- Email: [email protected]
On repeated myopic use of the inverse elasticity pricing rule
Accepted Manuscript On repeated myopic use of the inverse elasticity pricing rule Kenneth Fjell, Debashis Pal
PII: DOI: Reference:
S0165-1765(18)30485-3 https://doi.org/10.1016/j.econlet.2018.11.028 ECOLET 8286
To appear in:
Economics Letters
Received date : 25 July 2018 Revised date : 26 October 2018 Accepted date : 23 November 2018 Please cite this article as: K. Fjell and D. Pal, On repeated myopic use of the inverse elasticity pricing rule. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.11.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Convergence on profit maximizing price for sufficiently convex demand
Non-constant marginal cost parallel shifts the convergence condition for demand
Also convergence for an alternative expression if sufficiently convex demand
*Title Page
On Repeated Myopic Use of the Inverse Elasticity Pricing Rule Kenneth Fjell* NHH Norwegian School of Economics and
Debashis Pal University of Cincinnati
Revised: October 26, 2018
JEL codes: C61, D40, L11 Keywords: microeconomics; markup pricing; elasticity
*Corresponding author: Kenneth Fjell, NHH Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway. Email: [email protected]. Phone: +4797068386.
Acknowledgements: We are grateful to Robert W. Scapens, Trond E. Olsen, Joseph E. Harrington and an anonymous referee for very helpful comments.
Abstract We examine the effects of repeated myopic use of the inverse elasticity pricing rule. By myopic, we mean ignoring that elasticity and marginal cost may vary with output and price. It is known that myopic use of the rule leads to price changes which are too large relative to the optimal price change (Fjell, 2003). While some microeconomics textbooks suggest that the rule may be used repeatedly to reach optimal price, they are vague about the conditions for when this works. We show that repeated myopic use leads to convergence if demand is sufficiently convex, and specify an exact condition.
2
*Manuscript Click here to view linked References
1. Introduction A well-known rule for marking up marginal cost to ensure optimal or profit maximizing price is the so called inverse elasticity rule (e.g. Browning and Zupan, 2002; Mansfield, 1994; Nicholson and Snyder, 1985; Pindyck and Rubinfeld, 2005): (1)
where
is the price elasticity and mc is the marginal cost.
Existing literature states that the relation (1) holds exactly at the profit-maximizing price. In addition, some microeconomics textbooks also suggest that the rule may be used repeatedly to reach the profit-maximizing price. However, the conditions ensuring that repeated use will converge to the profit-maximizing price are only vaguely described. For instance, Browning and Zupan (2002) state that: “If you know your demand elasticity and marginal cost, this expression can be used to calculate the profit-maximizing price. This formula has one difficulty: it holds exactly only at the point of profit maximization, and because marginal cost and elasticity may vary with output, you may need to use this expression repeatedly [italics added] to locate the profitmaximizing price. However, if cost and elasticity vary only a little [italics added] over the range of output you are considering, this formula can approximate the profit-maximizing price quite closely.” (p. 302). In another influential microeconomics textbook, by Pindyck and Rubinfeld (2005), the rule in (1) is referred to as “A Rule of Thumb for Pricing.” (p. 344). Similar to Browning and Zupan (2002) they also suggest that the rule can be used repeatedly to reach the profit-maximizing price, but only vaguely indicate under what circumstances this process may converge.
3
We look for conditions such that repeated myopic use of the inverse elasticity rule gives rise to the profit-maximizing price. By myopic, we ignore that elasticity and marginal cost both may vary with output and price. Besanko et al. (2017) suggest that the rule can be used to determine whether price should be raised or lowered based on local estimates, though not by how much. Fjell (2003) finds that when elasticity and marginal cost depend on price, a one-time price adjustment based on the rule typically leads to a price which bypasses (overshoots) the profit maximizing price. In contrast to a one-time price adjustment, we study the impact of repeated use of the myopic adjustment process. Related to our problem is the use of heuristics based on a few observations to make conjectures about demand and costs. For instance, one could make assumptions about functional form of demand and cost, and then fit this to recent observations to make estimates of the entire functions, and subsequently maximize profit based on this (Baumol and Quandt, 1964; Laitinen, 2009). We also use local (point) information of demand and cost, yet we do not make conjectures about the functional form of demand. Rather, we explore the conditions for which repeated myopic use of the inverse elasticity pricing rule gives rise to the profit-maximizing price. Our approach is more analogous to that of Ray and Gramlich (2016) who show that a full-cost pricing heuristic can converge on the profit-maximizing price when the firm can estimate its equilibrium profit.1 As us, they neither have nor seek information on the entire demand and cost curves, but simply assume local knowledge of these.
1
They use the term “equilibrium income” (p. 27).
4
2. Analysis We first assume that marginal cost, mc, is strictly positive and constant.2 The concluding section states the result for non-constant marginal cost. We look for conditions such that when equation (1) is applied repeatedly in a myopic manner, prices converge to the profit-maximizing price, where at each step the new price,
is given by equation (2) below. (2)
where
denotes the price elasticity of demand at
Also, we only consider prices in the elastic range of demand, i.e.
. We assume that
.
since the rule in (1) yields
negative prices otherwise.3 Further, we assume that the firm accurately estimates elasticity and marginal cost at each of the prices that emerges from application of (2).4 Although it is a standard assumption that is implicitly made in the literature, it is important to note that inaccurate estimation of slopes may result in errors in the convergence analysis. Define (2) is
. A sufficient condition for convergence of the prices generated by (see e.g. Chiang, 1984). Hence, repeated myopic use of the inverse elasticity
rule will converge towards the profit-maximizing price if the following condition holds for all prices
2
If then the optimal price, p*, is such that . This implies that price cannot be too low. Specifically, , which is positive. 4 To estimate current point elasticity, it would be sufficient to have marginal knowledge of demand, i.e. to know the slope at the current price. Ray and Gramlich (2016) argue that this can be estimated through arbitrarily small price movements around the current price. 3
5
(3) From the straight bracket in (3) we see that the relationship between current and myopically prescribed price (i.e.
) is negative if
i.e. that demand becomes more elastic the higher
the price. For negatively sloped phase lines, the iteration path will be one of oscillation (Chiang, 1984). Thus, repeated myopic use will yield prices which oscillate around (or bypass) the profitmaximizing price, and converge to it if (3) holds.5 For constant elasticity demand functions, for which
, the left side of (3) reduces to zero and
we see that the condition in (3) always holds; we always get convergence. Further, since we then also have that
, we get convergence in one iteration since new price does not depend on
the old price, as pointed out by Fjell (2003). If elasticity is not constant, we see from (3) that the smaller the impact of a price change is on price elasticity, i.e. the lower
is, the more likely it is that we get convergence, and vice versa. This is
in line with what Browning and Zupan (2002) and Pindyck and Rubinfeld (2005) suggest. Note that if (3) holds with equality, then we get an infinite price loop around the optimal price. In other words, price iterates from the initial to the new price and back again indefinitely.6 If the left hand side of (3) is greater than one, we get divergence. In other words, repeated myopic use leads to a price which oscillates further and further away from the profit-maximizing price.
5
Condition (3) presents a sufficient condition. A necessary condition is that (3) holds for all prices in an interval around the profit maximizing price. The necessary condition does not hold for a linear demand. 6
The condition for this is
, that is, the ratio between the new price and the initial price equals the ratio
between the markup (of a constant marginal cost) at the initial price and the markup at the new myopic price.
6
Note that
(4)
Substituting (4) into (3) yields the following sufficient condition for convergence: (5) We rewrite the condition in (5) as: (6) The condition in (6) can be expressed as the following condition on convexity of the demand curve: (7) If the condition in (7) hold for all relevant prices then repeated myopic use of (1) gives rise to a sequence of prices converging towards the profit-maximizing price. Note that the first term on either side corresponds to the convexity of constant elasticity demand functions . In addition,
is positive. Hence, the sufficient condition for
convergence is always satisfied for constant elasticity demand functions, in line with what Browning and Zupan (2002) state. Note also that the left hand side of (7) is positive as long as price is not too high. Specifically: (8)
7
Note that this upper price limit in (8) exceeds the profit-maximizing price given by (1). Thus, to satisfy the sufficient condition, demand must be strictly convex at and below the profit-maximizing price, which among other excludes linear demands. In sum, we thus have that as long as the demand curve is sufficiently convex, but not too convex, i.e. that its second derivative fulfills the condition in (7), then repeated myopic use of the inverse elasticity pricing rule will give rise to the optimal price.
3. Concluding Remarks Some mainstream microeconomics textbooks suggest that the inverse elasticity pricing rule may be used repeatedly to reach the profit-maximizing price (Browning and Zupan, 2002; Pindyck and Rubinfeld, 2005). However, they are vague about the conditions such that the convergence works. We show that repeated myopic use will generate convergence to the profit-maximizing price if demand is sufficiently convex, but not too convex, where the second derivative of the demand function satisfies:
. The sufficient
condition for convergence does not hold, among others, for linear demands. We have considered constant marginal cost. If the marginal cost is non-constant, then the sufficient condition for convergence becomes:
When marginal cost is non-constant, the last terms on both sides, which are identical, will be nonzero. The term may be interpreted as causing a “parallel shift” of the limits of the condition in (7); 8
“up” or towards more convex demand functions if marginal cost is increasing in quantity and “down” or towards less convex demand functions if marginal cost is decreasing in quantity. We focused on the inverse elasticity pricing rule, because it receives a lot of attention in intermediate microeconomics and managerial economics textbooks. There are alternative ways to express the profit maximization condition of a monopolist. For example, the profit maximizing price may be expressed as
. With constant marginal cost, the corresponding
sufficient condition for convergence turns out to be
Hence, we get the following
sufficient convergence condition in terms of curvature of demand:
(9) Thus, demand must again be strictly convex, but not too convex, for repeated myopic use of this rule to converge, also. Neither rule generate convergence for linear demand. In contrast, if , then both processes converge, yet the sequence of prices and the rate of convergence are distinct. It will be interesting to find a profit-maximization rule that gives rise to convergences to the profit-maximizing price, but the inverse elasticity rule fails. Interestingly, the convergence of repeated myopic use of the inverse elasticity pricing rule seems related to the question of cost pass-through. Pass-through
may be expressed in terms of
elasticities. It can be shown that under some restrictions on demand
, where price is
defined as net of marginal cost (Weyl and Fabinger, 2013). Whether a monopolist passes through less than or more than an increase in marginal cost depends on how
9
varies with
If
, the myopic inverse elasticity pricing rule gives rise to convergence to the profit maximizing price. Further investigation in this regard is likely to offer valuable insights.
4. References Baumol, W. J., & Quandt, R. E. (1964). Rules of thumb and optimally imperfect decisions. The American economic review, 54(2), 23-46. Besanko, D. A., Dranove, D. S., Shanley, M., & Schaefer, M. (2017). Economics of Strategy. Wiley. Browning, E. K., & Zupan, M. A. (2002). Microeconomics: Theory & applications, 7th ed.. Wiley. Chiang, A. C. (1984). Fundamental methods of mathematical economics. McGraw-Hill. Fjell, K. (2003). Elasticity based pricing rules: a cautionary note. Applied Economics Letters, 10(12), 787791. Laitinen, E. K. (2009). From complexities to the rules of thumb: towards optimisation in pricing decisions. International Journal of Applied Management Science, 1(4), 340-366. Mansfield, E. (1994). Applied microeconomics. WW Norton & Company. Nicholson, W., & Snyder, C. (1985). Microeconomic Theory: Basic Principles and Extensions, Dryden Press. Pindyck, R. S., & Rubinfeld, D. L. (2005). Microeconomics (6th edn). Upper Saddle River, NJ: Pearson Prentice Hall. Ray, K., & Gramlich, J. (2016). Reconciling Full-Cost and Marginal-Cost Pricing. Journal of Management Accounting Research, 28(1), 27-37. Weyl, E.G., & Fabinger, M. (2013). Pass-Through as an Economic Tool: Principles of Incidence under Imperfect Competition. Journal of Political Economy, 121(3), 528-583.
10
PII: DOI: Reference:
S0165-1765(18)30485-3 https://doi.org/10.1016/j.econlet.2018.11.028 ECOLET 8286
To appear in:
Economics Letters
Received date : 25 July 2018 Revised date : 26 October 2018 Accepted date : 23 November 2018 Please cite this article as: K. Fjell and D. Pal, On repeated myopic use of the inverse elasticity pricing rule. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.11.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Convergence on profit maximizing price for sufficiently convex demand
Non-constant marginal cost parallel shifts the convergence condition for demand
Also convergence for an alternative expression if sufficiently convex demand
*Title Page
On Repeated Myopic Use of the Inverse Elasticity Pricing Rule Kenneth Fjell* NHH Norwegian School of Economics and
Debashis Pal University of Cincinnati
Revised: October 26, 2018
JEL codes: C61, D40, L11 Keywords: microeconomics; markup pricing; elasticity
*Corresponding author: Kenneth Fjell, NHH Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway. Email: [email protected]. Phone: +4797068386.
Acknowledgements: We are grateful to Robert W. Scapens, Trond E. Olsen, Joseph E. Harrington and an anonymous referee for very helpful comments.
Abstract We examine the effects of repeated myopic use of the inverse elasticity pricing rule. By myopic, we mean ignoring that elasticity and marginal cost may vary with output and price. It is known that myopic use of the rule leads to price changes which are too large relative to the optimal price change (Fjell, 2003). While some microeconomics textbooks suggest that the rule may be used repeatedly to reach optimal price, they are vague about the conditions for when this works. We show that repeated myopic use leads to convergence if demand is sufficiently convex, and specify an exact condition.
2
*Manuscript Click here to view linked References
1. Introduction A well-known rule for marking up marginal cost to ensure optimal or profit maximizing price is the so called inverse elasticity rule (e.g. Browning and Zupan, 2002; Mansfield, 1994; Nicholson and Snyder, 1985; Pindyck and Rubinfeld, 2005): (1)
where
is the price elasticity and mc is the marginal cost.
Existing literature states that the relation (1) holds exactly at the profit-maximizing price. In addition, some microeconomics textbooks also suggest that the rule may be used repeatedly to reach the profit-maximizing price. However, the conditions ensuring that repeated use will converge to the profit-maximizing price are only vaguely described. For instance, Browning and Zupan (2002) state that: “If you know your demand elasticity and marginal cost, this expression can be used to calculate the profit-maximizing price. This formula has one difficulty: it holds exactly only at the point of profit maximization, and because marginal cost and elasticity may vary with output, you may need to use this expression repeatedly [italics added] to locate the profitmaximizing price. However, if cost and elasticity vary only a little [italics added] over the range of output you are considering, this formula can approximate the profit-maximizing price quite closely.” (p. 302). In another influential microeconomics textbook, by Pindyck and Rubinfeld (2005), the rule in (1) is referred to as “A Rule of Thumb for Pricing.” (p. 344). Similar to Browning and Zupan (2002) they also suggest that the rule can be used repeatedly to reach the profit-maximizing price, but only vaguely indicate under what circumstances this process may converge.
3
We look for conditions such that repeated myopic use of the inverse elasticity rule gives rise to the profit-maximizing price. By myopic, we ignore that elasticity and marginal cost both may vary with output and price. Besanko et al. (2017) suggest that the rule can be used to determine whether price should be raised or lowered based on local estimates, though not by how much. Fjell (2003) finds that when elasticity and marginal cost depend on price, a one-time price adjustment based on the rule typically leads to a price which bypasses (overshoots) the profit maximizing price. In contrast to a one-time price adjustment, we study the impact of repeated use of the myopic adjustment process. Related to our problem is the use of heuristics based on a few observations to make conjectures about demand and costs. For instance, one could make assumptions about functional form of demand and cost, and then fit this to recent observations to make estimates of the entire functions, and subsequently maximize profit based on this (Baumol and Quandt, 1964; Laitinen, 2009). We also use local (point) information of demand and cost, yet we do not make conjectures about the functional form of demand. Rather, we explore the conditions for which repeated myopic use of the inverse elasticity pricing rule gives rise to the profit-maximizing price. Our approach is more analogous to that of Ray and Gramlich (2016) who show that a full-cost pricing heuristic can converge on the profit-maximizing price when the firm can estimate its equilibrium profit.1 As us, they neither have nor seek information on the entire demand and cost curves, but simply assume local knowledge of these.
1
They use the term “equilibrium income” (p. 27).
4
2. Analysis We first assume that marginal cost, mc, is strictly positive and constant.2 The concluding section states the result for non-constant marginal cost. We look for conditions such that when equation (1) is applied repeatedly in a myopic manner, prices converge to the profit-maximizing price, where at each step the new price,
is given by equation (2) below. (2)
where
denotes the price elasticity of demand at
Also, we only consider prices in the elastic range of demand, i.e.
. We assume that
.
since the rule in (1) yields
negative prices otherwise.3 Further, we assume that the firm accurately estimates elasticity and marginal cost at each of the prices that emerges from application of (2).4 Although it is a standard assumption that is implicitly made in the literature, it is important to note that inaccurate estimation of slopes may result in errors in the convergence analysis. Define (2) is
. A sufficient condition for convergence of the prices generated by (see e.g. Chiang, 1984). Hence, repeated myopic use of the inverse elasticity
rule will converge towards the profit-maximizing price if the following condition holds for all prices
2
If then the optimal price, p*, is such that . This implies that price cannot be too low. Specifically, , which is positive. 4 To estimate current point elasticity, it would be sufficient to have marginal knowledge of demand, i.e. to know the slope at the current price. Ray and Gramlich (2016) argue that this can be estimated through arbitrarily small price movements around the current price. 3
5
(3) From the straight bracket in (3) we see that the relationship between current and myopically prescribed price (i.e.
) is negative if
i.e. that demand becomes more elastic the higher
the price. For negatively sloped phase lines, the iteration path will be one of oscillation (Chiang, 1984). Thus, repeated myopic use will yield prices which oscillate around (or bypass) the profitmaximizing price, and converge to it if (3) holds.5 For constant elasticity demand functions, for which
, the left side of (3) reduces to zero and
we see that the condition in (3) always holds; we always get convergence. Further, since we then also have that
, we get convergence in one iteration since new price does not depend on
the old price, as pointed out by Fjell (2003). If elasticity is not constant, we see from (3) that the smaller the impact of a price change is on price elasticity, i.e. the lower
is, the more likely it is that we get convergence, and vice versa. This is
in line with what Browning and Zupan (2002) and Pindyck and Rubinfeld (2005) suggest. Note that if (3) holds with equality, then we get an infinite price loop around the optimal price. In other words, price iterates from the initial to the new price and back again indefinitely.6 If the left hand side of (3) is greater than one, we get divergence. In other words, repeated myopic use leads to a price which oscillates further and further away from the profit-maximizing price.
5
Condition (3) presents a sufficient condition. A necessary condition is that (3) holds for all prices in an interval around the profit maximizing price. The necessary condition does not hold for a linear demand. 6
The condition for this is
, that is, the ratio between the new price and the initial price equals the ratio
between the markup (of a constant marginal cost) at the initial price and the markup at the new myopic price.
6
Note that
(4)
Substituting (4) into (3) yields the following sufficient condition for convergence: (5) We rewrite the condition in (5) as: (6) The condition in (6) can be expressed as the following condition on convexity of the demand curve: (7) If the condition in (7) hold for all relevant prices then repeated myopic use of (1) gives rise to a sequence of prices converging towards the profit-maximizing price. Note that the first term on either side corresponds to the convexity of constant elasticity demand functions . In addition,
is positive. Hence, the sufficient condition for
convergence is always satisfied for constant elasticity demand functions, in line with what Browning and Zupan (2002) state. Note also that the left hand side of (7) is positive as long as price is not too high. Specifically: (8)
7
Note that this upper price limit in (8) exceeds the profit-maximizing price given by (1). Thus, to satisfy the sufficient condition, demand must be strictly convex at and below the profit-maximizing price, which among other excludes linear demands. In sum, we thus have that as long as the demand curve is sufficiently convex, but not too convex, i.e. that its second derivative fulfills the condition in (7), then repeated myopic use of the inverse elasticity pricing rule will give rise to the optimal price.
3. Concluding Remarks Some mainstream microeconomics textbooks suggest that the inverse elasticity pricing rule may be used repeatedly to reach the profit-maximizing price (Browning and Zupan, 2002; Pindyck and Rubinfeld, 2005). However, they are vague about the conditions such that the convergence works. We show that repeated myopic use will generate convergence to the profit-maximizing price if demand is sufficiently convex, but not too convex, where the second derivative of the demand function satisfies:
. The sufficient
condition for convergence does not hold, among others, for linear demands. We have considered constant marginal cost. If the marginal cost is non-constant, then the sufficient condition for convergence becomes:
When marginal cost is non-constant, the last terms on both sides, which are identical, will be nonzero. The term may be interpreted as causing a “parallel shift” of the limits of the condition in (7); 8
“up” or towards more convex demand functions if marginal cost is increasing in quantity and “down” or towards less convex demand functions if marginal cost is decreasing in quantity. We focused on the inverse elasticity pricing rule, because it receives a lot of attention in intermediate microeconomics and managerial economics textbooks. There are alternative ways to express the profit maximization condition of a monopolist. For example, the profit maximizing price may be expressed as
. With constant marginal cost, the corresponding
sufficient condition for convergence turns out to be
Hence, we get the following
sufficient convergence condition in terms of curvature of demand:
(9) Thus, demand must again be strictly convex, but not too convex, for repeated myopic use of this rule to converge, also. Neither rule generate convergence for linear demand. In contrast, if , then both processes converge, yet the sequence of prices and the rate of convergence are distinct. It will be interesting to find a profit-maximization rule that gives rise to convergences to the profit-maximizing price, but the inverse elasticity rule fails. Interestingly, the convergence of repeated myopic use of the inverse elasticity pricing rule seems related to the question of cost pass-through. Pass-through
may be expressed in terms of
elasticities. It can be shown that under some restrictions on demand
, where price is
defined as net of marginal cost (Weyl and Fabinger, 2013). Whether a monopolist passes through less than or more than an increase in marginal cost depends on how
9
varies with
If
, the myopic inverse elasticity pricing rule gives rise to convergence to the profit maximizing price. Further investigation in this regard is likely to offer valuable insights.
4. References Baumol, W. J., & Quandt, R. E. (1964). Rules of thumb and optimally imperfect decisions. The American economic review, 54(2), 23-46. Besanko, D. A., Dranove, D. S., Shanley, M., & Schaefer, M. (2017). Economics of Strategy. Wiley. Browning, E. K., & Zupan, M. A. (2002). Microeconomics: Theory & applications, 7th ed.. Wiley. Chiang, A. C. (1984). Fundamental methods of mathematical economics. McGraw-Hill. Fjell, K. (2003). Elasticity based pricing rules: a cautionary note. Applied Economics Letters, 10(12), 787791. Laitinen, E. K. (2009). From complexities to the rules of thumb: towards optimisation in pricing decisions. International Journal of Applied Management Science, 1(4), 340-366. Mansfield, E. (1994). Applied microeconomics. WW Norton & Company. Nicholson, W., & Snyder, C. (1985). Microeconomic Theory: Basic Principles and Extensions, Dryden Press. Pindyck, R. S., & Rubinfeld, D. L. (2005). Microeconomics (6th edn). Upper Saddle River, NJ: Pearson Prentice Hall. Ray, K., & Gramlich, J. (2016). Reconciling Full-Cost and Marginal-Cost Pricing. Journal of Management Accounting Research, 28(1), 27-37. Weyl, E.G., & Fabinger, M. (2013). Pass-Through as an Economic Tool: Principles of Incidence under Imperfect Competition. Journal of Political Economy, 121(3), 528-583.
10