dq²

Implementing the Model in Excel

The model is put to work in an Excel workbook that is available here. (The workbook might not open for users of Internet Explorer. If this is the case, right-click on the link at the end of the previous sentence and select the "Save Target As..." option.) 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 C' = 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 a a third sheet. Each sheet requires the discovery of a value that optimizes some function. In the first sheet, the optimizing condition is that R' = C', the profit-maximizing rule for the monopolist in the absence of a tax. In the second sheet the optimizing condition is that p = C', the condition for efficient resource allocation. Finally, in the third sheet the optimizing condition is that R' = C' + t, where t is the per-unit tax rate.

Equilibrium, No Tax. The figure at the right shows the results of implementing Solver given the parameter values in Table 1. These values were selected 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.

Before looking at the next two sheets, consider some aspects of this one. The parameter values are entered in the upper left-hand side of this sheet. The user should not enter them directly into similar cells in the other two sheets. These values are copied to the other sheets to ensure comparability of the output across sheets..

Solver is required to find the value of q that minimizes the absolute difference between marginal cost and marginal revenue. 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 (R' and C'). 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 derived from equations (1) and (2), copied from Table 2, are as follows:

	MR = α + (1-ρ)βq^-ρ- (μ+1)γq^μ	3
and
	MC = λ + (1-ε)δq^-ε +(ν +1)κq^ν.	4

One cannot simply set MR = MC and solve for q. The solutions involve arbitrary powers, for which no general analytical solution is available. As we have seen, however, Excel's "Solver" tool makes it unnecessary to solve this set of equations analytically. Instead, the solution is achieved to a very close approximation with a numerical method.

The solution is achieved as follows: In a cell (Cell K28 in the dialog box at the right), define the objective function to minimize as the absolute value of the difference between R' (MR) and C' (MC) with the cell to the right of "Profit-maximizing q, q* =" as the cell whose value can change (Cell F14 here). Solver finds a value for which MR and MC (R' and C') are closest to each other. Some other points of interest in Table 2 are these:

At q*, both p' and c' are negative. The value of p' is always negative; c' is negative when q* is less than the quantity at which average cost is minimized.

At q*, the marginal revenue curve is becoming steeper (MR is increasing at an increasing rate), and the marginal cost curve is becoming steeper (MR is increasing at an increasing rate).

At q*, both the demand curve and the average cost curve are becoming flatter.

The two terms at the bottom of Table 3 relate to the effect that a tax will have on quantity and price respectively. For the demand curve shown in Figure 2, the effect of the tax will be to reduce output (no surprise) and to increase price by more than the tax—by about $2.54 per $1 tax rate. This value is approximate for any discrete tax rate, as we shall see below;.

The last result above is consistent with Bishop (p. 199). The effects of a tax on quantity and price are as follows:

	^dq/_dt = ¹/_{(R" - C")},	5
the effect of a tax on quantity, and
	^dp/_dt = ^p'/_{(R" - C")},	6

the effect of a tax on price. In the present case p' = -804.56 while R" - C" = -316.32, so price rises by more than the tax rate.

The graph is provided to provide a quick view of the relevant curves and the critical values (q, MR, MC, and p). Clicking on "Better graph" will provide access to a graph that shows more detail. The "Graphing" link shows some values that are used to draw the graph; they have no analytical importance. The "Graph Axes" cell contains a note that instructs the user on changing the graph's axes in case the selected parameter values do not show up well on the graph as it is currently constituted.

Efficiency. The second sheet provides a benchmark for the monopoly case, either with or without a tax imposed on the monopolist's product. In this sheet, part of which appears at the right, the efficiency-maximizing quantity is identified, and it is compared to the monopoly result. Efficient Quantity

For the demand curve employed here, the monopoly quantity (3.68, from the output above) is less than one-half the efficient quantity.

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 emphasize that the demand curve is not necessarily representative.

As noted above, finding the efficient quantity involves using Solver once more. This time, the objective function is to minimize the absolute value of the vertical distance between the marginal cost curve and the demand curve.

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. Furthermore, the model predicts that at the new margin, an additional $1 tax hike would cause a price rise of $3.66. With excise tax

Also of note is the size of the added deadweight loss when the tax is imposed. The addition to DWL is $2,109.81, which dwarfs the amount of tax revenue raised. Thus it costs the private sector $605.89 + 2109.81 or
$2715.70 to deliver $605.89 to the government. This is a striking example of what Bishop means when he says (p. 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.

Solver is invoked once more. This time Solver finds the value of q that minimizes the absolute value of the vertical distance between MR and (MC + t). The function to be minimized is ( |R' - C'- t| ). To execute the model, t is added to MC, so the C' function under "With Tax" is the same function as in the cell to its left, but with t added. (We must also add t to the average cost function, c.) This problem could have been solved by subtracting t from R' rather than adding it to C'. Doing this would have resulted in a reported price of $4474.21 - $500. (We must also subtract t from the p function.)

Figure 3 shows the effects of the tax on average cost, marginal cost, quantity and price.

Down to Cases

The remainder of this paper consists of a set of exercises for a few illustrative cases. The number of cases that Equations (1) and (2) can generate is limitless. This section considers these:

linear demand with U-shaped and constant cost curves,
constant-elasticity demand with constant cost,
upward-sloping marginal revenue with U-shaped and constant cost curves, and
downward-sloping marginal cost curve with a linear demand curve.

Brief answers are here.

Linear Demand Parameters

Linear Demand Curve. The table at the right contains a set of parameters that results in a linear demand curve. Change any parameters except the one set equal to zero. Other values may be substituted, as long as β = 0 and μ = 1.

In this familiar case, R" = 2p' (the MR curve has twice the slope of the demand curve). From equation (6),

^dp/_dt = ^p'/_{(R" - C")},

we can conclude that

^dp/_dt > 1

only if -p' > -C". 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 that the demand curve. Note

1	a	Enter the values above into the "EquilibriumWithNoTax" worksheet and determine the effect of a tax. This exercise will involve a series of steps. First, enter the values and run Solver.(For now, leave the "Firm's Cost" parameters as they were in the initial case.) If Solver is not enabled on your computer's Excel, use the "Tools/Add-Ins" menu option to install it. Run Solver again in the "OptimalQuantity" and "EquilibriumWithTax" worksheets. Discuss the size of the deadweight loss due to the monopolist's not producing a quantity such that MC = p. Also, discuss the size of the added deadweight loss due to a tax.
	b	Now consider the case of a horizontal MC curve, coupled with the linear demand curve. Set λ = 2000, δ = 0, and κ = 0 in the "EquilibriumWithNoTax" sheet. Run Solver. Run Solver again in the "EquilibriumWithTax" sheet. By what fraction of the tax does price increase? Will this fraction always pertain when the demand curve is linear and the MC curve is horizontal? Explain. Hint: Note

Constant-elasticity demand curve. When α = γ = 0 the demand curve exhibits the same price elasticity of demand at every quantity. For the illustration to make sense, the demand curve must be elastic. Since the marginal revenue is MR = p(1 - 1/e), where e is the price elasticity of demand, the MR curve must lie below the demand curve. More precisely MR = p*(1 - 1/e) at each q. The use of this case to illustrate the possibility that dp/dt > 0 appears in Mixon.

2	a	Return to the "EquilibriumWithNoTax" sheet. . Leave the cost coefficients as they were set for 1(b). Now set α = γ = 0, β = 5000, and ρ = 0.2. This corresponds to setting the price elasticity of demand equal to 5. Run Solver. Confirm that the MR curve is 0.8 as high as the demand curve, i. e., that MR = 0.8P. Why 0.8?
	b	Run Solver again in the "EquilibriumWithTax" sheet. By what fraction of the tax does price increase? Will this fraction always pertain when the demand curve exhibits constant elasticity and the MC curve is horizontal? Explain.

An odd case? 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.

Smith, Formby, and Layson (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).

Equation 1 can generate a marginal revenue curve that is initially upward-sloping and then downward sloping: ρ > 0 and μ > 1 will generate such a function. [Note] For the current illustration, ρ = 1.5 and μ = 0.5, which generates a marginal revenue curve that slopes upward throughout.

3	a	Place the values at the right in the appropriate cells in the "EquilibriumWithNoTax" sheet, and execute Solver. (Solver can be sensitive to the initial value. Setting a high initial value can cause Solver to look at negative values of q. In such cases, it can fail to converge. Set the initial value in this case below 12. See the note on selecting an initial value in the "EquilibriumWithNoTax" sheet.) What does R" = 4.48 mean? Why should the firm not produce more than q = 12.65 units even though the marginal revenue of subsequent units is higher than at q = 12.65 units
	b	Impose a tax of $200 on this commodity. What is the predicted effect on price, using the value of dp/dt in the "EquilibriumWithNoTax" sheet? What is the actual effect?

Another odd case? Downward-sloping MC curve. Bishop (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."

4	a.	Place the values at the right in the appropriate cells in the "EquilibriumWithNoTax" sheet, and execute Solver. This firm experiences dramatic economies of scale. As a result, ^dp/_dt = ___________. Why is dp/dt greater than 1.0 in this case? That is, how does the existence of a downward-sloping MC lead to this result?
	b.	In the "OptimumQuantity" sheet run Solver. What quanttity/price combination must occur for economic efficiency to be realized? What difficulty would be required if that combination were imposed?
	c.	Return to the "EquilibriumWithNoTax" sheet. Change the value of α to shift the demand curve. Select values in the range of 3500 to 3700 or so. How can it be that increasing the demand causes the price to fall, while decreasing demand causes it to rise?

A final case: Convex demand curve. This case is much like the linear case in terms of its predictions. In particular, dp/dt < 1. As with other cases, the deadweight loss of a tax on the monopolist's output is quite high.

5.	a.	Place the values at the right in the appropriate cells in the "EquilibriumWithNoTax" sheet, and execute Solver. The demand for this firm's product is convex from below. (Brad: Do I have the terminology right?) What does p' = -27.69 mean? What does p" = -10.00 mean? What was p" in the case of the linear demand curve? What was p" in the case of the connstant-elasticity demand curve?
	b.	Impose a tax of $200 on this commodity. What is the predicted effect on price, using the value of dp/dt in the "EquilibriumWithNoTax" sheet? What is the actual effect? What is the size of the deadweight loss relative to the amount of revenue raised by the tax?

Notes

[Re: Deadweight Loss]
The workbook is predominantly about the effect of a tax on the monopolist's price. Nevertheless, information about deadweight loss is provided. The sheet "OptimalQuantity" reports the deadweight loss due to the monopoly's charging a price above MC in the absence of a tax. The sheet "WithTax" reports the added deadweight loss due to the imposition of a tax. Back to top

[Re: Linear Case]

p'/(R" - C') = ^p'/_{(2p' - C")} > 1.

Both numerator and denominator are negative, so multiplying both sides by the denominator reverses the inequality, yielding

p' < 2p' - C", or -p' < -C".

If the MC curve is upward-sloping, -C" is negative, so this condition cannot hold. See Bishop (p. 201) for more on this case.

Back to text

[Re: "An Odd Case?"]
It can be instructive to try various coefficient values and see how the MR & MC change curves by doing the following:

Place coefficient values in the appropriate cells in the "EquilibriumWithNoTax" sheet. Do not change the values in the "WithTax" sheet.
Run Solver.
Then move to the "EquilibriumWithTax" sheet and run Solver again.

One problem can occur. Sometimes Solver fails to find a value for q. It returns a nonsensical negative value. If this happens, refer to the thumbnail graph on the "EquilibriumWithTax" sheet. Set a value of q, and compare MR & MC. If MR > MC (MR < MC), select a larger (smaller) q.

The horizontal axis is set so that it will change as q changes. In contrast, the vertical axis shows values over a fixed range, currently $0 - $5000. If no line appears for the selected coefficients, change the vertical axis range. As a rule of thumb, select the 3rd or 4th values of p from the table at the right of the graph (Cell N5 or N6) as the maximum value.

Back to text

References

Robert L. Bishop, "The Effects of Specific and Ad Valorem Taxes," Quarterly Journal of Economics, 82 (1968), 198 - 218. Back to text

John P. Formby, Stephen Layson, and W. James Smith, "The Law of Demand, Positive Sloping Marginal Revenue, and Multiple Profit Equilibria," Economic Inquiry 20 (1982), 303 - 311.

J. Wilson Mixon, Jr. “On the Incidence of Taxes on a Monopolist's Price: A Pedagogical Note.” Journal of Economic Education 17 (1986): 201 – 203. Back to text

Figures

Figure 2. Profit Maximization
Back to Table 1
Back to text

Figure 3. Effects of Excise Tax
Back to Table 1
Back to text

Authors:

J. Wilson Mixon, Jr.
Berry College
Mount Berry, Georgia 30149 USA
Email: wmixon@berry.edu

Bradley N. Hopkins
Berry College
Mount Berry, Georgia 30149 USA