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Abstract

Textbooks commonly misrepresent the effects of market entry and exit by consumers and producers as parallel shifts of linear market demand and supply curves. Such portrayals are inconsistent with the underlying premise that a market-level curve is the horizontal summation of individual-level curves. A more accurate depiction is a pivot: entry flattens (and exit steepens) a market-level curve. Reconceiving the effects of entry and exit as pivoting, rather than shifting, the curve has important implications for own-price elasticities, consumer surplus, producer surplus, the incidence of taxation, and the competitive market adjustment to a long run equilibrium.

JEL codes: A22, B40, D01

Keywords

market entry pivot elasticity consumer surplus producer surplus tax incidence long run equilibrium

Introduction

Microeconomics textbooks, at both the principles and intermediate levels, commonly portray individual supply and demand curves as linear and include some version of the following three statements regarding demand, with analogous statements regarding supply.¹

* The market demand curve is the horizontal summation of individual buyers’ demand curves.

* When buyers enter (exit) the market, the market demand curve shifts outward (inward).

* The own-price elasticity of market demand depends on the nature of the good, the availability of substitutes, consumers’ incomes, and the length of time buyers have for adjusting purchases.

This paper shows that if the first claim is true, then the second is false and the third is deficient.

The following section demonstrates that, because market-level curves are constructed as the horizontal sums of individual-level curves, it is impossible for entry or exit to shift a downward-sloping market demand or upward-sloping market supply curve in the mathematical sense. Rather, as shown next, entry and exit cause a market-level curve to pivot, either around its vertical axis or another pivot point, flattening with entry and steepening with exit. Consequently, a complete list of the factors influencing the own-price elasticity of market demand (supply) should include the number of consumers (producers) in the marketplace. The Implications section demonstrates why the difference between a shift and a pivot is important: in addition to the inconsistencies between lessons, the failure to properly account for the effect of entry and exit generates incorrect results regarding elasticity, consumer surplus, producer surplus, tax incidence, and competitive market adjustment to a long run equilibrium. The penultimate section briefly considers nonlinear functions, and the final section concludes.

The Impossibility of Shifts

By definition, an inward or outward shift of a linear curve requires the intercept to change while the slope remains constant (Soper, 2004); thus, a shift connotes a parallel movement. Most textbooks correctly describe the market demand (supply) curve as the horizontal summation of individual demand (supply) curves, which are almost always assumed to be linear. That construction implies that neither entry nor exit can cause the curve to shift. To demonstrate this, we offer the following theorem.²

Theorem

If all consumers have negatively-sloped linear demand functions, then entry flattens and exit steepens the slope of the market demand curve.

Proof

Let each of n consumers have the demand function

P = a_{i} - b_{i} q_{i}^{D}

(1)

where

q_{i}^{D}

denotes the quantity demanded by the ith buyer, P is the market price,

a_{i} > 0 \forall i

, and

b_{i} > 0 \forall i

. The quantity demanded in the market (

Q^{D}

) is the sum of the individual quantities demanded,

Q^{D} = \sum q_{i}^{D} = \sum (\frac{a_{i}}{b_{i}}) - P \sum (\frac{1}{b_{i}})

(2)

so the market demand function is

P = [\frac{\sum (a_{i} / b_{i})}{\sum (1 / b_{i})}] - [\frac{1}{\sum (1 / b_{i})}] Q^{D}

(3)

which has the slope

\partial P / \partial Q^{D} = - [\frac{1}{\sum_{i = 1}^{n} (1 / b_{i})}]

. As n rises, the slope of the curve becomes less negative, and as n declines, the slope becomes more negative:

\frac{- 1}{\sum_{i = 1}^{n - 1} (1 / b_{i})} < \frac{- 1}{\sum_{i = 1}^{n} (1 / b_{i})} < \frac{- 1}{\sum_{i = 1}^{n + 1} (1 / b_{i})} .

(4)

Thus, as consumers enter or exit, the market demand curve cannot shift in a parallel manner; it becomes flatter with entry and steeper with exit. An analogous argument applies to upward-sloping supply curves.

Entry and Exit Cause Pivots

We next examine the effects of entry and exit on the vertical intercept of a linear curve. First, note that a shift would imply that the vertical intercept of a demand (supply) curve necessarily rises (falls) with entry and falls (rises) with exit. That, however, need not be the case.

Consumer Entry

To isolate the effect of entry on demand, suppose that there are initially n consumers, each having the demand curve $P = a - b_{1} q^{D}$ , so that the current quantity demanded in the market is $Q^{D} = n q^{D} = n (a - P) / b_{1}$ , and the existing market demand curve is

P = a - (b_{1} / n) Q^{D}

(5)

Now let z new consumers enter the market, each of whom has the individual demand function $P = (a + x) - b_{2} q^{D}$ where $x$ is a shift parameter, which may be positive, negative, or zero, and $b_{2}$ is positive but may be greater than, less than, or equal to $b_{1}$ . If the new consumers were the only ones in the market, their collective demand curve would be

P = (a + x) - (b_{2} / z) Q^{D} .

(6)

Consider three cases. First, suppose x > 0, so that the new entrants are willing to pay more than incumbent consumers for their first unit of the good. Then for $P > a$ , only the demand of the new consumers is relevant, but where $P \leq a$ , the demand of all consumers is relevant, and the total quantity demanded becomes

Q^{D} = \frac{n (a - P)}{b_{1}} + \frac{z (a + x - P)}{b_{2}} .

(7)

The new market demand curve is then a combination of two linear functions, equations (6) and (7):

\begin{array}{l} P = & {\begin{cases} \begin{array}{l} (a + x) - (b_{2} / z) Q^{D} & \forall a < P \end{array} \\ [a + \frac{z b_{1} x}{n b_{2} + z b_{1}}] - [\frac{b_{1} b_{2}}{n b_{2} + z b_{1}}] Q^{D} \forall P \leq a \end{cases} \end{array}

(8)

In this case, the market demand curve shifts upward and pivots, or kinks outward, at $P = a$ , as in Panel A of Figure 1, where the original market demand curve is represented by $D_{1}$ , the demand of the new consumers is $D_{2}$ , and the new market demand curve is highlighted as $D^{'}$ .³

Figure 1.

Market demand pivots outward in response to consumer entry. (A) x > 0 (B) x < 0 (C) x = 0.

As a second case, suppose x < 0, so that the new consumers are not willing to pay as much as the incumbent consumers for their first unit of the good. Then at the highest prices ( $P > a + x)$ , only the original market demand curve equation (5) is relevant, while at lower prices ( $P \leq a + x)$ , the incumbents and the new entrants are all willing to buy. Then the market demand curve pivots outward, or kinks, at $P = a + x$ , such that the new market demand function is

\begin{array}{l} P = & {\begin{cases} a - (b_{1} / n) Q^{D} \forall a + x < P \\ [a + \frac{z b_{1} x}{n b_{2} + z b_{1}}] - [\frac{b_{1} b_{2}}{n b_{2} + z b_{1}}] Q^{D} \forall P \leq a + x \end{cases} \end{array}

(9)

This is shown in panel B of Figure 1, where the original market demand curve is again labeled $D_{1}$ , the demand curve of the new consumers is $D_{2}$ , and the new market demand curve is $D^{'}$ . Notice that in this scenario, as $D_{1}$ becomes $D^{'}$ , the upper portion of the original market demand curve does not change, and thus, two possibilities regarding equilibrium arise. If market supply intersects demand at a price below the pivot point (a + x), then the equilibrium price and quantity both increase as result of the entry of new consumers. However, if supply intersects demand at a price above the pivot point, then despite the entry of new consumers, the equilibrium price and quantity remain unchanged, contrary to conventional wisdom.

For the third case, suppose $x = 0$ : the demand curve of the new consumers has the same vertical intercept as that of the incumbents, though it may again be steeper or flatter than the incumbents’ demand curve. Then as the new consumers enter, the market demand curve pivots outward at its vertical intercept $(a)$ , and from equation (7), the equation of the new curve is

P = a - [\frac{b_{1} b_{2}}{n b_{2} + z b_{1}}] Q^{D} .

(10)

This case is shown in Panel C of Figure 1. Note that in the limit, entry makes the slope approach zero and the market demand curve becomes nearly horizontal. Thus, the equilibrium price ( $P^{*}$ ) can never exceed the original vertical intercept ( $a$ ) as demand pivots upward (counterclockwise) along an upward-sloping supply curve. This is in contrast to depictions of entry as shifting the demand curve outward by increasing a, in which case there is no upper bound on $P^{*}$ .

A Classroom Example

As a simple numerical example suitable for an undergraduate course, Table 1 shows the partial demand schedules of two consumers (or, alternatively, two equally-sized cohorts of consumers) having linear demand functions for a particular good. At a price of $100, neither consumer would buy any units of this good, so none is demanded in the market; at a price of $80, the first consumer would buy 10 units and the second would buy 5, so the quantity demanded in the market is 15 units, and so forth. The first demand function is

P = 100 - 2 q_{1}^{D}

and the second is

P = 100 - 4 q_{2}^{D}

, somewhat steeper than the first. When combined, the market demand curve is

P = 100 - 1.33 Q^{D}

, which is flatter than either of the individual demand curves. We can immediately generalize from the example: if more such consumers enter the market, the market demand curve will become flatter still. The appendix provides a worksheet with a similar example that facilitates an experiential learning exercise for students.

Table 1.

Market Demand as the Horizontal Sum of Individual Demands.

Price	Quantity Demanded by Consumer 1	Quantity Demanded by Consumer 2	Quantity Demanded in the Market
100	0	0	0
80	10	5	15
60	20	10	30
40	30	15	45
20	40	20	60

The general intuition can be explained as follows. If the price is so high that each consumer would buy, say, only one unit of the good, then when new consumers enter the market, the quantity demanded in the market increases only by the number of new consumers; but if the price is low, so that each consumer would buy multiple units, then when new consumers enter, the quantity demanded in the market increases by a multiple of the number of new consumers. For this reason, entry flattens the curve, and by the same logic, when consumers exit the market, the market demand curve becomes steeper, as discussed in more detail below.⁴

Surprisingly, some textbooks correctly illustrate the market demand curve as being flatter than each individual demand curve, but fail to explicitly mention the difference in slope, and subsequently proceed to both describe in words and portray in diagrams the effect of entry (exit) by consumers as a parallel outward (inward) shift of the market demand curve.

Because we teach students that the market-level curve is the horizontal summation of individual curves, consistency requires that we depict market entry and exit as altering the slope and elasticity of the market-level curve, rather than shifting it in a parallel manner.

Consumer Exit

The analysis above indicates that if all consumers have linear demand curves, then the market demand curve will be nonlinear, unless all the individual demand curves have the same vertical intercept. Reversing the logic, if the market demand curve is linear, then either the individual demand curves are all linear and have the same vertical intercept, or at least one of the component curves has a higher vertical intercept and is kinked inward. These possibilities are depicted in Figure 2. In Panel A, curve $D_{1}$ is linear, while curve $D_{2}$ has a higher vertical intercept and is kinked inward, so that their horizontal summation creates the linear market demand curve D. Now if group 1 exits the market, so that only group 2 remains, curve D is replaced by $D_{2}$ ; the market demand curve pivots inward, becoming steeper at low prices, as in Panel A. If group 2 exits, so that only group 1 remains, D is replaced by $D_{1}$ ; the market demand curve shifts downward and becomes steeper everywhere (effectively, pivoting at a negative quantity), as in Panel B. In Panel C, both groups of consumers have linear demand curves with the same vertical intercept; if either group exits, the market demand curve D pivots inward (clockwise) around its vertical intercept, reversing Panel C of Figure 1.

Figure 2.

Market demand pivots inward in response to consumer exit.

Market Supply

The analysis of supply is analogous to that of demand, so we will only briefly summarize it. An individual producer’s short run supply curve is the portion of its marginal cost curve that lies above average variable cost (AVC); the firm shuts down at prices below AVC, so the supply curve is discontinuous before it reaches the horizontal axis. Nonetheless, the individual producer’s supply function is often written as a simple linear equation with a positive slope and either a positive, zero, or negative vertical intercept.

Again, there are three cases. If new entrants are able to bring their initial units of the good to the market at lower cost than the incumbent producers, so that the vertical intercept of the entrants’ collective supply curve is lower than that of the incumbents’ supply curve, then at the lowest prices, only the supply of the new entrants is relevant, while at higher prices, the supply of all producers is relevant. This is depicted in Panel A of Figure 3, where $S_{1}$ denotes the original market supply curve, $S_{2}$ is the collective supply curve of new entrants, and $S^{'}$ denotes the new market supply curve. In this case, the market supply curve shifts down and pivots out.

Figure 3.

Market supply pivots outward in response to producer entry.

Conversely, if new producers are only able to bring their first units of output to the market at higher prices than incumbent producers, then at the lowest prices, only the original market supply curve, $S_{1}$ , is relevant, and the curve pivots outward at the price at which the new producers offer their first units of output to the market, as shown in Panel B of Figure 3. Note that in this case, it is possible for market demand to intersect market supply below the pivot price, so that market entry leaves the equilibrium unchanged.

The third possibility is that the new producers have the same costs as incumbents, so the supply curves of the two groups have the same vertical intercept; then entry causes the market supply curve to pivot clockwise around the vertical intercept, as in Panel C of Figure 3.⁵

Finally, consider the exit of producers. If the original market supply curve is linear, then either all component supply curves have the same intercept, or one of the component curves has a lower vertical intercept and is nonlinear. In Panel A of Figure 4, curve $S_{1}$ is linear while $S_{2}$ has a lower vertical intercept and kinks inward, so that their horizontal summation generates the linear market supply curve S. If group 1 exits, so that only group 2 remains, the market supply curve pivots inward as it moves to the position of $S_{2}$ . In Panel B, group 2 exits so that only group 1 remains; in this case, the market supply curve shifts upward and becomes steeper (effectively, pivoting at a negative quantity).⁶ And in panel C, both component curves have the same vertical intercept; if either group exits, the market supply curve simply pivots inward at its vertical intercept, becoming steeper everywhere, and essentially reversing Panel C of Figure 3.

Figure 4.

Market supply pivots inward in response to producer exit.

Though we have presented multiple scenarios for completeness, it may be simplest to present entry and exit in introductory courses as pivoting the market demand and supply curves around their vertical intercepts, as depicted in Panel C of each diagram above. Although this assumes a common intercept for demand curves and a common intercept for supply curves, it is more plausible than suggesting that entry or exit causes parallel shifts of the market-level curve.⁷

Implications: Elasticity, Tax Incidence, Consumer Surplus, Producer Surplus, and Market Adjustment

In this section, we demonstrate that the traditional depiction of entry or exit as a shift gives false impressions of the changes in elasticity, tax incidence, consumer surplus, producer surplus, and the market adjustment mechanism.

Arc Elasticity of Demand

Although slope and elasticity are not the same, for linear functions there is a close relationship between the two, such that a more elastic curve has a flatter slope, and a less elastic curve is steeper. There are, of course, several different methods of calculating a price elasticity; to compare the overall elasticity of two different demand curves, we adopt Lerner’s (1933) arc elasticity metric as recommended by Seldon (1986), in which percentage changes are calculated relative to the lower values of price and quantity. To distinguish arc elasticities from point elasticities, we use $ε$ to denote the former and $η$ to denote the latter. And to simplify the mathematics, we will assume here that new entrants are homogeneous with incumbents insofar as their demand equations go, so that the market demand curve is $P = a - (b / n) Q^{D}$ , where $a > 0$ , $b > 0$ , and n is the total number of consumers in the market.

In the positive quadrant, where $Q \geq 0$ and $P \geq 0$ , the market demand curve extends from $(0, a)$ to $(a n / b, 0)$ . To avoid division by zero, we consider two sets of coordinates that are arbitrarily close to the vertical and horizontal intercepts: $(ϵ, (\frac{a n - b ϵ}{n}))$ and $((\frac{a n - ϵ n}{b}), ϵ)$ respectively, where $ϵ > 0$ is arbitrarily close to zero. Then the own-price arc elasticity of demand throughout the positive quadrant is the inverse of the slope of the demand curve:

- ε^{D} = - [\frac{(\frac{a n - ϵ n}{b}) - ϵ}{ϵ}] / [\frac{ϵ - (\frac{a n - b ϵ}{n})}{ϵ}] = n / b .

(11)

Equation (11) confirms that as consumers enter the market and $n$ rises, the absolute value of the own-price elasticity of market demand increases (and exit causes the opposite effect). However, if market entry is misrepresented as an outward parallel shift of the market demand curve (an increase in a), there is no effect on the arc elasticity measure. Thus, the manner in which the entry of new consumers is described makes an important difference in terms of elasticity.

The list of factors affecting the price elasticity of demand that appears in textbooks typically includes the nature of the good (e.g., whether it is a necessity or a luxury), the availability of substitutes, consumers’ incomes (or equivalently, the share of consumers’ budgets taken by the good in question), and the length of time that consumers have for adjusting their purchases. These are correct insofar as they apply to individual demand curves, but the discussion of elasticity usually follows the presentation of market demand, leaving the impression that this is a comprehensive list of factors affecting elasticity at the market-level. Moreover, empirical examples of price elasticities of demand taken from various industries are often presented as part of the lesson, confirming the impression that the list applies to the elasticity of market demand. (Indeed, an individual consumer’s price elasticity should not matter in a competitive market). According to equation (11), the litany of factors influencing the own-price elasticity of market demand should also include entry and exit by consumers.

Arc Elasticity of Supply

To see the effect of entry and exit on the price elasticity of supply, suppose that there are m producers, each having the individual supply function $P = c + k q^{S}$ , where $k \geq 0$ .⁸ The quantity supplied to the market is $Q^{S} = m q^{S}$ , and substitution gives the market supply function as $P = c + (k / m) Q^{S}$ . The vertical intercept, c, can be positive, zero, or negative, and this determines whether the curve is elastic, unitary, or inelastic, respectively (Horowitz, et al., 2013; Voon & Edwards, 1991).

First, consider $c > 0$ ; in this case, the market supply curve intersects the vertical axis at a positive price, so we examine the arc elasticity between an arbitrarily small quantity of output ( $Q = ϵ)$ and an arbitrarily large quantity ( $Q = θ)$ , where $0 < ϵ ≪ θ$ . The supply prices at these output levels are $P = c + (k ϵ / m)$ and $P = c + (k θ / m)$ , respectively. The overall arc elasticity of supply between these coordinates is then

ε^{S} = [\frac{(θ - ϵ)}{ϵ}] / [\frac{(\frac{c m + k θ}{m}) - (\frac{c m + k ϵ}{m})}{(\frac{c m + k ϵ}{m})}] = 1 + \frac{m c}{k ϵ} .

(12)

Notice that with $c > 0$ , the curve is price elastic ( $ε^{S} > 1$ ). As market entry by new producers (an increase in m) pivots the market supply curve clockwise, making it flatter, it becomes even more elastic. In contrast, if the entry of producers is misrepresented by an outward shift of the supply curve (a reduction of c holding the slope constant), the effect is just the opposite: the curve becomes less elastic; indeed, if the shift is sufficient to make c < 0, it becomes inelastic.

Another difference is also apparent. If $c > 0$ , then as the market supply curve becomes flatter in response to the entry of new producers, the equilibrium price falls but must always equal or exceed $c$ . This differs from outward shifts of supply, which reduce $c$ and ultimately leave no lower bound (other than zero) on the equilibrium price.

For $c < 0$ , the market supply curve intersects the horizontal axis at a positive quantity. In this case, consider the arc elasticity between an arbitrarily low price ( $P = ϵ$ ) at which $Q = (ϵ - c) m / k$ , and an arbitrarily high price ( $P = θ$ ) at which $Q = (θ - c) m / k$ , where $0 < ϵ ≪ θ$ . Using the same type of metric as above, the arc elasticity of supply with respect to price is

ε^{S} = [\frac{[\frac{(θ - c) m}{k}] - [\frac{(ϵ - c) m}{k}]}{[\frac{(ϵ - c) m}{k}]}] / [\frac{θ - ϵ}{ϵ}] = \frac{ϵ}{ϵ - c} .

(12A)

Here, a change in m alters the slope but not the overall arc elasticity of the supply curve. In contrast, changes in the intercept do affect the elasticity; in particular, if the market supply curve shifts outward (inward), it becomes more (less) inelastic. Indeed, if exit is portrayed as a parallel inward shift of the supply curve such that an originally negative c increases into positive values, then it is possible for an inelastic supply curve to become elastic if enough producers exit the market—a rather counterintuitive result.

Finally, if $c = 0$ , then $ε^{S} = 1$ . Here again, a change in $m$ does not alter $ε^{S}$ , but a shift that decreases $c$ makes the curve inelastic and a shift that increases $c$ makes the curve elastic.

Thus, as with demand, the manner in which the entry (or exit) of producers is portrayed has an important effect on how the elasticity of the resulting market supply curve is perceived. The factors listed by textbooks as affecting the price elasticity of supply ought to include entry and exit by producers, but typically do not.⁹

Point Elasticities at Equilibrium

The two subsections above used arc elasticity measures to examine the overall price elasticities of demand and supply curves in isolation. Of course, as most textbooks explain, the elasticity changes at various points along a linear demand curve, being elastic at high prices and inelastic at low prices. Here, we consider the difference between a pivot and a shift of each curve as it affects price elasticity at the point of equilibrium $(Q^{*}, P^{*})$ , and for this purpose we use the point elasticity measure, $η = (\partial Q / Q^{*}) / (\partial P / P^{*})$ .

With the linear demand and supply equations above, the equilibrium quantity is

Q^{*} = [(a - c) n m] / [b m + k n]

(13)

and the equilibrium price is

P^{*} = (a k n + b c m) / (b m + k n)

(14)

The absolute value of the price elasticity of demand at the point of equilibrium is

- η^{D} = \frac{n P^{*}}{b Q^{*}} = \frac{P^{*}}{(a - P^{*})} = \frac{a k n + b c m}{(a - c) b m} .

(15)

Equation (15) indicates that a demand curve that passes through any given equilibrium point exhibits greater elasticity when there is a greater value of n. Comparing a pivot of demand (an increase in n) to a shift of demand (an increase in a) that would result in the same new equilibrium, the pivot increases the point elasticity of demand more than the shift.¹⁰

We perform a similar analysis of supply. At equilibrium, the point elasticity of supply with respect to price is

η^{S} = \frac{m P^{*}}{k Q^{*}} = \frac{P^{*}}{(P^{*} - c)} = \frac{a k n + b c m}{(a - c) k n} .

(16)

Like the arc elasticity of supply, the point elasticity depends on c; in particular, $η^{S} \frac{>}{<} 1$ as $c \frac{>}{<} 0$ . And at any given equilibrium $(Q^{*}, P^{*})$ , supply exhibits greater elasticity or less inelasticity when m is larger and the supply curve is flatter than otherwise. Thus, comparing a clockwise pivot of supply (an increase in m) against an outward shift of supply (a decrease in c) that would result in the same new equilibrium, the pivot results in a more elastic curve (if $c > 0$ ) or a less inelastic curve (if $c \leq 0$ ) than the shift. The reverse is also true: a counterclockwise pivot of the supply curve resulting from market exit results in a less elastic or more inelastic curve than an inward shift that reaches the same new equilibrium.

Additionally, as noted above, when $c > 0$ , the equilibrium price is bounded by $c \leq P^{*}$ . Equation (15) indicates that $- η^{D} > 1$ when $P^{*} > (a / 2)$ . Thus, if $c \geq (a / 2)$ , then $- η^{D} > 1$ : equilibrium will only occur on the elastic portion of the market demand curve, even as the entry of new producers pivots the market supply curve clockwise and pushes P* down toward c. This differs notably from portraying the entry of producers as an outward shift of supply, which reduces $c$ and can push equilibrium onto the inelastic portion of the demand curve.

Tax Incidence

Most textbooks rightly explain that the incidence of a tax falls primarily on the side of the market with the least elastic behavior. Because market entry or exit by consumers (producers) can change the own-price elasticity of demand (supply), it can also affect the incidence of taxes. In particular, the consumers’ share of the tax burden is related to the elasticities in the neighborhood of equilibrium according to

B^{D} = η^{S} / (η^{S} + - η^{D})

(17)

and the producers’ share of the tax burden is

B^{S} = - η^{D} / (η^{S} + - η^{D})

(18)

(For derivations, see Swinton and Thomas (2001) or Eisenhauer (2022)). Taking the ratio of equations (17) to (18) and substituting from equations (15) and (16) gives the relative tax burdens as the absolute ratio of the slopes of the demand and supply curves.¹¹ That is,

B^{D} / B^{S} = (b / n) / (k / m) .

(19)

Holding b and k constant, equation (19) shows that changes in tax incidence depend entirely on changes in n and m. In particular, entry on one side of the market redistributes the tax incidence toward the other side. The opposite is also true: exit by buyers or sellers forces the remaining members of that group to bear a greater tax burden. Consider, for example, the Great Resignation—the large-scale withdrawal of workers from the labor force (Faccini, et al. 2022). This should cause the labor supply curve to pivot inward, becoming steeper and redistributing the incidence of payroll taxes away from employers and toward the remaining employees. By contrast, a parallel shift of the labor supply curve would leave the tax incidence unchanged.

New Consumers and Consumer Surplus

With linear supply and demand curves, the consumer surplus (CS) at equilibrium is the area of the triangle bounded by the vertical axis, the demand curve, and the equilibrium price. Thus,

C S = 0.5 (a - P^{*}) Q^{*}

(20)

where

a \geq P^{*}

. Changes in CS occur according to the total derivative

d C S = 0.5 Q^{*} d a + 0.5 (a - P^{*}) d Q^{*} - 0.5 Q^{*} d P^{*} .

(21)

Notice that, because $d a = 0$ when demand pivots, but $d a > 0$ as demand shifts outward, the change in CS from a counterclockwise pivot of demand is always less than the change in CS from an outward shift of demand that results in the same new equilibrium.

Consider three distinct cases. First, if supply is horizontal (i.e., perfectly elastic), so that $d P^{*} = 0$ when demand changes, then an outward shift of demand causes $d C S = 0.5 Q^{*} d a + 0.5 (a - P^{*}) d Q^{*} > 0$ , whereas an outward pivot of demand causes $d C S = 0.5 (a - P^{*}) d Q^{*} > 0$ , which is also positive but smaller than the effect of a shift. Second, if supply is vertical (i.e., perfectly inelastic), $d Q^{*} = 0$ as demand changes; then an outward shift of demand implies $d P^{*} = d a$ , so that $d C S = 0$ . But if demand pivots outward when supply is vertical, CS declines: $d C S = - 0.5 Q^{*} d P^{*} < 0$ . Third, if supply is positively-sloped, an outward shift of demand causes $d a > d P^{*}$ , so CS increases: $d C S = 0.5 Q^{*} (d a - d P^{*}) + 0.5 (a - P^{*}) d Q^{*} > 0$ . By contrast, the change in CS due to an upward pivot of the demand curve in this case depends on the relative price elasticities of supply and demand at equilibrium:

d C S = 0.5 (a - P^{*}) d Q^{*} - 0.5 Q^{*} d P^{*} \frac{>}{<} 0 as

η^{S} = P^{*} d Q^{*} / Q^{*} d P^{*} \frac{>}{<} P^{*} / (a - P^{*}) = - η^{D} .

(22)

As equation (22) shows, the change in CS due to a pivot may be positive, zero, or even negative; in any event, it will be less positive than the change resulting from an outward shift of demand to the same new equilibrium. Moreover, if the pivot makes market demand more elastic than market supply ( $- η^{D} > η^{S}$ ), CS will decline with additional entry by consumers. There is a simple, intuitive explanation for this phenomenon: the additional competition among consumers raises $P^{*}$ toward the maximum price that any buyers will pay, eroding the CS benefits that accrued to earlier consumers. In the limit, as n approaches infinity, the demand curve becomes completely elastic, and consumer surplus is entirely redistributed to producers as producer surplus.

New Producers and Producer Surplus

For a linear supply curve with $c \geq 0$ , the producer surplus (PS) at equilibrium is the area of the triangle bounded by the vertical axis, the supply curve, and the equilibrium price.¹² Thus,

P S = 0.5 (P^{*} - c) Q^{*}

(23)

where

P^{*} \geq c

. Changes in PS occur according to

d P S = 0.5 {(P}^{*} - c) d Q^{*} + 0.5 Q^{*} d P^{*} - 0.5 Q^{*} d c .

(24)

Notice that an outward shift of the market supply curve implies $d c < 0$ , whereas an outward pivot of supply implies $d c = 0$ ; thus, the increase in PS is always smaller for a clockwise pivot than for an outward shift that results in the same new equilibrium.

Consider three scenarios. First, if demand is perfectly elastic, so that $d P^{*} = 0$ , an outward shift of supply causes $d P S = 0.5 {(P}^{*} - c) d Q^{*} - 0.5 Q^{*} d c > 0$ ; but an outward pivot of supply due to the entry of new producers causes $d P S = 0.5 {(P}^{*} - c) d Q^{*} > 0$ . Both increase PS, but the increase is exaggerated if the change is portrayed as a shift of the supply curve. Second, if demand is perfectly inelastic, then $d Q^{*} = 0$ ; any shift of supply would imply $d c = d P^{*}$ and thus, $d P S = 0$ . By contrast, an outward pivot of supply resulting in the same new equilibrium would reduce producer surplus: $d P S = 0.5 Q^{*} d P^{*} < 0$ . Third, if demand is downward-sloping, an outward shift of supply implies $d c < 0$ and $d P^{*} < 0$ , but the change in c is of greater absolute magnitude than the change in $P^{*}$ , so that $d c < d P^{*}$ . This yields an unambiguous increase in producer surplus: $d P S = 0.5 (P^{*} - c) d Q^{*} + 0.5 Q^{*} (d P^{*} - d c) > 0$ . Alternatively, if the entry of new producers is correctly treated as a clockwise pivot of the supply curve, then

d P S = 0.5 (P^{*} - c) d Q^{*} + 0.5 Q^{*} d P^{*} \frac{>}{<} 0 as

- η^{D} = - P^{*} d Q^{*} / Q^{*} d P^{*} \frac{>}{<} P^{*} / (P^{*} - c) = η^{S} .

(25)

As equation (25) reveals, the effect on PS from additional sellers entering the market is positive so long as supply is less elastic than demand; when supply is more elastic than demand, a further counterclockwise pivoting of the market supply curve reduces PS (Miller, et al., 1988). In the limit, as m goes to infinity and the market supply curve approaches perfect elasticity, PS approaches zero; just as competition among producers pushes economic profit to zero, it can also eliminate producer surplus.

Market Adjustment to Long Run Equilibrium

Producer entry and exit provide the mechanism by which firms in competitive markets earn zero economic profits ( $π = 0$ ) in the long run. Each price-taking firm sets the initial equilibrium price, $P^{*}$ , equal to marginal cost (MC) in order to maximize profit; if $P^{*}$ exceeds average total cost (ATC), then $π > 0$ in the short run, and new producers are enticed to enter the market. If the new firms have the same technology and costs as incumbents, as usually assumed, their entry pivots the market supply curve outward, reducing the equilibrium price until it equals the minimum ATC and all firms experience $π = 0$ . But if the market supply curve is portrayed as shifting out, its vertical intercept falls, implying that at least some entrants have different production costs than incumbents. This fundamentally alters the analysis, by making the process dependent upon cost differentials rather than entry per se. Moreover, with a variety of cost structures, it is less likely that all firms will break even at the same equilibrium price.¹³

Table 2 summarizes the results of this section. Because the pivot model is consistent with the basic premise that market-level curves are the horizontal summation of individual-level curves, its implications for elasticities, consumer and producer surplus, tax incidence, and the long run adjustment mechanism are also more consistent with market fundamentals. It therefore seems more reasonable and no less convenient to describe entry and exit as pivoting the market supply and demand curves around their vertical intercepts than to perpetuate a flawed pedagogy based on shifts.

Table 2.

Summary of Differences Between Shifts and Pivots as Depictions of Market Entry by Homogeneous Agents.^a

Entry by	Condition	Parameter	Pivot	Shift
Consumers		$P^{*}$	$P^{*} \leq a$	No upper bound on $P^{*}$
		$- ε^{D}$	Increase	No effect
	At equilibrium	$- η^{D}$	Larger increase	Smaller increase or decrease
		$B^{D}$	Decrease	No change
		$B^{S}$	Increase	No change
	Horizontal supply curve	CS	Smaller increase	Larger increase
	Vertical supply curve	CS	Decrease	No change
	Upward-sloping supply $η^{S} > - η^{D}$	CS	Smaller increase	Larger increase
	Upward-sloping supply $η^{S} = - η^{D}$	CS	No change	Increase
	Upward-sloping supply $η^{S} < - η^{D}$	CS	Decrease	Increase
Producers	$c > 0$	$P^{*}$	$c \leq P^{*}$	$0 \leq P^{*}$
	$c > 0$	$ε^{S}$	Increase	Decrease
	$c \leq 0$	$ε^{S}$	No change	Decrease
	At equilibrium, with $c > 0$	$η^{S}$	Increase	Decrease
	At equilibrium, with $c \leq 0$	$η^{S}$	Smaller decrease	Larger decrease
	At equilibrium, with $c \geq (a / 2)$	$- η^{D}$	$- η^{D} > 1$	$- η^{D} \frac{>}{<} 1$
		$B^{D}$	Increase	No change
		$B^{S}$	Decrease	No change
	Horizontal demand curve	PS	Smaller increase	Larger increase
	Vertical demand curve	PS	Decrease	No change
	Downward-sloping demand $η^{S} < - η^{D}$	PS	Smaller increase	Larger increase
	Downward-sloping demand $η^{S} = - η^{D}$	PS	No change	Increase
	Downward-sloping demand $η^{S} > - η^{D}$	PS	Decrease	Increase
	Long run	$π$	$π = 0$ certain	$π = 0$ uncertain

^a $P^{*}$ : equilibrium price; $ε^{D}$ , $η^{D}$ , $ε^{S}$ , $η^{S}$ : arc and point price elasticities of demand and supply, respectively; $B^{D}$ , $B^{S}$ : consumers’ and producers’ shares of taxes, respectively; CS: consumer surplus; PS: producer surplus; $π$ : economic profit.

Nonlinear Functions

Linear supply and demand functions are certainly predominant in undergraduate textbooks and other pedagogical works. Indeed, the emphasis that most texts place on discussing the changing elasticity along a linear demand curve suggests the centrality of the linear form for educational purposes. And linear functions remain fairly common in empirical research, because linear regression techniques are often employed in estimation.¹⁴ Even so, as a final caveat, we briefly consider alternative functional forms. Certainly, an isoelastic function behaves quite differently—by definition, it will not become more or less elastic as agents enter or exit the market. However, the isoelastic function is not only more complex to teach, it is also a rather special case: it is not clear that constant own-price elasticity generally applies to most consumers or most producers in most markets. Additionally, there are other nonlinear functional forms whose own-price elasticities will respond to market entry and exit. As one example, suppose the individual-level demand function is $P = α e^{- γ q^{D}}$ where $γ > 0$ and $α > 0$ . Then for n such consumers, $Q^{D} = (n / γ) \ln (α / P)$ or $\ln P = \ln α - (γ / n) Q^{D}$ . The price elasticity of demand in this case is $η^{D} = - n / γ Q < 0$ , so this nonlinear curve also becomes increasingly elastic for any given Q as n rises. Likewise, if the individual supply function is $P = {β e}^{λ q^{S}}$ where $λ > 0$ and $0 < β < α$ , then $q^{S} = (1 / λ) \ln (P / β)$ . For m such producers, $Q^{S} = (m / λ) \ln (P / β)$ or $\ln P = \ln β + (λ / m) Q^{S}$ , so the price elasticity of supply is $η^{S} = m / (λ Q)$ ; as m increases, the price elasticity of supply increases for any Q. In short, the changes in slope and elasticity that result from market entry and exit are not unique to linear functions.

Conclusion

With the exception of economics majors, most college students will only study economics at the principles level (Bosshardt & Walstad, 2017; Bosshardt and Watts, 2008). And although descriptions of the market are more detailed and comprehensive at the intermediate level, they are usually not fundamentally different from those taught earlier in introductory courses. Thus, the manner in which we portray economic models in these courses influences—one might say determines—the way that our students will think of the economy for the rest of their lives. It is imperative, therefore, that we provide simple but consistent and reasonably accurate models that describe reality as nearly as possible. The traditional practice of depicting entry or exit as shifting a linear market supply or demand curve is a convenient fiction, but it is unrealistic and inconsistent with the lesson regarding the construction of the market-level curve as a horizontal summation of individual agents’ curves. As a consequence, that practice also leads to erroneous conclusions regarding the effects of entry and exit on own-price elasticity, tax incidence, consumer surplus, producer surplus, and importantly, the competitive market adjustment to long run equilibrium. This paper therefore proposes that entry and exit be taught as pivoting the market supply and demand curves rather than shifting them.

Certainly, any pedagogical innovation must be presented in a manner that respects student abilities and time constraints in the course. Principles courses are already crowded with subject matter and there is an obvious opportunity cost to introducing new material. At that level, rather than treating market entry as a new topic, it can simply be noted as the logical conclusion to the customary demonstration of horizontal summation: entry flattens and exit steepens the market curve. The intuition can be stated as easily as in the classroom example above. Consequently, entry and exit can be included as factors affecting the elasticity of a market-level curve, and an even clearer distinction emerges naturally between these factors and those that change a market curve by altering individual curves. Any further analysis is more likely to be useful in an intermediate course, where students generally have greater interest in economic details and more familiarity with mathematics. There, the implications regarding consumer and producer surplus, tax incidence, and so forth can be explored as thoroughly as the instructor wishes.¹⁵

Supplemental Material

Supplemental Material - Shift or Pivot? Market Entry and Exit With Linear Supply and Demand Curves

Supplemental Material for Shift or Pivot? Market Entry and Exit With Linear Supply and Demand Curves by Joseph G. Eisenhauer in The American Economist.

Footnotes

Acknowledgments

I thank participants at the 87^th annual Midwest Economics Association conference, as well as this journal’s Associate Editor Laura Ahlstrom and an anonymous referee for helpful comments. Any errors are my own.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Joseph G. Eisenhauer

Supplemental Material

Supplemental material for this article is available online.

Notes

Author Biography

Joseph G. Eisenhauer is the Dean of the College of Business Administration at the University of Detroit Mercy. A Past President and Distinguished Fellow of the New York State Economics Association, he was educated at the State University of New York at Buffalo and the Wharton School of Business at the University of Pennsylvania. His research focuses on the economics of risk, ethical decision-making, statistics, and economic measurement.

References

Allen

W. B.

Weigelt

Doherty

Mansfield

(2009). Managerial economics: Theory, applications, and cases (7th ed.). W. W. Norton.

Alperovich

Weksler

(1996). A class of utility functions yielding linear demand functions, The American Economist, 40(1), 20–23. https://doi.org/10.1177/056943459604000103

Asarta

C. J.

Chambers

R. G.

Harter

(2021). Teaching methods in undergraduate introductory economics courses: Results from a sixth national quinquennial survey. The American Economist, 66(1), 18–28. https://doi.org/10.1177/0569434520974658

Bosshardt

Walstad

W. B.

(2017). Economics and business coursework by undergraduate students: Findings from baccalaureate and beyond transcripts. The Journal of Economic Education, 48(1), 51–60. https://doi.org/10.1080/00220485.2016.1252299

Bosshardt

Watts

(2008). Undergraduate students’ coursework in economics. The Journal of Economic Education, 39(2), 198–205. https://doi.org/10.3200/JECE.39.2.198-205

Brue

S. L.

McConnell

C. R.

(2007). Essentials of economics. McGraw-Hill Irwin.

Cheung

S. L.

(2005). A classroom entry and exit game of supply with price-taking firms. The Journal of Economic Education, 36(4), 358–367. https://doi.org/10.3200/JECE.36.4.358-368

Edwards

G. W.

Voon

J. P.

(1997). The calculation of research benefits with linear and nonlinear specifications of demand and supply: Reply. American Journal of Agricultural Economics, 79(4), 1368–1371. https://www.jstor.org/stable/1244293

Eisenhauer

J. G.

(2022). Price elasticity, tax incidence, and sales volume: A simple model. Journal for Economic Educators, 22(2), 29–41. https://libjournals.mtsu.edu/index.php/jfee/article/view/2290

10.

Faccini

Melosi

Miles

2022. The effects of the ‘Great Resignation’ on labor market slack and inflation. Chicago Fed Letter, 465. https://www.chicagofed.org/publications/chicago-fed-letter/2022/465

11.

Goolsbee

Levitt

Syverson

(2020). Microeconomics (3rd ed.). Worth.

12.

Graves

P. E.

Sexton

R. L.

Lee

D. R.

(1996). Slope versus elasticity and the burden of taxation. The Journal of Economic Education, 27(3), 229–232. https://doi.org/10.1080/00220485.1996.10844912

13.

Harter

Asarta

C. J.

(2022). Teaching methods in undergraduate intermediate theory, statistics and econometrics, and other upper-division economics courses: Results from a sixth national quinquennial survey. The American Economist, 67(1), 132–146. https://doi.org/10.1177/05694345211037904

14.

Horowitz

J. B.

Karls

M. A.

Sesmero

van Cott

T. N.

(2013). Innovation, parallel shifts of supply, and welfare. Agricultural Economics Review, 14(1), 37–44. https://doi.org/10.22004/ag.econ.253536

15.

Hutchinson

(2017). Principles of microeconoimcs. University of Victoria.

16.

Karagiannis

Furtan

W. H.

(2002). The effects of supply shifts on producers’ surplus: The case of inelastic linear supply curves. Agricultural Economics Review, 3(1), 5–11. https://doi.org/10.22004/ag.econ.26430

17.

LaFrance

J. T.

(1985). Linear demand functions in theory and practice. Journal of Economic Theory, 37(1), 147–166. https://doi.org/10.1016/0022-0531(85)90034-1

18.

Lerner

A. P.

(1933). The diagrammatical representation of elasticity of demand. The Review of Economic Studies, 1(1), 39–44. https://doi.org/10.2307/2967436

19.

Lindner

R. K.

Jarrett

F. G.

(1978). Supply shifts and the size of research benefits. American Journal of Agricultural Economics, 60(1), 48–58. https://doi.org/10.2307/1240160

20.

Mankiw

N. G.

(2007). Essentials of economics (4th ed.). Thomson South-Western.

21.

Mathis

S. A.

Koscianski

(2002). Microeconomic theory: An integrated approach. Prentice Hall.

22.

Miller

G. Y.

Rosenblatt

J. M.

Hushak

L. J.

(1988). The effects of supply shifts on producers’ surplus, American Journal of Agricultural Economics, 70(4), 886–891. https://www.jstor.org/stable/pdf/1241930.pdf

23.

Nieswiadomy

(1986). A note on comparing the elasticities of demand curves. The Journal of Economic Education, 17(2), 125–128. https://doi.org/10.1080/00220485.1986.10845155

24.

Salvatore

(2007). Managerial economics in a global economy (6th ed.). Oxford University Press.

25.

Seldon

J. R.

(1986). A note on the teaching of arc elasticity. The Journal of Economic Education, 17(2), 120–124. https://doi.org/10.1080/00220485.1986.10845154

26.

Soper

(2004). Mathematics for economics and business (2nd ed.). Blackwell.

27.

Swinton

J. R.

Thomas

C. R.

(2001). Using empirical point elasticities to teach tax incidence. The Journal of Economic Education, 32(4), 356–368. https://doi.org/10.1080/00220480109596114

28.

Voon

J. P.

Edwards

G. W.

(1991). The calculation of research benefits with linear and nonlinear specifications of demand and supply functions. American Journal of Agricultural Economics, 73(2), 415–420. https://doi.org/10.2307/1242725

29.

Won

Kober

Densing

(2022). Nonlinear inverse demand curves in electricity market modeling. Energy Economics, 107(March), 105809. https://doi.org/10.1016/j.eneco.2022.105809

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