Economic Modelling 29 (2012) 586–600
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Monopoly, economic efficiency and unemployment Bo Zhao ⁎ School of International Trade & Economics, University of International Business & Economics, No. 10, Huixin Dongjie, Chaoyang District, Beijing 100029, China
a r t i c l e
i n f o
Article history: Accepted 1 January 2012 JEL classification: L12 J64 Keywords: Monopoly Efficiency loss Unemployment Efficiency wage
a b s t r a c t The objective of this paper is to analyze the efficiency consequences of monopoly from the perspective of an efficiency-wage model of unemployment based on Shapiro and Stiglitz (1984). An important feature of our model is that a firm can raise the probability that a shirking worker is detected by increasing its effort or investment in the monitoring of workers. Using this model we study how a monopolist's decision with regard to employment, output and monitoring is affected by exogenous variables such as job separation rate, technological advances, market size, and unemployment benefits. Furthermore, by comparing with the competitive equilibrium we find that monopoly is associated with higher unemployment rate, smaller output, and less monitoring. Surprisingly, however, monopoly does not necessarily lead to lower welfare level. © 2012 Elsevier B.V. All rights reserved.
1. Introduction It is well-known that monopoly causes inefficient allocation of resources. As illustrated by the standard textbook model of monopoly, deadweight losses arise because a monopolist sets its price above the marginal cost of production. In addition, productive inefficiencies and rent seeking activities have also been cited as reasons for efficiency losses of monopoly. However, there is one area of potential efficiency losses of monopoly that so far has rarely been explored in microeconomic theory, that is, the effects of monopoly on unemployment. Since unemployment represents unutilized labor resource, it can be argued that an increase in unemployment rate, ceteris paribus, causes additional efficiency losses. Given that output is an increasing function of labor, reduction in output by a monopoly will normally cause a reduction in labor employed in the monopolized industry. To the extent that the surplus labor released by the monopolized industry is not entirely absorbed by other industries in the economy, more unemployment will result. Therefore, it seems plausible that monopoly may cause higher rate of unemployment. Indeed, it has been argued by some economists (see Layard et al., 2005; Geroski et al., 1995 for examples) that competition in the product market reduces unemployment. However, while the issue of unemployment figures prominently in other fields of economics such as macro and labor economics, microeconomic analysis of monopoly is still confined to an equilibrium framework that, by its implicit assumption of a perfectly flexible labor market, is incapable of handling unemployment.
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The main objective of this paper is to analyze the efficiency and employment consequences of monopoly from the perspective of an efficiency wage model of unemployment based on Shapiro and Stiglitz (1984). In this model a monopolist has to offer a wage high enough to induce workers to expend efforts on the job. An important feature of our model is that a monopolist can raise the probability that a shirking worker is detected by increasing its effort or investment in the monitoring of workers. 1 Using this model we study how the monopolist's decision with regard to employment, output and monitoring are affected by exogenous variables such as job separation rate, technological advances, market size, and unemployment benefits. Furthermore, we examine the efficiency consequences of monopoly by comparing the monopoly equilibrium with the competitive equilibrium. In this regard, the most important finding from our analysis is that while monopoly is associated with higher unemployment rate, smaller output, and less monitoring, it does not necessarily lead to lower welfare level. This result is surprising in light of the common belief about the welfare losses of monopoly. The rest of this paper is organized as follows. Section 2 presents the model and characterizes the monopoly equilibrium. In Section 3 we analyze how the monopoly equilibrium is affected by various parameters of the model, and in Section 4 we compare the monopoly equilibrium with the benchmark of a competitive equilibrium. In Section 5 we present the results from the simulations of our model with specific functional forms. Section 6 extends the one-firm model to an M-industry model. And conclusions of this paper are in Section 7.
1 Shapiro and Stiglitz (1984) discuss informally the case of endogenous monitoring. They indicate that in general it is not possible to ascertain whether the competitive equilibrium entails too much or too little employment.
B. Zhao / Economic Modelling 29 (2012) 586–600
2. The model Consider an industry served by a monopolist. The demand for the good produced by the monopolist is represented by p = αP(Q), where p is the price and Q is the output with P′(Q) b 0, and α > 0 measures the market size. Additionally, we assume that 2P′(Q) + P′′(Q) ⋅ Q b 0 to ensure that the monopolist's marginal revenue is decreasing in output. The monopolist produces the good according to the production function Y = sF(eL) with the standard assumptions that F′(⋅) > 0 and F′′(⋅) b 0, where Y is the output, L is the number of workers employed, e is the effort level expended by the representative worker, and s is an exogenous technology parameter. Hence, eL represents the effective amount of labor employed by the firm. To incorporate unemployment into the model, we use the efficiency wage model of Shapiro and Stiglitz (1984). Specifically, we assume that workers can shirk (i.e. exerting no effort) on the job. The firm, on the other hand, cannot perfectly observe workers' effort. Assume there are N identical workers who, because of their specialized skills, can find work only in this industry. Each worker has a utility function U(w, e) = w − e, where w is the received wage and e is the worker's effort level on the job. For simplicity we assume that the level of effort, e, takes on only two values, zero and some positive constant. A value of 0 means that no effort is supplied (that is, the worker chooses to shirk), and e > 0 means the worker does not shirk. Thus, if the worker is employed and does not shirk, his utility is w − e; but if he shirks on the job, his utility is w. If a worker is caught shirking on the job, he is fired immediately. In addition, a worker may be separated from his job for reasons other than shirking. The natural separation rate, denoted by b, is defined as the ratio of job separations to the number of workers employed, per > 0, the ununit of time. An unemployed worker receives a utility w employment benefit. The flow out of the unemployment pool is determined by new hires. The job acquisition rate (or the accession rate), denoted by a, is defined as the ratio of new hires to the number of workers unemployed per unit of time. Each worker in the unemployment pool has the same opportunity to be hired, independent of the reason that has caused his unemployment (shirking or natural separation). Firms can only monitor workers imperfectly. In other words, if a worker shirks, there is some probability, denoted by q, that the worker will be caught and fired. In the standard Shapiro and Stiglitz efficiency wage model, the detection probability q is taken as exogenous. In this model, we endogenize q by assuming that q is a function of the effort and/or investment by the firm in monitoring the workers. Let m denote the level of monitoring. We assume that q(m) with q′(m) > 0 and q′′(m) b 0. The monitoring of workers is costly to the firm, and the firm has to pay a wage high enough to discourage workers from shirking. Using the same procedure presented in Section 1, we obtain the equilibrium efficiency wage rate as
þeþ w ¼w
e ða þ b þ r Þ qðmÞ
where π is the profit from the firm and H(m) is the cost of monitoring workers with H′(⋅) > 0 and H′′(⋅) > 0. 3 Note that ∂ w*/∂ m =− eq ′(m)(a + b + r)/[q(m)] 2 b 0, in words, the no-shirking wage w* is decreasing in m. The firm can lower the wage paid to the workers without inducing shirking by investing more in monitoring. Since monitoring is costly to the firm, it faces a trade-off between monitoring costs and the wage paid to workers. In the steady state of the labor market, the flow into the unemployment pool per unit time is equal to the flow out of the unemployment pool per unit time. That is bL ¼ aðN−LÞor
a ¼ bL=ðN−LÞ:
ð3Þ
Taking into consideration the efficiency wage, the monopolist's optimization problem is written as 4: þeþ max π ¼ αP ðsF ðeLÞÞ⋅sF ðeLÞ− w fm; Lg
e ða þ b þ r Þ L−H ðmÞ: qðmÞ
ð4Þ
The first-order conditions are q′ðmÞ eða þ b þ r ÞL−H′ðmÞ ¼ 0 ½qðmÞ2
ð5Þ
2
αs e⋅P ′ðsF ðeLÞÞF ′ðeLÞF ðeLÞ þ αse⋅ P ðsF ðeLÞÞF ′ðeLÞ þeþ − w
e ða þ b þ r Þ ¼ 0: qðmÞ
ð6Þ
Eq. (5) implies that in equilibrium, the owner of the firm sets the marginal benefit of monitoring equal to the marginal cost of monitoring. In Eq. (6) the term [αs2e ⋅ P′(sF(eL))F′(eL)F(eL) + αse ⋅ P(sF(eL))F′(eL)] is the monopolist's marginal revenue product (MRP).5 Then, Eq. (6) indicates that the monopolist will choose the employment level such that the MRP equals the wage rate. It is easy to derive that ( ) ∂2 π 2½q′ðmm Þ2 q′′ðmm Þ eLm ⋅ða þ b þ r Þ−H′′ mm b 0 ¼ − þ 2 3 2 ½qðmm Þ ½qðmm Þ ∂m ðmm ;Lm Þ ð7Þ b 0 by assumption 2 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ ∂ π 2 2 2 ¼ αs e ½ F ′ ð eL Þ 2P′ sF eL þ sF eLm P ′′ sF eLm m m 2 ∂L ðmm ;Lm Þ i h 2 2 2 þ αs e F ′′ eLm P′ sF eLm F eLm þ αse ⋅P sF eLm b 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > 0 by FOC
ð8Þ ∂2 π ∂2 π q′ðmm Þ ¼ eða þ b þ r Þ > 0: ðLm ;mm Þ ¼ ∂m∂L ðmm ;Lm Þ ∂L∂m ½qðmm Þ2
ð9Þ
ð1Þ
where r is the intertemporal discount rate, and interest rate is often used as a proxy of it. Additionally, please note that we do not distinguish the firm's wage and the economy-wide wage since we assume that there is only one (representative) firm in this model. And the firm's objective function is 2: π ¼ p ⋅Y−w ⋅L−H ðmÞ
587
ð2Þ
2 In this paper, we assume that the manager is also the owner of the firm. That is, there is no principal–agent problem between the manager and the owner.
3 It should be noted that, in this paper, the monitoring cost is assumed to be independent of the number of workers. Alternatively, for future research, it may take the form of h(m)L. 4 The literature in industrial economics as well as in labor economics would favor a model design that sequential decisions are made so that the firm initially commits itself to wages (and monitoring) and that the product market decisions are made contingent on the compensation schemes, and then firms adjust their wage offers according to the situations of product and labor markets, while workers adjust their expectations of labor compensation. Since the efficiency wage is offered in our model, we can reasonable to ignore the wage adjusting process and therefore it seems that wages (as well as monitoring investments) are determined simultaneously with employment (and production). 5 For a monopolist, MRP = MR ⋅ MP. Since the marginal revenue is decreasing in output (∂ MR/∂ Q b 0) and ∂ MP/∂ L = se2F′′(eL) b 0, we have ∂ MRP/∂ L = ∂ MR/∂ Q ⋅ MP2 + ∂ MP/∂ L ⋅ MR b 0, that is, the MRP is decreasing in the amount of labor employed.
588
B. Zhao / Economic Modelling 29 (2012) 586–600
To ensure that the second-order conditions for a maximum are satisfied, we assume
Lemma 2. At the equilibrium point (mm * , Lm * ), the slope of L = Am(m) is greater than the slope of L = Bm(m), that is, A′m(mm * ) > B′m(mm * ).
" #2 " # ∂2 π ∂2 π ∂2 π ⋅ > : ∂m∂L ∂L2 ∂m2 ðm ; L Þ ðm ; L Þ
Proof. Denote ΦL*, Φm * , ΩL* and Ωm * as the respective values at the equilibrium point (m*m, L*m). Then, A′m(mm * ) =−Φ*m /ΦL* and B′m(m*m) = −Ω*m/Ω L* .
m
m
m
ð10Þ
m
Using the condition for the labor market equilibrium (3), we can state the two equilibrium conditions as
eL⋅q′ðmÞ bN þ r −H′ðmÞ ¼ 0 ½qðmÞ2 N−L
ð11Þ
2
αs e⋅P ′ðsF ðeLÞÞF ′ðeLÞF ðeLÞ þ αse⋅P ðsF ðeLÞÞF ′ðeLÞ þeþ − w
e bN þr qðmÞ N−L
¼ 0:
ð12Þ
It is useful for later analysis to establish the properties of these two conditions in the space of (m, L). Let L = Am(m) denote the functional relationship implied by Eq. (11); and L = Bm(m) the functional relationship implied by Eq. (12). Thus, the intersection of these two curves is the equilibrium, denoted by (mm * , Lm * ). Lemma 1. Both curves, L = Am(m) and L = Bm(m), are strictly upward sloping in the (m, L) space. Proof. To simplify presentation, let Φ(m, L) = 0and Ω(m, L) = 0 denote Eqs. (11) and (12), respectively. And let Φj and Ωj (j ¼ m; L; b; s; α; w) denote the partial derivative of Φ and Ω with respect to j. Then, the slopes of the two curves at the equilibria can be presented as A′m(m*) = − Φm/ΦL and B′m(m*) = − Ωm/ΩL. It is easy to obtain
eq′ðmÞ bN ebNL⋅q′ðmÞ þ r þ ΦL ¼ ½qðmÞ2 N−L ½qðmÞ2 ðN−LÞ2 ( Φm ¼
−
)
2½q′ðmÞ2 q′′ðmÞ bN eL⋅ þ r −H′′ðmÞ þ N−L ½qðmÞ3 ½qðmÞ2
ΩL ¼ αs2 e2 ½F ′ðeLÞ2 ½2P ′ðsF ðeLÞÞ þ sF ðeLÞP ′′ðsF ðeLÞÞ þ αse2 F ′′ðeLÞ½P ′ðsF ðeLÞÞF ðeLÞ þ P ðsF ðeLÞÞ − ebN½qðmÞ−1 ðN−LÞ−2 Ωm ¼
ð13Þ
ð14Þ
ð15Þ
2 ΦL > ∂ π=∂m∂Lðmm ;Lm Þ > 0
ð17Þ
2 2 Φm ¼ ∂ π=∂m ðmm ;Lm Þ b0
ð18Þ
2 2 ΩL b ∂ π=∂L ðmm ;Lm Þ b 0
ð19Þ
2 Ωm ¼ ∂ π=∂L∂mðmm ;Lm Þ > 0:
ð20Þ
Furthermore, ΦL −∂2 π=∂m∂Lðmm ;Lm Þ ⋅Ωm
ebNLm ⋅q′ðmm Þ eq′ðmm Þ bN þr ⋅ 2 2 2 N−L ½qðmm Þ ðN−Lm Þ ½qðmm Þ m
¼
e2 bNLm ⋅½q′ðmm Þ2 bN ⋅ þ r ½qðmm Þ4 ðN−Lm Þ2 N−Lm
¼ and
ΩL −∂2 π=∂L2 ðmm ;Lm Þ ⋅Φm
)
( ebN 2eLm ½q′ðmm Þ2 bN > þ r ⋅ N−nLm ½qðmm Þ3 qðmm ÞðN−Lm Þ2
e2 bNLm ⋅½q′ðmm Þ2 bN > ⋅ þr : ½qðmm Þ4 ðN−Lm Þ2 N−Lm That is, h
eq′ðmÞ bN þ r : ½qðmÞ2 N−L
ð16Þ
By the assumptions that q′(⋅) > 0, q′′(⋅) b 0, H′(⋅) > 0 and H′′(⋅) > 0, we know that ΦL > 0, Φm b 0 and Ωm > 0. Now, consider the sign of ΩL. Note that in Eq. (15), the term [2P′(sF(eL))+ sF(eL)P′′(sF(eL))] is negative since the monopolist's marginal revenue is assumed to be decreasing in output; and the first order condition (11) implies that the term [P′(sF(eL))F(eL) + αse2 ⋅ P(sF(eL))] is positive. These imply that ΩL b 0. Then, A′m ðmÞ ¼ −Φm =ΦL > 0 and
Substitute the labor market steady state condition a = bN/(N − L) into the second order conditions (7)–(9), and compare them with Φ*L , Φm * , ΩL* and Ωm * to obtain
B′m ðmÞ ¼ −Ωm =ΩL > 0:
That is, both L = Am(m) and L = Bm(m) are strictly upward slopping. Intuitively, L = Am(m) is upward slopping since higher level of monitoring reduces the wage rate and thus motives the monopolist to hire more workers; on the other hand, L = Bm(m) is upward slopping because higher level of employment lowers the MRP of labor and this drives the monopolist to invest more on monitoring to reduce the wage rate in order to equate it to the decreased MRP. QED
h i i 2 2 2 ΩL −∂ π=∂L ðmm ;Lm Þ ⋅Φm > ΩL −∂ π=∂m∂Lðmm ;Lm Þ ⋅Ωm :
ð21Þ
Using Eqs. (10) and (17)–(21), we have h
∂2 π=∂L2 ⋅∂2 π=∂m2
i ðmm ;Lm Þ
h i2 > ∂2 π=∂m∂L
ΩL ⋅Φm > ΦL ⋅Ωm Φm =ΦL bΩm =ΩL A′m ðmm Þ > B′m ðmm Þ:
⇒ ⇒
⇒
ðmm ;Lm Þ
⇒−Φm =ΦL > −Ωm =ΦL
* , Lm * ), the slope of L = Am(m) is That is, at the intersection (mm greater than the slope of L = Bm(m). QED Lemmas 1 and 2 imply that the monopoly equilibrium can be illustrated as in Fig. 1. 3. Properties of the monopoly equilibrium In this section we conduct comparative statics to analyze how the monopoly equilibrium is affected by various parameters of the model. The parameters we will examine are separation rate (b), technology Þ. shock (s), market size (α), and unemployment benefits ðw
B. Zhao / Economic Modelling 29 (2012) 586–600
L
589
L
( )
( )
( ) L*m
L*m
~
L*m
m*m
O
m
O
Fig. 1. The monopoly equilibrium.
m Fig. 2. Effect of separation rate on monopoly equilibrium.
3.1. Separation rate
3.2. Technology shock
First, conducting comparative statics with respect to b, we obtain,
Using the same procedure of comparative statics as above, we can show that
dm ΦL ⋅Ωb −ΩL ⋅Φb ¼ db Δ
ð22Þ
dm Φ ⋅Ω −ΩL ⋅Φs ¼ L s ds Δ
dL ΩL ⋅Φb −Φm ⋅Ωb ¼ db Δ
ð23Þ
where,
and
dL Ω ⋅Φ −Φm ⋅Ωs ¼ m s ds Δ
ð25Þ
Φs ¼ 0
where
Φb ¼
n 2 o : Ωs ¼ αe⋅F ′ eL ⋅ 3P′ sF eL ⋅sF eL þ P′′ sF eL ⋅ sF eL þ P sF eL
eL Nq′ðm Þ eN ; Ωb ¼ − qðm ÞðN−L Þ ½qðm Þ2 ðN−L Þ
ð24Þ
* − ΦL* ⋅ Ωm * is positive according to Lemma 2. and Δ ≡ ΩL* ⋅ Φm Then, d m*/d b > 0 if and only if
eq′ðm Þ bN eN ∂MRP eL Nq′ðm Þ >0 þ r ⋅ ⋅ − qðm ÞðN−L Þ ∂L ½qðm Þ2 N−L ½qðm Þ2 ðN−L Þ
e bN ∂MRP f− þ r −L ⋅ >0 qðm Þ N−L ∂L
∂MRP e bN > f þr : ∂L qðm ÞL N−L f−
On the other hand, (
−
)
½q′ðm Þ2 q′′ðm Þ bN þ þ r −H′′ m b0: eL⋅ N−L ½qðm Þ3 ½qðm Þ2
Therefore, Proposition 1. An increase in separation rate reduces the equilibrium employment and thus raises the equilibrium unemployment rate, but has an ambiguous effect on the level of monitoring. To be more specific, it raises the level of monitoring if and only if
∂MRP e bN ∂L > qðm ÞL N−L þ r : Proposition 1 is illustrated in Fig. 2. An increase in b causes the curve L = Am(m) to shift out and the curve L = Bm(m) to shift down. The new equilibrium E~ m lies below the original equilibrium Em. However, E~ m may lie on the right or the left of Em, depending on the condition derived above.
2
2
Ωs > 0f3P ′ðsF ðeL ÞÞ⋅sF ðeL Þ þ P ′′ðsF ðeL ÞÞ⋅½sF ðeL Þ þ P ðsF ðeL ÞÞ > 0
ΦL ⋅Ωb −ΩL ⋅Φb > 0
Ωm ⋅Φb −Φm ⋅Ωb ¼
Recall ΦL* > 0, Φm * b 0and Δ > 0, then d m*/d s > 0 and d L*/d s > 0 if and only if
fP þ sF ⋅ð3P ′ þ sF P ′′Þ > 0⇔P þ 3sF ⋅P ′ þ ½sF P ′′ > 0 P sF ⋅P ′′ f− − > 3⇔ε p þ σ p > 3 sF ⋅P ′ P ′
where, εp* is the elasticity of demand with respect to price at the equilibrium point, and σ*p is the elasticity of the slope of the demand function at the equilibrium point. The sign of the latter is determined by the concavity or convexity of the demand function. Proposition 2. A positive technology shock increases the equilibrium employment (thus reducing the equilibrium unemployment rate) and raises the monopolist's monitoring level if and only if εp* + σp* > 3. Graphically, a change in s does not affect the curve L = Am(m) because s does not appear in Eq. (11). On the other hand, as the derivation below shows, an increase in s causes L = Bm(m) to shift up if and only if εp* + σp* > 3. In Eq. (12),
e bN þeþ þr ↑ m↓fqðmÞ↓f w qðmÞ N−L h i 2 f αs e⋅P ′ðsF ðeLÞÞF ′ðeLÞF ðeLÞ þ αse⋅P ðsF ðeLÞÞF ′ðeLÞ ↑ h i f∂ αs2 e⋅P ′ðsF ðeLÞÞF ′ðeLÞF ðeLÞ þ αse⋅P ðsF ðeLÞÞF ′ðeLÞ =∂s > 0 fP ðsF ðeLÞÞ þ 3s⋅F ðeLÞ⋅P ′ðsF ðeLÞÞ þ ½s⋅F ðeLÞ2 ⋅P ′′ðsF ðeLÞÞ > 0 fεp þ σ p > 3: As shown in Fig. 3, if ε*p + σp* > 3, an increases in s causes the curve L = Bm(m) to shift up, which implies that the levels of both employment and monitoring are higher in the new equilibrium. On the other hand, if εp* + σp* b 3, the curve L = Bm(m) shifts down as s increases, which implies that the levels of both employment and monitoring are lower in the new equilibrium.
590
B. Zhao / Economic Modelling 29 (2012) 586–600
L
L ~
L*m ~
L*m L*m L*m Lˆ *m
m*m
mˆ *m
O
~* m m
Note from Proposition 2 that in the special case of linear demand function (i.e. P′ ′(⋅) = 0 thus σp = 0), the unemployment rate is decreasing in technology shocks if and only if εp* > 3. Intuitively, a positive technology shock increases the amount of output produced by a given number of productive workers. That increase in output will cause a relatively small fall in the price if the demand is highly elastic. In such a situation, the firm is willing to hire additional workers to take advantage of the higher marginal product of labor brought about by the positive technology shock. With more workers, the marginal gain from an additional unit of monitoring is higher, and hence the monopolist invests more on monitoring. The opposite is true if the demand elasticity is small. 6 3.3. Market size Straightforward comparative statics exercise shows that
and
dL Ωm ⋅Φα −Φm ⋅Ωα ¼ dα Δ
ð26Þ
where,
2
Ωα ¼ s e⋅ P ′ðsF ðeLÞÞF ′ðeLÞF ðeLÞ þ se⋅ P ðsF ðeLÞÞF ′ðeLÞ: Note that ΦL* > 0, Φm* b 0and Δ > 0, then d m*/d α > 0 and d L*/ d α > 0 if and only if
3.4. Unemployment benefits It is straightforward to derive: dm ΦL ⋅Ωw −ΩL ⋅Φw ¼ dw Δ
and
dL Ωm ⋅Φw −Φm ⋅Ωw ¼ dw Δ
ð27Þ
where, Φw ¼ 0 and Ωw ¼ −1. Then it is clear that dm =dwb0 and dL =dwb0 since ΦL* > 0, Φm* b 0 and Δ > 0. In other words,
Proposition 4. An increase in unemployment benefits results in a decrease in the level of employment (thus an increase in the unemployment rate) and a reduction in the level of monitoring. only As shown in Fig. 5, an increase in unemployment benefits w will affects the curve L = Bm(m). For any given Lm, an increase in w force m to go up to satisfy Eq. (12). That is, the curve L = Bm(m)will shift down if the unemployment benefits increase. ↑w
Φα ¼ 0
m
Fig. 4. Effect of market size on monopoly equilibrium.
Fig. 3. Effect of technology shock on monopoly equilibrium.
dm ΦL ⋅Ωα −ΩL ⋅Φα ¼ dα Δ
~* m m
m*m
O
m
m ÞÞF ′ðeLm ÞF ðeLm Þ þ αse⋅P ðsF ðeLm ÞÞF ′ðeLm Þ ða þ b þ rÞ f qðmm Þ↑ f mm ↑:
Z→ αs e⋅P ′ðsF ðeL Given Lm unchanged
þeþ ⇒ w
e qðmm Þ
2
The result that more generous unemployment benefits increases unemployment is not surprising. This finding is also supported by ample empirical evidence. 7 What is new in this analysis is that more generous unemployment benefits also cause the monopolist to invest less in monitoring, which further reduces the level of employment.
Ωα > 0fP ′ðsF ðeLÞÞ⋅ sF ðeLÞ þ P ðsF ðeLÞÞfεp > 1: This condition is satisfied because a monopolist always operates at a point where the elasticity of demand is greater than one. Proposition 3. A larger market size raises the levels of both employment and monitoring. As shown in Fig. 4, an increase in αdoes not affect the curve L = Am(m) but causes the curve L = Bm(m) to shift up. Consequently, an increase of market size leads to higher levels of employment and monitoring. 6 Indeed, some of the recent empirical researches have uncovered evidence that positive technology shocks can raise the level of unemployment. Basu et al. (2004) and Galí (1999) find that positive technology shocks typically led to a short-run decline in employment during the post-World War II period in the United States.
4. Efficiency consequences of monopoly The objective of this section is to analyze the efficiency consequences of monopoly in the presence of unemployment caused by efficiency wage considerations. We will do so by comparing the monopoly equilibrium with that of perfect competition. 4.1. Perfect competition: the benchmark case In a perfectly competitive industry, firms are price-takers. Therefore, the representative firm's optimization problem in our
7 The impact of a high benefit on unemployment is well documented by Layard et al. (2005). It is also confirmed by many other empirical studies, such as Nickell et al. (2005).
B. Zhao / Economic Modelling 29 (2012) 586–600
þeþ αse⋅P ðsF ðeLÞÞ⋅F ′ðeLÞ− w
L
~
L*m
~* m m
O
m*m
Lemma 3. Both curves, L = Ac(m) and L = Bc(m), are strictly upward sloping in the (m, L) space.
competitive benchmark is to choose the level of monitoring mc and the number of workers Lc to:
ð28Þ
fm;Lg
Using the production function and the non-shirking wage equation, we can re-write the firm's optimization problem as: þeþ max p ⋅sF ðeLÞ− w fm;Lg
e qðmÞ
ða þ b þ r Þ L−H ðmÞ :
þeþ p seF ′ðeLÞ− w
e qðmÞ
ða þ b þ r Þ ¼ 0:
c
c
ð39Þ
ð31Þ
^ ¼ eq′ðmÞ bN þ r : Ω m 2 N−L ½qðmÞ
ð41Þ
ð34Þ
ð35Þ
c
Equilibriums in the product and the labor markets require that p ¼ α⋅P ðsF ðeLÞÞ and bL = a(N − L)or a = bL/(N − L). Using these conditions we rewrite the conditions that characterize the competitive equilibrium as: q′ðmÞ bN þ r −H′ðmÞ ¼ 0 ⋅ 2 N−L ½qðmÞ
)
2½q′ðmÞ2 q′′ðmÞ bN þ r −H′′ðmÞ þ eL⋅ 3 2 N−L ½qðmÞ ½qðmÞ
ð40Þ
To ensure that the second-order conditions for a max are satisfied, we make the additional assumption that
c
−
^ ¼ αs2 e2 ⋅P ′ðsF ðeLÞÞ⋅½F ′ðeLÞ2 þ αse2 ⋅P ðsF ðeLÞÞ⋅F ′′ðeLÞ Ω L −ebN½qðmÞ−1 ðN−LÞ−2
( ) ∂2 πc q′′ðmc Þ 2½q′ðmc Þ2 ¼ − eða þ b þ r ÞLc −H′′ mc b0 ð33Þ ∂m2 ðmc ;Lc Þ ½qðmc Þ2 ½qðmc Þ3
" #2 " # ∂2 πc ∂2 π c ∂2 πc ⋅ > : ∂m∂L ∂L2 ∂m2 ðm ;L Þ ðm ;L Þ
(
ð38Þ
ð30Þ
ð32Þ
∂2 πc ∂2 πc q′ðmc Þ ¼ ¼ eða þ b þ r Þ > 0: ∂L∂m ðmc ;Lc Þ ∂m∂L ðmc ;Lc Þ ½qðmc Þ2
^ ¼ eq′ðmÞ bN þ r þ ebNL⋅q′ðmÞ Φ L 2 N−L ½qðmÞ ½qðmÞ2 ðN−LÞ2
^ ¼ Φ m
Condition (30) implies that in the competitive equilibrium the firm invests in monitoring to the point where the marginal benefit of monitoring equals its marginal cost, while Eq. (31) indicates that the firm will set employment at the level such that the value of marginal product equals to the wage rate. It is straightforward to derive that ∂2 πc 2 ¼ p se F ′′ eLc b0 ∂L2 ðmc ;Lc Þ
^ ðm; LÞ ¼ 0and Ω ^ ðm; LÞ ¼ 0 denote Eqs. (36) and (37), reProof. Let Φ ^ j and Ω ^ j (j = m, L) denote the partial derivative of spectively. And let Φ ^ and Ω ^ with respect to j. Then, the slopes of the two curves at Φ ^ m =Φ ^ L and B′c ðm Þ ¼ the equilibria can be presented as A′c ðm Þ ¼ −Φ ^ ^ −Ω m =Ω L . It is easy to obtain
ð29Þ
The first-order conditions are: q′ðmÞ eða þ b þ r ÞL−H′ðmÞ ¼ 0 ½qðmÞ2
ð37Þ
m
Fig. 5. Effect of unemployment benefit on monopoly equilibrium.
max πc ¼ p Y−w L−H ðmÞ:
e bN þr ¼ 0: qðmÞ N−L
These two conditions jointly determine the equilibrium levels of employment Lc* and monitoring mc* chosen by the competitive firm. In what follows we explore the properties of Eqs. (36) and (37) in the (m, L) space. These properties will be useful in the comparison of competitive and monopoly equilibriums. Let L = Ac(m) be the curve implied by Eq. (36), and L = Bc(m) the curve represented by Eq. (37). The intersection of these two curves is (mc*, Lc*), representing the levels of monitoring and employment in the competitive equilibrium.
L*m
eL⋅
591
ð36Þ
The assumptions q′(⋅) > 0, q′′(⋅) b 0, H′(⋅) > 0 and H′′(⋅) > 0 imply ^ L > 0, Φ ^ m b0, Ω ^ L b0 and Ω ^ m > 0. Then, that Φ ^ =Φ ^ > 0 and B′ ðmÞ ¼ −Ω ^ =Ω ^ > 0: A′c ðmÞ ¼ −Φ m L c m L That is, both L = Ac(m) and L = Bc(m) are strictly upward slopping. QED Lemma 4. At the equilibrium point (mc*, Lc*), the slope of L = Ac(m) is greater than the slope of L = Bc(m), that is, A′c(mc*) > B′c(mc*). ^ , Φ ^ , Ω ^ and Ω ^ as the respective values at Proof. Denote Φ L m L m ^ and ^ =Φ the equilibrium point (mm * , Lm * ). Then, A′m mm ¼ −Φ m L ^ ^ B′m mm ¼ −Ω m =Ω L . Substitute the labor market steady state condition a = bN/(N − L) into the second order conditions (32)–(34), and ^ , Φ ^ , Ω ^ and Ω ^ to obtain then compare them with Φ L
^ > ∂2 π =∂m∂L Φ c L
ðmc ; Lc Þ
^ ¼ ∂2 π =∂m2 Φ m c
ðmc ; Lc Þ
^ b∂2 π =∂L2 Ω L c
ðmc ; Lc Þ
^ ¼ ∂2 π =∂m∂L Ω m c
>0
b0
b0
ðmc ; Lc Þ
m
L
m
ð42Þ
ð43Þ
ð44Þ
> 0:
ð45Þ
592
B. Zhao / Economic Modelling 29 (2012) 586–600
L
Furthermore, "
#
^ −∂2 π =∂m∂L Φ L c
^ ⋅Ω m
L*c
ebNLc ⋅q′ðmc Þ eq′ðmc Þ bN þr ⋅ 2 2 2 N−L ½qðmc Þ ðN−Lc Þ ½qðmc Þ c 2 2
e bNLc ⋅½q′ðmc Þ bN ¼ ⋅ þr : ½qðmc Þ4 ðN−Lc Þ2 N−Lc
L*m
ðmc ; Lc Þ
¼
and "
#
^ −∂2 π =∂L2 Ω c L
O
^ ⋅Φ m
ðmc ; Lc Þ
)
( 2
2eLm ½q′ðmc Þ bN > þr ⋅ N−nLc ½qðmc Þ3
e2 bNLc ⋅½q′ðmc Þ2 bN > ⋅ þr : N−Lc ½qðmc Þ4 ðN−Lc Þ2
That is, ^ −∂2 π =∂L2 Ω L c
# ðmc ; Lc Þ
" ^ ^ −∂2 π =∂m∂L ⋅Φ m > Φ L c
# ^ : ⋅Ω m
ðmc ; Lc Þ
ð46Þ
Using Eqs. (45) and (52)–(56), we obtain h
2
2
2
2
∂ πc =∂L ⋅∂ πc =∂m
i ðmc ; Lc Þ
h i2 2 > ∂ πc =∂m∂L
ðmc ; Lc Þ
^ =Φ ^ > Φ ^ ⋅Ω ^ ⇒Φ ^ bΩ ^ =Ω ^ ⇒Ω^ L ⋅Φ m L m m L m L ^ =Φ ^ > −Ω ^ =Ω ^ ⇒A′ ðm Þ > B′ ðm Þ: ⇒−Φ c c m L m L c c That is, at the intersection (mc*, Lc*), the slope of L = Ac(m) is greater than the slope of L = Bc(m). QED 4.2. Comparison of monopoly with competitive benchmark To determine the efficiency consequences of monopoly, we compare the monopoly equilibrium with that under perfect competition. Proposition 5. Relative to a perfectly competitive market, a monopolist employs fewer workers and invests less in the monitoring of workers. Proof. Note from Eqs. (11) and (36) that L = Ac(m) and L = Am(m) are exactly the same curve in the (m–L) space. Now, we consider the relationship between L = Bc(m) and L = Bm(m). þeþ ð37Þ: L ¼ Bc ðmÞ : se⋅P ðsF ðeLÞÞ⋅F ′ðeLÞ− w
m*c
m
Fig. 6. Monopoly equilibrium vs. competitive equilibrium.
ebN qðmc ÞðN−Lc Þ2
"
m*m
e bN þr qðmÞ N−L
Since q′(⋅) > 0, the above inequality implies that mc b mm. That is, for any given L, the corresponding value of m on the curve L = Bc(m) is always smaller than the corresponding value of m on the curve L = Bm(m). In other words, the curve L = Bm(m) must be everywhere below the curve L = Bc(m). Therefore, we can draw the three curves, L = A(m), L = Bc(m) and L = Bm(m), in the m–L space in the same diagram as shown in Fig. 6. Because Bm(m) is completely below Bc(m) and A(m) is upward sloping, the monopoly equilibrium Em must lie on the left and below to the competitive equilibrium Ec. From Fig. 6 we see that both the level of employment and the level of monitoring are lower in the case of monopoly. QED Since the quantity produced is an increasing function of employment, Proposition 5 in turn implies that the monopolist produces a smaller output and accordingly sets a higher price than a competitive firm. On the surface, these output and price effects of monopoly appear to be the same as in the standard monopoly case. However, there are more factors at play in this model. Intuitively, a monopolist has a tendency to restrict output because it faces a downward sloping demand curve. As a result of this tendency, employment level falls, which tends to push down the non-shirking wage. In our model this has an additional effect because it induces the monopolist to reduce the level of monitoring, causing a further fall in employment. Indeed, from Fig. 6 we can see that holding the level of monitoring (at the corresponding competitive monitoring level) and hence the detection rate q(m) constant, monopoly would reduce the employ⌢ ment level from Lc* to L m . But the lower investment in monitoring by the monopolist causes the employment level to drop further ⌢ from L m to Lm *. Note that the efficiency wage received by workers is actually
þeþ w ¼w
e bN þr qðmÞ N−L
ð47Þ
¼0
2
ð11Þ: L ¼ Bm ðmÞ : s e⋅P ′ðsF ðeLÞÞ⋅F ′ðeLÞ⋅F ðeLÞ
e bN þeþ þ se⋅P ðsF ðeLÞÞ⋅F ′ðeLÞ− w þr ¼ 0: qðmÞ N−L
which implies that the wage rises with employment but decreases with monitoring, ∂ w*/∂ L > 0 and ∂ w*/∂ m b 0. Then, Proposition 5 suggests that the effect of monopoly on the wage rate is ambiguous since monopoly causes the levels of both employment and monitoring to fall. 8
Since P′(⋅) b 0 and F′(⋅) > 0, we find that for any given L, se⋅ P ðsF ðeLÞÞ⋅F ′ðeLÞ > s2 e⋅ P ′ðsF ðeLÞÞ ⋅F ′ðeLÞ⋅F ðeLÞ þ se⋅ P ðsF ðeLÞÞ⋅F ′ðeLÞ
e bN e bN þeþ þeþ þr >w þr ⇒w qðmc Þ N−L qðmm Þ N−L
⇒qðmc Þ b qðmm Þ:
8 In the literature there is no consensus regarding the question whether product market power has positive effect on the wage. For example, Nickell (1999) argues that an overall rise in market power throughout the economy will lead to both higher unemployment and lower wages. In another paper, however, Nickell et al. (1994) show empirically that product market power does have a positive impact on wages.
B. Zhao / Economic Modelling 29 (2012) 586–600
5. Simulations In this section, we conduct numerical simulations using a version of our model with specific functional forms. The objectives of these simulations are to provide concrete examples for the ambiguous results in the general model, and more importantly, to quantify the welfare effects of monopoly. Specifically, the model is simulated to compare the consumer surplus, producer surplus and total surplus under monopoly and the competitive benchmark.
τ ρτ
A−Bs L
⋅αsρL
ρ−1
593 τþ1 ρτþρ−1
−Bαρτs
L
m þθ þ1þ m − w ða þ b þ r Þ ¼ 0: δmm
ð54Þ Using the labor market steady state condition a = bL/(N − L), and rearranging the above equations, we have the following equilibrium conditions for the monopoly case γþ1
θ½ðN−LÞr þ NbL−δβγ ðN−LÞm
¼0
ð55Þ
m þ θ bN τ ρτ ρ−1 τþ1 ρτþρ−1 þ1þ ⋅ αsρL þr ¼ 0: A−Bs L −Bαρτs L − w δm N−L
5.1. Functional forms
ð56Þ Suppose that the inverse demand function is given by: τ P ðQ Þ ¼ α⋅ A−B ⋅ Q
ð48Þ
where α, A, B and τ are all positive parameters. Note that this function degenerates into a linear demand function when τ = 1. The production function takes the form ρ
Y ¼ F ðLÞ ¼ s ⋅ L
ð49Þ
where 0 b ρ b 1. In the above production function the worker's effort is normalized to one, e = 1. The monitoring technology is represented by δ⋅m mþθ
In the competitive benchmark the firm's optimization problem can be written as: mþθ ρ γ þ1þ ða þ b þ r Þ L−βm max πc ¼ p ⋅sL − w δm fm;Lg
ð57Þ
The first order conditions are θða þ b þ r ÞL γ−1 −βγm ¼0 δm2 mþθ ρ−1 þ1þ ða þ b þ r Þ ¼ 0: − w p ⋅sρL δm
ð58Þ ð59Þ
ð50Þ
Substituting the product market clearing condition and the labor h τ i market steady state condition, p ¼ P ðF ðLÞÞ ¼ α ⋅ A−B⋅ sLρ and
where 0 b δ ≤ 1 and θ > 0. The monopolist's monitoring cost function is
a = bL/(N − L), into the first order conditions, we obtain the following equilibrium conditions that characterize the competitive equilibrium:
qðmÞ ¼
γ
H ðmÞ ¼ β ⋅ m
with β > 0; γ > 1:
ð51Þ
γþ1
θ½ðN−LÞr þ NbL−δβγ ðN−LÞm
¼0
ð60Þ
Using these functions, we solve the monopoly equilibrium and the competitive benchmark as follows. The monopolist's optimization problem can be presented as
mþθ bN τ ρτ ρ−1 þ1þ ⋅ þr ¼ 0: A−Bs L ⋅αsρL − w δm N−L
h ρ τ i mþθ ρ γ þ1þ ða þ b þ r Þ L−βm : ⋅ sL − w max π ¼ α ⋅ A−B⋅ sL δm fm;Lg
5.2. Monopoly equilibrium vs. competitive benchmark
ð52Þ The first order conditions yield θða þ b þ r ÞL γ−1 −βγm ¼0 δm2
ð53Þ
ð61Þ
In this section, we present the simulation results that illustrate the properties of the monopoly equilibrium. Those of the competitive benchmark are also presented to illustrate the effects of monopoly. Table 1 presents the simulation results from a particular set of parameter values. They confirm that relative to the competitive benchmark, monopoly in the product market indeed leads to a higher
Table 1 Competitive equilibrium vs. monopoly equilibrium functions, parameters and results. Parameters and values
Separation rate (b)
Interest rate (r)
Technology shocks (s)
Market size (α)
Unemployment benefit (w)
Total labor (N)
0.05
0.05
1.0
1.0
0.5
1000
Functions
Demand function p = 100 − 0.1Q1.5
Results
Employment
Production function
Monitoring tech.
Monitoring cost
F(L) = L0.65
0:8m qðmÞ ¼ m þ 15
H(m) = 6m3
Monitoring level
Lc*
Lm *
mc*
mm *
672.29
324.04
3.451
2.543
Equilibrium price
Percentage change of employment (Lc* − Lm * )/Lc*
Change of unemployment rate (um * − uc*)
51.80%
34.83%
Equilibrium output
Efficiency wage
Competitive (pc*)
Monopoly (pm *)
Competitive (Yc*)
Monopoly (Ym* )
Competitive (wc*)
Monopoly (wm *)
42.8698
71.9564
68.851
42.844
2.8538
2.5689
594
B. Zhao / Economic Modelling 29 (2012) 586–600
Table 2 Effects of separation rate functions and parameters. Parameters and values
Separation rate (b)
Interest rate (r)
Technology shocks (s)
Market size (α)
Unemployment benefit (w)
Labor force (N)
0.0001–0.3
0.05
1.0
1.0
0.5
500
Functions
Demand function P(Q) = 100 − 0.1Q1.5
Production function
Monitoring tech.
Monitoring cost
F(L) = L0.5
0:8m qðmÞ ¼ m þ 15
H(m) = 6m3
unemployment rate and a lower level of monitoring. As expected, the monopolist produces a smaller quantity and charges a higher price. Interestingly, in this particular case the monopolist offers a lower efficiency wage despite the lower level of monitoring. However, as has been observed in the discussion of the general model, monopoly may also lead to a higher wage rate. For example, if the value of s is changed from 1.0 to 2.0, the equilibrium wages become wm * = 2.6507 under monopoly and wc* = 2.5707 under perfect competition. These examples illustrate that, in general, monopoly has an ambiguous effect on the wage. Recall from Proposition 1 that a higher separation rate b raises the unemployment rate but has an ambiguous effect on the monopolist's monitoring level. The following simulations, based on the parameters in Table 2, illustrate that this is indeed the case. From Fig. 7, we see that the equilibrium unemployment rate under monopoly indeed rises with b but the increasing rate falls as b becomes larger. Incidentally, we can see from the diagram that the unemployment rate under the competitive equilibrium displays the same pattern. The gap of unemployment rate between monopoly and the competitive benchmark, Δu* ≡ um * − u*, c falls as b increases. Fig. 8 illustrates that the higher separation rate can either raise or reduce the monopolist's monitoring. Note that the critical value of b is defined by dmm =db b¼b ¼ 0. For bb b , a higher separation rate raises In adthe monopolist' monitoring level. The opposite is true for b > b. dition, we see from Fig. 8 that the relationship between the separation rate and the monitoring level under perfect competition displays the same pattern, with b being the critical value for the competitive case. Recall from Proposition 2 that a positive technology shock may have either a positive or negative effect on both unemployment rate and monitoring level. Simulations based parameters in Table 3 confirm this general result. These simulations are illustrated in Figs. 9 and 10. They show that under monopoly, unemployment rate decreases and monitoring level increases with s if the value of s is below the critical value s, and the opposite is true if s is above s. Incidentally, the unemployment and the level of monitoring in the competitive benchmark displays the same pattern, but its critical value, s , is larger than s.
5.3. Welfare effects of monopoly 5.3.1. The measures of welfare Given that this is a partial equilibrium model, we will use the total surplus as the measure of welfare. Note that compared with the standard textbook model of monopoly, we have an additional group of agents in our model, the workers. In principle, the measure of total welfare should also take into consideration the utility of workers. This raises an additional issue of how to treat the unemployment benefits. If the unemployment benefits are financed by lump sum taxes on consumers, they are merely a wealth transfer and as such should not be included in the total surplus. Alternatively, if there are no government transfer payments and the unemployment benefits merely represent the value that an unemployed worker obtain from home production, the unemployment benefits should be included in the total surplus. Therefore, in our simulations we use total surplus measures that include both cases. In addition, for completeness we also consider the conventional measure of total surplus that takes into consideration the welfare of consumers and the firms only (i.e. excluding the workers). Therefore, in our welfare analysis we use the following three measures of total surplus: ➣ Type I: Use the traditional definition of the consumer surplus, producer surplus and total surplus. Consumer surplus: CS ≡ ∫ 0Q*P(Q)d Q − P(Q*) ⋅ Q* Producer surplus: PS ≡ P(Q*) ⋅ Q* − w*L* − H(m*) Total surplus: TS ≡ CS + PS = ∫ 0Q*P(Q)d Q − w*L* − H(m*) ➣ Type II: Assume that workers are also consumers and thus include the workers' utility as a part of the consumer surplus. Recall that the worker's utility is defined as U L ðw; eÞ ¼
w −e w
if employed if unemployed
m* u*
2.6
80
2.5
/
2.4 60
/
2.3 2.2
40
2.1 20
b 0
0
0.05
0.1
0.15
0.2
0.25
0.3
Fig. 7. Effect of separation rate on unemployment rate.
0.05
0.1
b 0.15
0.2
0.25
0.3
b Fig. 8. Effect of separation rate on the level of monitoring.
b
B. Zhao / Economic Modelling 29 (2012) 586–600
595
Table 3 Effects of technology shocks functions and parameters. Parameters and values
Separation rate (b)
Interest rate (r)
Technology shocks (s)
Market size (α)
Unemployment benefit (w)
Labor force (N)
0.05
0.05
0.2–3.0
1.0
0.5
1000
Functions
Demand function P(Q) = 100 − 0.1Q1.5
Production function
Monitoring tech.
Monitoring cost
F(L) = L0.65
0:8m qðmÞ ¼ m þ 15
H(m) = 6m3
PS ≡ P Q ⋅Q −w L −H m
It is clear that ∫0Qc*P(Q)d Q >∫0Qm*P(Q)d Q and H(mc*) > H(mm * ) since we have L*c > L m * and mc* > mm * . And, as discussed in Section 4.2, the competitive efficiency wage wc* may be greater or less than the monopoly efficiency wage wm * . Therefore, the term wc* ⋅ Lc* may be greater or less than the term wm * ⋅ Lm * . That is to say, monopoly may generate a higher Type I total surplus than competition does. SpecifiI cally, TSm > TScI if and only if
Q N−L −H m : TS ≡CS þ PS ¼ ∫0 P ðQ ÞdQ −e⋅L þ w⋅
wc ⋅Lc −wm ⋅Lm þ H mc −H mm > |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Moreover, assume that the unemployment benefits come from home production, so that the unemployment benefits are included in the calculation of surpluses. Q N−L CS ≡ ∫0 P ðQ ÞdQ−P Q ⋅Q þ w −e ⋅L þ w⋅
➣ Type III: This measure is the same as Type II except that we assume that the unemployment benefits are transfers and thus are not included in the calculations of surpluses.
PS ≡ P Q ⋅Q −w L −H m
In countries where unemployment benefits take the form of income transfer from governments to unemployed workers, type III total surplus is the most appropriate type of welfare measure among the three presented here. Note that the monopolist's costs of monitoring effort and/or investment are included in the calculation of producer surplus.
I
Q m
and
Difference of gross
ΔW
ΔH
ΔCSg
ð62Þ
consumer surplus
II Q N−Lm −H mm : and TSm ¼ ∫0 m P ðQ ÞdQ −e⋅Lm þ w⋅ * and mc*> m m * , ∫0Qc*P(Q)d Q>∫0Q m* P(Q)d Q, e ⋅ Lc* > e ⋅ L m *, Since Lc* > L m H(mc*)>H(m m * ) and w⋅ N−Lc bw⋅ N−Lm . Then, compared to monopoly, perfect competition possibly results in a lower Type II total surplus. II And TSm > TScII if and only if e⋅ Lc −Lm þ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
5.3.2. General comparisons of welfare For Type I total surplus, we have ¼
Difference of monitoting cost
II Q N−Lc −H mc TSc ¼ ∫0 c P ðQ ÞdQ−e⋅Lc þ w⋅
Q TS ≡CS þ PS ¼ ∫0 P ðQ ÞdQ −e⋅L −H m :
Q ∫0 c P ðQ ÞdQ −wc ⋅Lc −H mc
Difference of total wages
Eq. (62) indicates that monopoly may be superior to perfect competition in terms of Type I total surplus if and only if the difference of total production cost ( ΔW þ ΔH) is greater than the difference of gross consumer surplus (ΔCSg ). Similarly, for Type II total surplus, we have
Q CS ≡ ∫0 P ðQ ÞdQ−P Q ⋅Q þ w −e ⋅L
I TSc
Q
∫Q cm P ðQ ÞdQ : |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
Lc −Lm w⋅ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
þ H mc −H mm > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Q
∫Q cm P ðQ ÞdQ : |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
Difference of
Difference of total
Difference of
Difference of gross
total efforts
unemployment benefits ΔW
monitoting cost
consumer surplus
ΔH
ΔCSg
exerted by workers ΔE
TSm ¼ ∫0 P ðQ ÞdQ −wm ⋅Lm −H mm :
ð63Þ
m*
u*
3.50
80
3.25 60
3.00 2.75
40 2.50
/
2.25
20
/ 0
s
0.5
s
0
s~
1.0
1.5
2.0
2.5
3.0
s
0.5
1.0
1.5
2.0
2.5
3.0
* c
du / ds Fig. 9. Effects of technology shocks on monitoring.
Fig. 10. Effects of technology shocks on unemployment rate.
s
596
B. Zhao / Economic Modelling 29 (2012) 586–600
Eq. (63) implies that monopoly may generate a higher Type II total surplus than perfect competition does in case that the difference of ; the firm/manager: total pay-outs (the workers: ΔE; the society: ΔW ΔH) is greater than the difference of gross consumer surplus. Finally, for Type III total surplus, we have III TSc III TSm
¼ ¼
Q ∫0 c P ðQ ÞdQ −e⋅Lc −H mc Q m
∫0 P ðQ ÞdQ −e⋅Lm −H
mm
∫ 0Qc*P(Q)d
and
TS (Type I) 4000 3200 2400
:
1600
Q > ∫ 0Qm* P(Q)d
It is easy to obtain Q , e ⋅ Lc* > e ⋅ Lm * , H(mc*)> H(mm * ) because Lc* > Lm * and mc* > mm * . Thus, monopoly may also lead to a higher Type III total surplus than perfect competition does. Specifically, III TSm > TScIII if and only if Q þ H mc −H mm > ∫Q cm P ðQ ÞdQ : e⋅ Lc −Lm |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} Difference of
Difference of
Difference of gross
total efforts exerted by workers
monitoting cost ΔH
consumer surplus ΔCSg
ΔE
(a) Effects of b on TS
ð64Þ
Perfect Competition Monopoly
800 0
0.05
0.1
0.15
0.2
0.25
0.3 b
(b) Effect of b on (TSc–TSm)
TSc−TSm (Type I) 750 600 450
The left hand side of Eq. (64) is the sum of the difference of worker's efforts put into production (ΔE) and the difference of monitoring cost (ΔH). And the right hand side is the difference of gross consumer surplus (ΔCSg ). These general analyses imply that, no matter which measure is used, monopoly may possibly generate a higher total welfare than perfect competition. 5.3.3. Simulations of welfare effects of monopoly Our simulation results show that the total surplus under monopoly is not necessarily lower than that under perfect competition. This is true for all the three measures of total surplus. In what follows we present a number of examples to illustrate. Figs. 11 and 12 show that monopoly may improve welfare as defined by the textbook version of total surplus (i.e. Type I). Fig. 11 is based on the simulation results using the parameter values in Table 2. It shows that if separation rate is big enough (b > 0.24), monopoly generates higher total surplus than competition does. Fig. 12 is based on the simulations results using the parameter values in Table 3. From this graph we see that for s in a particular interval (0.3 ≤ s ≤ 0.8), the total surplus under monopoly is larger than that under perfect competition. To illustrate that monopoly may improve welfare as measured by Type II and Type III total surplus, we present the simulation results for the case where s = 0.35. Figs. 13 and 14 show how the Type II total surplus varies as the value changes. From Fig. 13 we see that Type II total surplus of either α or w under monopoly exceeds that under perfect competition if the market size parameter α is greater than 42. The same can be said about Fig. 14 is lower than 0.948. for the case where unemployment benefits w The same simulations are done by using Type III total surplus, and the results are qualitatively similar to those using Type II total surplus. As we can see from Figs. 15 and 16, monopoly improves welfare is relatively small. if, ceteris paribus, s is relatively large or w These numerical simulations, together with the general analysis in Section 5.3.2, demonstrate that Proposition 6. Monopoly is not always dominated by perfect competition in terms of economic efficiency if an efficiency wage is offered and unemployment is taken into consideration.
300 150 0
0.24 0.05
0.1
0.15
0.2
0.25
Fig. 11. Effects of separation rate on welfare (Type I).
endogeneity in the welfare effects of monopoly. We will do so by first making an observation on the general model and then present a number of examples from the simulation exercises.
(a) Effects of s on TS TS (Type I) 5000 4000 3000
Perfect Competition Monopoly
2000 1000 0
0.5
1
1.5
2
2.5
3
s
2
2.5
3
s
(b) Effect of s on (TSc–TSm)
TSc−TSm (Type I) 1250 1000 750 500 250
0.3 0
5.4. The role of endogenous detection rate q
0.8 0.5
1
1.5
-250 An important feature of our model is that the probability of detection q is endogenous. In this sub-section we explore the role of this
0.3 b
-150
Fig. 12. Effects of technology shocks on welfare (Type I).
B. Zhao / Economic Modelling 29 (2012) 586–600
(a) Effects of a on TS
597
(a) Effects of a on TS
TS (Type II)
TS (Type III)
6000
6000
Perfect Competition Monopoly
4800
4800
3600
3600
2400
2400
s = 0.35
1200 0.5
0
1
1.5
2
2.5
Perfect Competition Monopoly
s = 0.35
1200 3
a
(b) Effect of a on (TSc–TSm)
0
0.5
1
1.5
2
2.5
3
a
(b) Effect of a on (TSc–TSm)
TSc−TSm (Type II)
TSc−TSm (Type III)
s = 0.35
120
200
80
s = 0.35
150
40
100
2.42
0
0.5
1
1.5
2
2.5
3
a
50
-40
0
-800
-50
-120
-100
2.63 0.5
1
1.5
2
2.5
3
a
Fig. 13. Effects of market size on welfare (Type II).
Fig. 15. Effects of market size on welfare (Type III).
To determine the role played by the endogenous detection rate, we consider a restricted version of our model where m and accordingly q are fixed at the level of the competitive equilibrium; in other
words, the value of m is fixed at mc*, and the value of q is fixed at qc*. As illustrated in Fig. 17, the level of employment in the equilibrium of this restricted model is lower than that in the competitive
(a) Effects of w on TS
(a) Effects of w on TS
TS (Type III)
TS (Type II)
6000
7000
5500
Perfect Competition Monopoly
6750 6500
5000
6250
4500
s = 0.35 α = 2.75
6000 5750
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5 w
3500
200
625
150
500
100
s = 0.35 α = 2.75
50
0.948 1
1.5
2
2.5
α = 2.75 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5 w
TSc−TSm (Type III)
TSc−TSm (Type II)
0.5
s = 0.35
4000
(b) Effect of w on (TSc–TSm)
(b) Effect of w on (TSc–TSm)
0
Perfect Competition Monopoly
3
3.5
4
4.5
5 w
375 250
0.61
-50 -100 Fig. 14. Effects of unemployment benefit welfare (Type II).
s = 0.35 α = 2.75
125 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-125 Fig. 16. Effects of unemployment benefit on welfare (Type III).
5 w
598
B. Zhao / Economic Modelling 29 (2012) 586–600
(a) Effects of s on TS
L
TS (Type II) L*c
2500 2000
L*m
1500
L*m
1000
α = 0.25 Perfect Competition Monopoly (endogenous q) Monopoly (exogenous q)
500 O
m*m
m*c
(m*m )
m 0
0.5
1
1.5
2
2.5
3
s
2
2.5
3
s
)
Fig. 17. Role of endogenous detection rate (q).
)
(b) Effect of s on (TSm–TSm)
equilibrium. It turns out we can make a similar statement about welfare measured in terms of Types II and III total surplus. From the definition of Type II total surplus, we find that for fixed m, ∂TSII =∂L ¼ þ eÞ > 0 for all L b Lc*. In determining the sign P ðsF ðeL ÞÞ⋅seF ′ðeL Þ−ðw of this derivative we have used the first-order condition of the com⌢ petitive firm's profit-maximization problem. Since L m bLc , we know that
⌢II > TS m . Similarly, we can show that for Type III total sur⌢III TSIII > TS . That is, when workers' utilities are included in
TSm−TSm (Type II) 15 10 5
TSIIc
pluses, m c the total surplus, monopoly with an exogenous detection rate always reduces welfare. This finding implies that when measured by Type II and Type III total surplus, the ambiguous welfare effect of monopoly in our model is driven by endogenous detection rate. Indeed, simulation results show that under some range of parameter values endogenizing q can increase Types II and III total surplus. One set of such examples are presented in Figs. 18 and 19, which are based on the parameter values in Table 3 except a different market size. They show that for a large enough s, Type II and Type III total surpluses are higher under monopoly with endogenous q than that with exogenous q. It should be pointed out that the above conclusions about Type II and Type III total surplus do not apply to Type I total surplus. The example in Fig. 20, from simulations using the parameter values in Table 3, illustrate the possibility that (i) even with exogenous detection rate, Type I total surplus may still be higher under monopoly than under perfect competition; and (ii) endogenous choice of monitoring (and hence endogenous detection rate) may improve the total surplus in the monopoly equilibrium despite the lower employ⌢ ment and smaller output. As we can see from Fig. 20, TS m > TSc if ⌢ 0.286 b s b 0.789. Furthermore, since TSm > TS m in this case, for s in this range the Type I total surplus under monopoly with endogenous q is larger than that under perfect competition.
α = 0.25
0
1.635 0.5
1
1.5
-5 -10 -15 Fig. 18. Comparison of Type II total surplus endogenous q vs. exogenous q.
(a) Effects of s on TS TS (Type III) 1250 1000 750
α = 0.25 Perfect Competition Monopoly (endogenous q) Monopoly (exogenous q)
500 250 0
0.5
1
1.5
2
2.5
3
s
2
2.5
3
s
)
6. Extension
pi ¼ αP ðQ i Þ; 9
i ¼ 1; 2;…; M
ð65Þ
Actually, I am currently working on another extension of the model to a more general case — Cournot competition with unemployment caused by efficiency wage, which is expected to find a new path to explore the relationship between competition and employment, as well as economic efficiency. 10 The properties of these functions are same as what we have assumed in Section 2.
(b) Effect of s on (TSm–TSm)
)
So far in the model, we have assumed that there is only one firm (the monopolist or the representative competitive firm) in the economy. Therefore, the wage rate offered by the firm is also the wage rate of the economy. In this section, we extend the one-firm model to an M-firm case: assume there are M monopolized industries in the economy. 9 All firms face identical downward slopping demand and have identical concave production technology represented by Eqs. (65) and (66), respectively. 10
TSm−TSm (Type III) 15 10
α = 0.25
5 0
1.895 0.5
1
1.5
-5 -10 -15 Fig. 19. Comparison of Type III total surplus endogenous q vs. exogenous q.
B. Zhao / Economic Modelling 29 (2012) 586–600
(a) Effects of s on TS
At equilibrium, wi = w− i for all i since all firms are identical. Then, we may derive wi from Eq. (67) as
TS (Type I) 5000
þeþ wi ¼ w
4000
Perfect Competition Monopoly (endogenous q) Monopoly (exogenous q)
1000 0.5
1
1.5
2
2.5
3
ð71Þ
q′ðmi Þ eðb þ rÞLi −H′ðmi Þ ¼ 0 ½qðmi Þ2
s
)
0
e ða þ b þ r Þ: qðmi Þ
Substituting the labor market steady state condition a = bMLi/ (N − MLi) and Eq. (71) into the first order conditions, we obtain
3000 2000
599
2
αs e⋅P ′ðsF ðeLi ÞÞF ′ðeLi ÞF ðeLi Þ
þeþ þ αse⋅P ðsF ðeLi ÞÞF ′ðeLi Þ− w
)
(b) Effect of s on (TSc–TSm)
ð72Þ
TSc−TSm (Type I)
e bN þr ¼ 0: qðmi Þ N−MLi
ð73Þ
Similarly, we may derive the equilibrium conditions for a competitive benchmark case as
1250 1000
q′ðmci Þ eðb þ r ÞLci −H′ðmci Þ ¼ 0 ½qðmci Þ2
750 500
þeþ αse⋅P ðsF ðeLci ÞÞF ′ðeLci Þ− w
250 -0
0.286 0.789 0.5
1
1.5
2
2.5
3
Fig. 20. Comparison of total surplus (Type I) perfect competition vs. monopoly with exogenous q.
i ¼ 1; 2;…; M:
ð66Þ
In this case, we need to distinguish firm i's wage rate to the aggregate wage rate. Enlightened by our discussion in Section 1, we may write the no-shirking wage offered by firm i as wi ¼
ðb þ r Þ ðw−i −eÞa þ w bþr þ 1þ e ða þ b þ r Þ qðmÞ
Using the same technique as in Section 4, it is not difficult to prove that, at equilibrium every monopolist employs fewer workers compared to the competitive benchmark case, L*i b Lci* . Therefore, the aggregate equilibrium employment under monopoly (L* ≡ ML*) i will be lower than the aggregate equilibrium employment under perfect competition (Lc* ≡ MLci * ), L* b Lc*. Moreover, relative to the competitive benchmark, every monopolist puts less effort or investment into monitoring, m*i b mci * . Additionally, it is easy to demonstrate that the properties of this M-industry model are the same with what we have derived for the one-firm model in Section 3. Furthermore, numerical simulations (to avoid verbosity, we do not present them here) also suggest that monopoly may generate higher welfare, measured by the three types of total surplus, than perfect competition does. 7. Conclusions
ð68Þ
The first-order conditions are q′ðmi Þ eðb þ r ÞLi −H′ðmi Þ ¼ 0 ½qðmi Þ2
ð75Þ
ð67Þ
where w− i represents the average wage rate of the economy other than firm i. Each monopolist chooses his own employment Li and monitoring mi to maximize its profit. That is, for firm i max πi ¼ αP ðsF ðeLi ÞÞ⋅sF ðeLi Þ fmi ;Li g ðb þ r Þ ðw−i −eÞa þ w bþr þ 1þ e Li −H ðmi Þ: − ða þ b þ r Þ qðmi Þ
e bN þr ¼ 0: qðmci Þ N−MLci
s
-250
yi ¼ sF ðeLi Þ;
ð74Þ
ð69Þ
2
αs e⋅P ′ðsF ðeLi ÞÞF ′ðeLi ÞF ðeLi Þ
ðb þ r Þ ðw−i −eÞa þ w bþr þ αse⋅P ðsF ðeLi ÞÞF ′ðeLi Þ− þ 1þ e ¼ 0: ða þ b þ r Þ qðmi Þ
ð70Þ
In this paper we have analyzed the efficiency and employment consequences of monopoly in the presence of unemployment caused by efficiency wage considerations. We have shown that in addition to a smaller output and a higher price, monopoly also leads to higher unemployment rate than the competitive equilibrium. It is worth noting that the effects of monopoly on the wage rate and total welfare, however, are ambiguous. Numerical simulations of the model indicate that under certain range of parameter values, monopoly generates higher total surplus than perfect competition. Therefore, by introducing an additional distortion (i.e. unemployment) into the model, it is no longer the case that monopoly is always dominated by perfect competition in terms of economic efficiency. Another contribution of this paper is that we have constructed an efficiency-wage model that takes into consideration the demand condition that the firm faces and the firm's choice of monitoring. The introduction of these additional considerations has generated some interesting results. For example, we have found that the effects of a technology shock on unemployment and the firm's monitoring depend on the properties of the product demand function; a positive technology shock that raises the marginal product of labor may in fact reduce the level of employment if product demand is not sufficiently elastic.
600
B. Zhao / Economic Modelling 29 (2012) 586–600
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