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.]
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:
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:
Bishop shows that for the special case of linear demand and cost curves, a result very much like that of the competitive case occurs:
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:
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:
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:
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:
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. 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.
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 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 > 1only 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
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.
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.
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."
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.
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