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Monopoly power, increasing returns to variety, and local indeterminacy
Review of Economic Dynamics 14 (2011) 384–388
Contents lists available at ScienceDirect
Review of Economic Dynamics www.elsevier.com/locate/red
Monopoly power, increasing returns to variety, and local indeterminacy Juin-jen Chang a,1 , Hsiao-wen Hung b,∗ , Chun-chieh Huang c a b c
Institute of Economics, Academia Sinica, Department of Economics, Fu-Jen Catholic University, Taiwan Department of Industrial Economics, Tamkang University, Tamsui, Taipei County 251, Taiwan Department of Economics, Fu-Jen Catholic University, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 2 September 2008 Revised 12 September 2009 Available online 14 October 2009 JEL classification: E00 E32 L16 Keywords: Local indeterminacy Increasing returns to variety Endogenous entry
The required degree of increasing returns-to-scale to satisfy the Benhabib–Farmer condition for local indeterminacy is too high to be empirically plausible. In the paper, we develop a natural extension of Benhabib and Farmer’s (1994) model by introducing aggregation increasing returns to variety. It is shown that an increase in the degree of monopoly power can create increasing returns to an expansion in variety which decreases reliance on the degree of increasing returns-to-scale in production for generating local indeterminacy. As the degree of monopoly power increases, the required degree of increasing returns for local indeterminacy decreases monotonically. As a result, our numerical analysis indicates that the required degree of increasing returns for local indeterminacy can be easily located in the empirically plausible range. © 2009 Elsevier Inc. All rights reserved.
1. Introduction Since the seminal paper of Benhabib and Farmer (1994), henceforth BF, there have been a vast number of articles in the macroeconomic literature that have established the existence of indeterminate equilibrium paths. BF argued that an indeterminate steady state can be generated in a one-sector model of business fluctuations driven by self-fulfilling beliefs, provided that the increasing returns are sufficiently strong. This argument has attracted widespread criticism due to the degree of increasing returns-to-scale needed to satisfy the Benhabib–Farmer condition for local indeterminacy being too high to be empirically plausible. In their model, the degree of increasing returns-to-scale required in production for generating local indeterminacy is about 1.5. However, Burnside (1996) found the estimate of internal returns to be about 1.33, although the regressions were even more sensitive to the instrument set. According to the estimation of Basu and Fernald (1997), for the U.S. private business economy the empirically plausible degree of increasing returns-to-scale in production is between 1.03 and 1.18. Obviously, the degree of increasing returns-to-scale required to satisfy the Benhabib–Farmer condition for local indeterminacy is far higher based on the empirically plausible values. To fix the question, subsequent research has incorporated various elements into the models in order to push down the required degree of increasing returns for local indeterminacy to empirically plausible values. By extending the one-sector model of BF to a multiple-sector model, Benhabib and Farmer (1996), Perli (1998), Weder (2000), and Harrison (2001) show that sector-specific externalities lead the minimum level of increasing returns needed for indeterminacy to become less stringent. Wen (1998) and Guo and Harrison (2001) incorporate endogenous capital utilization into RBC models that lead to
* 1
Corresponding author. Fax: +886 2 26209731. E-mail address: [email protected] (H.-w. Hung). We would like to thank an associate editor of this journal for his/her insightful comments and suggestions. The usual disclaimer applies.
1094-2025/$ – see front matter doi:10.1016/j.red.2009.10.007
© 2009 Elsevier Inc.
All rights reserved.
J.-j. Chang et al. / Review of Economic Dynamics 14 (2011) 384–388
385
indeterminacy for smaller magnitudes of external effects than the BF model. In addition, Guo and Lansing (2005) propose a model calibrated to match empirical evidence on the relative size of maintenance and repair activity and show that local indeterminacy can occur with a mild and empirically-plausible degree of increasing returns — of around 1.08. By following this line of research, this paper develops a natural and straightforward extension of the BF model by endogenizing the number of producers (and products). Endogenous entry results in aggregation increasing returns to variety and, consequently, generates an endogenous propagation mechanism for local indeterminacy. Such an extension is convincing and important. As shown by Dos Santos Ferreira and Dufourt (2006), different levels of economic activities sustained by different numbers of active firms are consistent with the conditions of free entry and contestability. Furthermore, the important contribution of product creation and destruction to aggregate output and business fluctuations has also been proved by the empirical study of Bernard et al. (2006). In this paper we show that endogenous entry under monopolistic competition leads to aggregation increasing returns to variety and this reduces the minimum value of the labor externality needed to generate indeterminacy. Of importance, if the degree of monopoly power increases, the required degree of increasing returns for local indeterminacy decreases monotonically. Thus, local indeterminacy can occur with a very mild degree of increasing returns as long as the price markup ratio (the price–marginal cost ratio) is high enough. Specifically, we estimate that if the markup ratio is calibrated to around 1.4, the required degree of increasing returns-to-scale can be pushed down to 1.01429. This value is located within the range of the empirically plausible degree of increasing returns-to-scale. A particular emphasis is that the calibrated markup value is plausible in practice. Hall (1988, 1990) examines U.S. industry data on output and the labor input and finds that the estimated value of the markup ratio is between 1.864 and 3.791. By using data on value added, the estimate of the markup ratio is greater than 1.5 for all 7 1-digit industries and for 17 of the 21 2-digit industry groups. Morrison’s (1990) estimate shows that the average annual markups of U.S. manufacturing industries are between 1.179 and 1.803. By extending Hall’s (1990) data set, Roeger (1995) finds that the estimated markup ratios range from 1.19 to 3.14. Some studies have examined the role of monopolistic competition as a possible source of sunspot equilibria in one-sector models. Although Farmer and Guo (1994) incorporate the presence of monopolistic competition in one of the versions of their models, it is only necessary to sustain increasing returns at the level of the firm, thus ensuring a non-negative profit in the presence of increasing returns to scale in production. Given that the number of producers is given exogenously, monopoly power does not play a role in generating local indeterminacy. Schmitt-Grohé (1997) has also considered the effect of entry and exit in his survey paper. However, the increasing returns to variety have been abstracted from his study. Besides, the role of monopolistic competition has also been highlighted in various two sector models. Peck and Shell (1991) prove that the sunspot equilibria exist in a pure exchange-economy in which agents have market power in both the commodity and securities markets. Chatterjee et al. (1993) analyze a two-good OLG model and show the possibility of sunspot equilibria in the presence of sufficiently strong complementarities between the two-good sectors. Galí (1994) argues that sunspot fluctuations may arise when the elasticity of substitution across goods in consumption differs from that across goods in production. Apparently, our paper is different from theirs both in terms of its indeterminacy mechanism and numerical intention. In a simple one-sector model we indicate how endogenous entry leads to increasing returns to variety and, as a result, how monopoly power reduces the minimum value of the labor externality needed to generate indeterminacy. 2. Monopolistic markup and equilibrium indeterminacy 2.1. Endogenous entry This study simply extends the BF model to allow for endogenous entry and we describe the model briefly.2 In the economy considered there are two types of goods: a homogeneous final good, which is the numeraire and is produced by competitive firms, as well as differentiated intermediate goods indexed by i = 1, . . . , N t , where N t is the number of intermediate goods at time t. Production differentiation allows monopolistic competition to exist between the intermediate-goods N producers. By defining y it as the quantity of input i, the final good production technology is given by: Y t = ( 0 t y λit di )1/λ ; λ ∈ (0, 1) which displays, as stressed by Devereux et al. (1996, 2000), increasing returns to an expansion in variety (specialization) with the degree of (1/λ): under endogenous entry the larger the number N t of intermediate goods, the higher the amount of final production obtainable from a given amount of initial resources. A final good firm solves the profit maximization problem to determine its demand for the ith intermediate good, i.e., p it = (Y t / y it )1−λ which has a constant price elasticity 1/(1 − λ). Obviously, λ not only measures the degree of monopoly power, but also determines the degree of returns to variety. In line with BF, intermediate-goods producer i uses capital kit aθ bθ and labor h it to produce a product using the technology: y it = kait hbit ( K t 1 H t 2 ) − φ ; a + b = 1, where φ is the unit fixed cost, K t and H t represent the aggregate levels of capital and labor, and θ1 and θ2 are the corresponding externality parameters. Subject to the demand function and the production function above, we can easily derive the first-order conditions for h it and kit . By focusing on a symmetric equilibrium, the zero-profit and the aggregate consistency conditions allow us to derive (1−λ)/λ the following price of intermediate goods, the wage rate and the rental rate, respectively: pt = N t , w t = bY t / H t , and 2
A detailed mathematical deduction is available upon request. We also refer the reader to our working paper for a complete derivation.
386
J.-j. Chang et al. / Review of Economic Dynamics 14 (2011) 384–388 a(1+θ )
b(1+θ )
a(1+θ )/λ
b(1+θ )/λ
1 2 1 2 rt = aY t / K t , where N t = [(1 − λ)/φ] K t Ht , Y t = N t yt = Ω K t Ht , and Ω = λ[(1 − λ)/φ](1−λ)/λ . It is clear from the above equations that the endogenous number of firms in the intermediate goods sector will lead to aggregation increasing returns to variety. Consider a standard RBC model with separable utility under which households face the following optimization problem: N ∞ 1+χ maxct ,ht ,kt 0 [ln ct − ht /(1 + χ )]e −ρ t dt, subject to k˙ t = w t ht + (rt − δ)kt + 0 t πit di − ct , where ct is consumption, ρ is the rate of time preference, πit are profits and δ is the capital depreciation rate. Solving the household’s problem above yields the first-order conditions in terms of the aggregate variables and, accordingly, we have the Keynes–Ramsay rule and the economy-wide resource constraint as follows:
C˙ t Ct
= rt − ρ − δ = a
Yt Kt
1/λ
− ρ − δ and K˙ t = Y t − δ K t − C t , a(1+θ1 )/λ
with the instantaneous relationship of labor H t = [bΩ K t
(1)
/C t ]1/[1+χ −b(1+θ2 )/λ] .
2.2. Local indeterminacy Eq. (1) constructs the 2 × 2 dynamic system in terms of C t and K t . As addressed in the literature on dynamic rational expectations models, e.g., Buiter (1984), the dynamic system will have a unique perfect-foresight equilibrium path (saddlepath stability) if the number of (positively) unstable roots equals the number of jump variables. In this dynamic system there is only one jump variable C t . Accordingly, BF argued that if the dynamic system has two roots with negative real parts, there exists a continuum of equilibrium trajectories that converges to the steady state (sink) and, accordingly, local indeterminacy emerges in the economy. That is, a necessary (but not sufficient) condition refers to a positive value of the determinant of the Jacobian. With this understanding, we establish the following proposition: Proposition 1. When the number of firms is determined endogenously, the necessary condition for indeterminacy is 1 + θ˜2 > λ(1 + χ )/b. However, if the number of firms is given exogenously, the necessary condition for indeterminacy turns out to be 1 + θˆ2 > (1 + χ )/b, which is the same as the Benhabib–Farmer condition. Proof. See the Appendix A.
2
Similar to BF and Farmer and Guo (1994), for local indeterminacy, the necessary condition implies that the equilibrium wage-hours locus needs to be positively sloped and steeper than the labor supply curve. However, in their model local indeterminacy requires an implausibly higher degree of increasing returns-to-scale in production (see Burnside, 1996, and Basu and Fernald, 1997). Proposition 1 clearly points out that the required degree of increasing returns-to-scale in production can be substantially reduced when free entry endogenizes the number of firms, leading to aggregation increasing returns to variety. Of importance, if the market is characterized by less competition (a lower value of λ), the degree of aggregation increasing returns to variety becomes larger and, as a result, the necessary condition for local indeterminacy is easier to satisfy. Due to λ < 1 (and hence θ˜2 < θˆ2 ), our model can escape from the widespread criticism incurred by the Benhabib–Farmer condition. We can glean the intuition for the necessary condition for generating local indeterminacy by examining the Keynes– Ramsey rule (1), which is rewritten as the following discrete-time function for ease of illustration:
C t +1 Ct
=
= r t +1 − ρ − δ ⎧ ⎨ AK {a(1+θ1 )(1+χ )/[1+χ −b(1+θ2 )]}−1 C −b(1+θ2 )/[1+χ −b(1+θ2 )] − ρ − δ, t +1
t +1
if N t +1 = 1,
⎩ BK {a(1+θ1 )(1+χ )/λ[1+χ −b(1+θ2 )/λ]}−1 C −b(1+θ2 )/λ[1+χ −b(1+θ2 )/λ] − ρ − δ, if N t +1 = t +1 t +1
(1−λ) a(1+θ1 ) b(1+θ2 ) H t +1 , φ K t +1
(2) b(1+θ2 )/[1+χ −b(1+θ2 )] (1+χ )/[1+χ −b(1+θ2 )]
b(1+θ2 )/λ[1+χ −b(1+θ2 )/λ]
Ω (1+χ )/[1+χ −b(1+θ2 )/λ] and the condition
where A = ab λ , B = ab a(1 +θ1 )(1 + χ )/λ[1 + χ − b(1 +θ2 )/λ] < 1 holds because sustained endogenous growth is not allowed in our analysis. Notice that we normalize the number of firms to unity when focusing on the case where the number of firms is exogenously given. Under such a situation, the necessary condition of BF for generating local indeterminacy will be recovered. Suppose that agents become optimistic about next period’s return on capital rt +1 . In acting upon this belief, the household will sacrifice consumption today (reducing C t ) for more investment (raising K t +1 ) and higher future consumption (C t +1 ). As a result, the value of the left-hand side of (2) increases. Under constant returns-to-scale in production, a higher capital stock is associated with a lower rate of return (∂(rt +1 )/∂ K t +1 < 0), and therefore agents’ expectations cannot be self-fulfilling because the value of the right-hand side of (2) falls. To validate the agents’ optimistic expectations as a self-fulfilling equilibrium, the righthand side of (2) should also increase, i.e., the next period’s return on capital must be increasing in K t +1 . As indicated in (2), this implies that the necessary condition requires that 1 + θ2 > (1 + χ )/b if the number of firms N t is exogenously given and normalized to 1 (there are constant returns to the quantity employed of a fixed variety of intermediate goods).
J.-j. Chang et al. / Review of Economic Dynamics 14 (2011) 384–388
387
However, if the number of firms is endogenously determined, monopolistic competition will generate increasing returns to an expansion in variety which reinforces the increased magnitude of the return rate of capital, as shown in (2). In generating local indeterminacy, this decreases reliance on the degree of increasing returns-to-scale in production, i.e., the necessary condition turns out to be 1 + θ2 > λ(1 + χ )/b. Given that λ < 1, our model can push down the required value of the labor externality for generating indeterminacy. It is important to further discuss how effectively aggregation increasing returns to variety can push down the minimum value of the labor externality in order to generate indeterminacy by means of a simple numerical analysis. For the sake of comparison, most parameters we set are identical to those in BF: ρ = 0.065, δ = 0.1, and a = 0.3 (and hence b = 0.7). Moreover, in line with the common specification, for example, Wen (1998), Bennett and Farmer (2000), Harrison (2001) and Guo (2004), the inverse of the labor supply elasticity is set as χ = 0, meaning that the labor supply elasticity is equal to infinity.3 Apparently, a lower degree of monopoly power (λ) or a higher degree of increasing returns-to-scale in production (the value of the labor externality θ2 ) is more likely to result in local indeterminacy. According to the estimation of Basu and Fernald (1997), the empirically plausible degree of increasing returns-to-scale in production is in the region of 1.03 to 1.18. If we follow the BF benchmark parameterization and set λ = 0.667, the required degree of increasing returns-to-scale in production will be reduced to around 0. This implies that local indeterminacy can occur without increasing returns-to-scale in production as long as aggregation increasing returns to variety are present. Judd (1997) set λ ∈ [0.6, 0.9]. Given this parameterization, we have the minimum values of increasing returns-to-scale in production as 1 + θ˜2 = 1.01429, 1 + θ˜2 = 1.0929 and 1 + θ˜2 = 1.1429, if λ = 0.71, λ = 0.76 and λ = 0.8, respectively.4 By focusing on the case where λ = 0.71(i.e., the price markup ratio is around p /mc = 1/λ = 1.4), our estimated result 1 + θ˜2 = 1.01429 is lower than the outcome in existing studies, e.g., 1.074 in Harrison (2001) and 1.064 in Benhabib and Farmer (1996). That is, the required degree of increasing returns-to-scale in production can be substantially reduced and, consequently, becomes empirically plausible when we consider the effect of aggregation increasing returns to variety. Of importance, as stressed in Section 1, is that many empirical studies show that a markup ratio of 1.4 is plausible in practice. See, for example, Hall (1988, 1990), Morrison (1990), and Roeger (1995). 3. Concluding remarks It is well known that the degree of increasing returns-to-scale required to satisfy the Benhabib–Farmer-Guo condition for local indeterminacy is too high to be empirically plausible. In this paper, we have developed a natural extension of the BF model by endogenizing the number of firms. By so doing, we have shown that monopolistic competition leads to aggregation increasing returns to variety and this reduces the minimum value of the labor externality needed to generate indeterminacy. In particular, we have indicated that, as the degree of monopoly power increases, the degree of increasing returns required for local indeterminacy decreases monotonically. Specifically, under an empirically convincing markup value of 1.4, the required degree of increasing returns-to-scale can be pushed down to 1.01429. This is not only located within the range of the empirically plausible degree of increasing returns-to-scale, but is also lower than the outcome in existing studies, e.g., 1.074 in Harrison (2001) and 1.064 in Benhabib and Farmer (1996). Appendix A It follows from (1) that, at the steady state equilibrium, the economy is characterized by K˙ t = C˙ t = 0, thereby determining a unique pair of stationary values { K ∗ , C ∗ }. Thus, we can compute the Jacobian matrix of (1) evaluated at the steady state. The trace and determinant of the Jacobian are given by:
Tr( J ) =
(ρ + δ)[b(1 + θ2 ) − (1 + θ1 )(1 + χ )] − δ, {[b(1 + θ2 )/λ] − (1 + χ )}λ
Det( J ) = −
ψ(ρ + δ)(1 + χ )[ρ + δ(1 − a)] , a{[b(1 + θ2 )/λ] − (1 + χ )}
where ψ = [a(1 + θ1 )/λ] − 1 < 0 in order to ensure that capital exhibits a positive and diminishing marginal product. If there are two roots with negative real parts in the dynamic system, the economy will be locally indeterminate. The necessary condition for indeterminacy requires that the determinant of the Jacobian must be positive, i.e.,1 + θ2 > λ(1 + χ )/b. On the other hand, if the effect of increasing returns to variety is absent (by setting N t = 1), our condition will be reduced to that of BF, i.e., 1 + θ2 > (1 + χ )/b. References Basu, S., Fernald, J.G., 1997. Returns to scale in U.S. production estimates and implications. Journal of Political Economy 105, 249–283. Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. Journal of Economic Theory 63, 19–41. Benhabib, J., Farmer, R.E.A., 1996. Indeterminacy and sector-specific externalities. Journal of Monetary Economics 37, 421–443. Bennett, R.L., Farmer, R.E.A., 2000. Indeterminacy with non-separable utility. Journal of Economic Theory 93, 118–143.
3 4
See Hansen (1985) for the relevant discussion. The calculated values in our numerical analysis satisfy the necessary and sufficient conditions for local indeterminacy.
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Bernard, A.B., Redding, S.J., Schott, P.K., 2006. Multi-product firms and product switching. NBER working paper no. 12293. Buiter, W.H., 1984. Saddlepoint problems in continuous time rational expectations models: A general method and some macroeconomic examples. Econometrica 52, 665–680. Burnside, C., 1996. Production function regressions, returns to scale, and externalities. Journal of Monetary Economics 37, 177–201. Chatterjee, S., Cooper, R., Ravikumar, B., 1993. Participation dynamics: Sunspots and cycles. NBER working paper no. 3438. Devereux, M.B., Head, A.C., Lapham, B.J., 1996. Monopolistic competition, increasing returns, and the effects of government spending. Journal of Money, Credit, and Banking 28, 233–254. Devereux, M.B., Head, A.C., Lapham, B.J., 2000. Government spending and welfare with returns to specialization. Scandinavian Journal of Economics 102, 547–561. Dos Santos Ferreira, R., Dufourt, F., 2006. Free entry and business cycles under the influence of animal spirits. Journal of Monetary Economics 53, 311–328. Farmer, R.E.A., Guo, J.-T., 1994. Real business cycles and the animal spirits hypothesis. Journal of Economic Theory 63, 42–72. Galí, J., 1994. Monopolistic competition, business cycles, and the composition of aggregate demand. Journal of Economic Theory 63, 73–96. Guo, J.T., 2004. Increasing returns, capital utilization, and the effects of government spending. Journal of Economic Dynamics and Control 28, 1059–1078. Guo, J.T., Harrison, S.G., 2001. Indeterminacy with capital utilization and sector-specific externalities. Economics Letters 72, 355–360. Guo, J.T., Lansing, K.J., 2005. Maintenance expenditures and indeterminacy under increasing returns to scale. Discussion paper. Hall, R.E., 1988. The relationship between price and marginal cost in U.S. industry. Journal of Political Economy 96, 921–947. Hall, R.E., 1990. Invariance properties of Solow’s productivity residual. In: Diamond, P. (Ed.), Growth, Productivity, Unemployment. MIT Press, Cambridge, MA, pp. 71–112. Hansen, G.D., 1985. Indivisible labor and the business cycle. Journal of Monetary Economics 16, 309–327. Harrison, S.G., 2001. Indeterminacy in a model with sector-specific externalities. Journal of Economic Dynamics and Control 25, 747–764. Judd, K.L., 1997. The optimal tax rate for capital income is negative. NBER working paper no. 6004. Morrison, C.J., 1990. Market power, economic profitability and productivity growth measurement: An integrated structural approach. NBER working paper no. 3355. Peck, J., Shell, K., 1991. Market uncertainty: Correlated and sunspot equilibria in imperfectly competitive economies. Review of Economic Studies 58, 1011– 1029. Perli, R., 1998. Indeterminacy, home production and the business cycle: A calibrated analysis. Journal of Monetary Economics 41, 105–125. Roeger, W., 1995. Can imperfect competition explain the difference between primal and dual productivity measures? Estimates for U.S. manufacturing. Journal of Political Economy 103, 316–330. Schmitt-Grohé, S., 1997. Comparing four models of aggregate fluctuations due to self-fulfilling expectations. Journal of Economic Theory 72, 96–147. Weder, M., 2000. Animal spirits, technology shocks and the business cycle. Journal of Economic Dynamics and Control 24, 273–295. Wen, Y., 1998. Capacity utilization under increasing returns to scale. Journal of Economic Theory 81, 7–36.
Contents lists available at ScienceDirect
Review of Economic Dynamics www.elsevier.com/locate/red
Monopoly power, increasing returns to variety, and local indeterminacy Juin-jen Chang a,1 , Hsiao-wen Hung b,∗ , Chun-chieh Huang c a b c
Institute of Economics, Academia Sinica, Department of Economics, Fu-Jen Catholic University, Taiwan Department of Industrial Economics, Tamkang University, Tamsui, Taipei County 251, Taiwan Department of Economics, Fu-Jen Catholic University, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 2 September 2008 Revised 12 September 2009 Available online 14 October 2009 JEL classification: E00 E32 L16 Keywords: Local indeterminacy Increasing returns to variety Endogenous entry
The required degree of increasing returns-to-scale to satisfy the Benhabib–Farmer condition for local indeterminacy is too high to be empirically plausible. In the paper, we develop a natural extension of Benhabib and Farmer’s (1994) model by introducing aggregation increasing returns to variety. It is shown that an increase in the degree of monopoly power can create increasing returns to an expansion in variety which decreases reliance on the degree of increasing returns-to-scale in production for generating local indeterminacy. As the degree of monopoly power increases, the required degree of increasing returns for local indeterminacy decreases monotonically. As a result, our numerical analysis indicates that the required degree of increasing returns for local indeterminacy can be easily located in the empirically plausible range. © 2009 Elsevier Inc. All rights reserved.
1. Introduction Since the seminal paper of Benhabib and Farmer (1994), henceforth BF, there have been a vast number of articles in the macroeconomic literature that have established the existence of indeterminate equilibrium paths. BF argued that an indeterminate steady state can be generated in a one-sector model of business fluctuations driven by self-fulfilling beliefs, provided that the increasing returns are sufficiently strong. This argument has attracted widespread criticism due to the degree of increasing returns-to-scale needed to satisfy the Benhabib–Farmer condition for local indeterminacy being too high to be empirically plausible. In their model, the degree of increasing returns-to-scale required in production for generating local indeterminacy is about 1.5. However, Burnside (1996) found the estimate of internal returns to be about 1.33, although the regressions were even more sensitive to the instrument set. According to the estimation of Basu and Fernald (1997), for the U.S. private business economy the empirically plausible degree of increasing returns-to-scale in production is between 1.03 and 1.18. Obviously, the degree of increasing returns-to-scale required to satisfy the Benhabib–Farmer condition for local indeterminacy is far higher based on the empirically plausible values. To fix the question, subsequent research has incorporated various elements into the models in order to push down the required degree of increasing returns for local indeterminacy to empirically plausible values. By extending the one-sector model of BF to a multiple-sector model, Benhabib and Farmer (1996), Perli (1998), Weder (2000), and Harrison (2001) show that sector-specific externalities lead the minimum level of increasing returns needed for indeterminacy to become less stringent. Wen (1998) and Guo and Harrison (2001) incorporate endogenous capital utilization into RBC models that lead to
* 1
Corresponding author. Fax: +886 2 26209731. E-mail address: [email protected] (H.-w. Hung). We would like to thank an associate editor of this journal for his/her insightful comments and suggestions. The usual disclaimer applies.
1094-2025/$ – see front matter doi:10.1016/j.red.2009.10.007
© 2009 Elsevier Inc.
All rights reserved.
J.-j. Chang et al. / Review of Economic Dynamics 14 (2011) 384–388
385
indeterminacy for smaller magnitudes of external effects than the BF model. In addition, Guo and Lansing (2005) propose a model calibrated to match empirical evidence on the relative size of maintenance and repair activity and show that local indeterminacy can occur with a mild and empirically-plausible degree of increasing returns — of around 1.08. By following this line of research, this paper develops a natural and straightforward extension of the BF model by endogenizing the number of producers (and products). Endogenous entry results in aggregation increasing returns to variety and, consequently, generates an endogenous propagation mechanism for local indeterminacy. Such an extension is convincing and important. As shown by Dos Santos Ferreira and Dufourt (2006), different levels of economic activities sustained by different numbers of active firms are consistent with the conditions of free entry and contestability. Furthermore, the important contribution of product creation and destruction to aggregate output and business fluctuations has also been proved by the empirical study of Bernard et al. (2006). In this paper we show that endogenous entry under monopolistic competition leads to aggregation increasing returns to variety and this reduces the minimum value of the labor externality needed to generate indeterminacy. Of importance, if the degree of monopoly power increases, the required degree of increasing returns for local indeterminacy decreases monotonically. Thus, local indeterminacy can occur with a very mild degree of increasing returns as long as the price markup ratio (the price–marginal cost ratio) is high enough. Specifically, we estimate that if the markup ratio is calibrated to around 1.4, the required degree of increasing returns-to-scale can be pushed down to 1.01429. This value is located within the range of the empirically plausible degree of increasing returns-to-scale. A particular emphasis is that the calibrated markup value is plausible in practice. Hall (1988, 1990) examines U.S. industry data on output and the labor input and finds that the estimated value of the markup ratio is between 1.864 and 3.791. By using data on value added, the estimate of the markup ratio is greater than 1.5 for all 7 1-digit industries and for 17 of the 21 2-digit industry groups. Morrison’s (1990) estimate shows that the average annual markups of U.S. manufacturing industries are between 1.179 and 1.803. By extending Hall’s (1990) data set, Roeger (1995) finds that the estimated markup ratios range from 1.19 to 3.14. Some studies have examined the role of monopolistic competition as a possible source of sunspot equilibria in one-sector models. Although Farmer and Guo (1994) incorporate the presence of monopolistic competition in one of the versions of their models, it is only necessary to sustain increasing returns at the level of the firm, thus ensuring a non-negative profit in the presence of increasing returns to scale in production. Given that the number of producers is given exogenously, monopoly power does not play a role in generating local indeterminacy. Schmitt-Grohé (1997) has also considered the effect of entry and exit in his survey paper. However, the increasing returns to variety have been abstracted from his study. Besides, the role of monopolistic competition has also been highlighted in various two sector models. Peck and Shell (1991) prove that the sunspot equilibria exist in a pure exchange-economy in which agents have market power in both the commodity and securities markets. Chatterjee et al. (1993) analyze a two-good OLG model and show the possibility of sunspot equilibria in the presence of sufficiently strong complementarities between the two-good sectors. Galí (1994) argues that sunspot fluctuations may arise when the elasticity of substitution across goods in consumption differs from that across goods in production. Apparently, our paper is different from theirs both in terms of its indeterminacy mechanism and numerical intention. In a simple one-sector model we indicate how endogenous entry leads to increasing returns to variety and, as a result, how monopoly power reduces the minimum value of the labor externality needed to generate indeterminacy. 2. Monopolistic markup and equilibrium indeterminacy 2.1. Endogenous entry This study simply extends the BF model to allow for endogenous entry and we describe the model briefly.2 In the economy considered there are two types of goods: a homogeneous final good, which is the numeraire and is produced by competitive firms, as well as differentiated intermediate goods indexed by i = 1, . . . , N t , where N t is the number of intermediate goods at time t. Production differentiation allows monopolistic competition to exist between the intermediate-goods N producers. By defining y it as the quantity of input i, the final good production technology is given by: Y t = ( 0 t y λit di )1/λ ; λ ∈ (0, 1) which displays, as stressed by Devereux et al. (1996, 2000), increasing returns to an expansion in variety (specialization) with the degree of (1/λ): under endogenous entry the larger the number N t of intermediate goods, the higher the amount of final production obtainable from a given amount of initial resources. A final good firm solves the profit maximization problem to determine its demand for the ith intermediate good, i.e., p it = (Y t / y it )1−λ which has a constant price elasticity 1/(1 − λ). Obviously, λ not only measures the degree of monopoly power, but also determines the degree of returns to variety. In line with BF, intermediate-goods producer i uses capital kit aθ bθ and labor h it to produce a product using the technology: y it = kait hbit ( K t 1 H t 2 ) − φ ; a + b = 1, where φ is the unit fixed cost, K t and H t represent the aggregate levels of capital and labor, and θ1 and θ2 are the corresponding externality parameters. Subject to the demand function and the production function above, we can easily derive the first-order conditions for h it and kit . By focusing on a symmetric equilibrium, the zero-profit and the aggregate consistency conditions allow us to derive (1−λ)/λ the following price of intermediate goods, the wage rate and the rental rate, respectively: pt = N t , w t = bY t / H t , and 2
A detailed mathematical deduction is available upon request. We also refer the reader to our working paper for a complete derivation.
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J.-j. Chang et al. / Review of Economic Dynamics 14 (2011) 384–388 a(1+θ )
b(1+θ )
a(1+θ )/λ
b(1+θ )/λ
1 2 1 2 rt = aY t / K t , where N t = [(1 − λ)/φ] K t Ht , Y t = N t yt = Ω K t Ht , and Ω = λ[(1 − λ)/φ](1−λ)/λ . It is clear from the above equations that the endogenous number of firms in the intermediate goods sector will lead to aggregation increasing returns to variety. Consider a standard RBC model with separable utility under which households face the following optimization problem: N ∞ 1+χ maxct ,ht ,kt 0 [ln ct − ht /(1 + χ )]e −ρ t dt, subject to k˙ t = w t ht + (rt − δ)kt + 0 t πit di − ct , where ct is consumption, ρ is the rate of time preference, πit are profits and δ is the capital depreciation rate. Solving the household’s problem above yields the first-order conditions in terms of the aggregate variables and, accordingly, we have the Keynes–Ramsay rule and the economy-wide resource constraint as follows:
C˙ t Ct
= rt − ρ − δ = a
Yt Kt
1/λ
− ρ − δ and K˙ t = Y t − δ K t − C t , a(1+θ1 )/λ
with the instantaneous relationship of labor H t = [bΩ K t
(1)
/C t ]1/[1+χ −b(1+θ2 )/λ] .
2.2. Local indeterminacy Eq. (1) constructs the 2 × 2 dynamic system in terms of C t and K t . As addressed in the literature on dynamic rational expectations models, e.g., Buiter (1984), the dynamic system will have a unique perfect-foresight equilibrium path (saddlepath stability) if the number of (positively) unstable roots equals the number of jump variables. In this dynamic system there is only one jump variable C t . Accordingly, BF argued that if the dynamic system has two roots with negative real parts, there exists a continuum of equilibrium trajectories that converges to the steady state (sink) and, accordingly, local indeterminacy emerges in the economy. That is, a necessary (but not sufficient) condition refers to a positive value of the determinant of the Jacobian. With this understanding, we establish the following proposition: Proposition 1. When the number of firms is determined endogenously, the necessary condition for indeterminacy is 1 + θ˜2 > λ(1 + χ )/b. However, if the number of firms is given exogenously, the necessary condition for indeterminacy turns out to be 1 + θˆ2 > (1 + χ )/b, which is the same as the Benhabib–Farmer condition. Proof. See the Appendix A.
2
Similar to BF and Farmer and Guo (1994), for local indeterminacy, the necessary condition implies that the equilibrium wage-hours locus needs to be positively sloped and steeper than the labor supply curve. However, in their model local indeterminacy requires an implausibly higher degree of increasing returns-to-scale in production (see Burnside, 1996, and Basu and Fernald, 1997). Proposition 1 clearly points out that the required degree of increasing returns-to-scale in production can be substantially reduced when free entry endogenizes the number of firms, leading to aggregation increasing returns to variety. Of importance, if the market is characterized by less competition (a lower value of λ), the degree of aggregation increasing returns to variety becomes larger and, as a result, the necessary condition for local indeterminacy is easier to satisfy. Due to λ < 1 (and hence θ˜2 < θˆ2 ), our model can escape from the widespread criticism incurred by the Benhabib–Farmer condition. We can glean the intuition for the necessary condition for generating local indeterminacy by examining the Keynes– Ramsey rule (1), which is rewritten as the following discrete-time function for ease of illustration:
C t +1 Ct
=
= r t +1 − ρ − δ ⎧ ⎨ AK {a(1+θ1 )(1+χ )/[1+χ −b(1+θ2 )]}−1 C −b(1+θ2 )/[1+χ −b(1+θ2 )] − ρ − δ, t +1
t +1
if N t +1 = 1,
⎩ BK {a(1+θ1 )(1+χ )/λ[1+χ −b(1+θ2 )/λ]}−1 C −b(1+θ2 )/λ[1+χ −b(1+θ2 )/λ] − ρ − δ, if N t +1 = t +1 t +1
(1−λ) a(1+θ1 ) b(1+θ2 ) H t +1 , φ K t +1
(2) b(1+θ2 )/[1+χ −b(1+θ2 )] (1+χ )/[1+χ −b(1+θ2 )]
b(1+θ2 )/λ[1+χ −b(1+θ2 )/λ]
Ω (1+χ )/[1+χ −b(1+θ2 )/λ] and the condition
where A = ab λ , B = ab a(1 +θ1 )(1 + χ )/λ[1 + χ − b(1 +θ2 )/λ] < 1 holds because sustained endogenous growth is not allowed in our analysis. Notice that we normalize the number of firms to unity when focusing on the case where the number of firms is exogenously given. Under such a situation, the necessary condition of BF for generating local indeterminacy will be recovered. Suppose that agents become optimistic about next period’s return on capital rt +1 . In acting upon this belief, the household will sacrifice consumption today (reducing C t ) for more investment (raising K t +1 ) and higher future consumption (C t +1 ). As a result, the value of the left-hand side of (2) increases. Under constant returns-to-scale in production, a higher capital stock is associated with a lower rate of return (∂(rt +1 )/∂ K t +1 < 0), and therefore agents’ expectations cannot be self-fulfilling because the value of the right-hand side of (2) falls. To validate the agents’ optimistic expectations as a self-fulfilling equilibrium, the righthand side of (2) should also increase, i.e., the next period’s return on capital must be increasing in K t +1 . As indicated in (2), this implies that the necessary condition requires that 1 + θ2 > (1 + χ )/b if the number of firms N t is exogenously given and normalized to 1 (there are constant returns to the quantity employed of a fixed variety of intermediate goods).
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However, if the number of firms is endogenously determined, monopolistic competition will generate increasing returns to an expansion in variety which reinforces the increased magnitude of the return rate of capital, as shown in (2). In generating local indeterminacy, this decreases reliance on the degree of increasing returns-to-scale in production, i.e., the necessary condition turns out to be 1 + θ2 > λ(1 + χ )/b. Given that λ < 1, our model can push down the required value of the labor externality for generating indeterminacy. It is important to further discuss how effectively aggregation increasing returns to variety can push down the minimum value of the labor externality in order to generate indeterminacy by means of a simple numerical analysis. For the sake of comparison, most parameters we set are identical to those in BF: ρ = 0.065, δ = 0.1, and a = 0.3 (and hence b = 0.7). Moreover, in line with the common specification, for example, Wen (1998), Bennett and Farmer (2000), Harrison (2001) and Guo (2004), the inverse of the labor supply elasticity is set as χ = 0, meaning that the labor supply elasticity is equal to infinity.3 Apparently, a lower degree of monopoly power (λ) or a higher degree of increasing returns-to-scale in production (the value of the labor externality θ2 ) is more likely to result in local indeterminacy. According to the estimation of Basu and Fernald (1997), the empirically plausible degree of increasing returns-to-scale in production is in the region of 1.03 to 1.18. If we follow the BF benchmark parameterization and set λ = 0.667, the required degree of increasing returns-to-scale in production will be reduced to around 0. This implies that local indeterminacy can occur without increasing returns-to-scale in production as long as aggregation increasing returns to variety are present. Judd (1997) set λ ∈ [0.6, 0.9]. Given this parameterization, we have the minimum values of increasing returns-to-scale in production as 1 + θ˜2 = 1.01429, 1 + θ˜2 = 1.0929 and 1 + θ˜2 = 1.1429, if λ = 0.71, λ = 0.76 and λ = 0.8, respectively.4 By focusing on the case where λ = 0.71(i.e., the price markup ratio is around p /mc = 1/λ = 1.4), our estimated result 1 + θ˜2 = 1.01429 is lower than the outcome in existing studies, e.g., 1.074 in Harrison (2001) and 1.064 in Benhabib and Farmer (1996). That is, the required degree of increasing returns-to-scale in production can be substantially reduced and, consequently, becomes empirically plausible when we consider the effect of aggregation increasing returns to variety. Of importance, as stressed in Section 1, is that many empirical studies show that a markup ratio of 1.4 is plausible in practice. See, for example, Hall (1988, 1990), Morrison (1990), and Roeger (1995). 3. Concluding remarks It is well known that the degree of increasing returns-to-scale required to satisfy the Benhabib–Farmer-Guo condition for local indeterminacy is too high to be empirically plausible. In this paper, we have developed a natural extension of the BF model by endogenizing the number of firms. By so doing, we have shown that monopolistic competition leads to aggregation increasing returns to variety and this reduces the minimum value of the labor externality needed to generate indeterminacy. In particular, we have indicated that, as the degree of monopoly power increases, the degree of increasing returns required for local indeterminacy decreases monotonically. Specifically, under an empirically convincing markup value of 1.4, the required degree of increasing returns-to-scale can be pushed down to 1.01429. This is not only located within the range of the empirically plausible degree of increasing returns-to-scale, but is also lower than the outcome in existing studies, e.g., 1.074 in Harrison (2001) and 1.064 in Benhabib and Farmer (1996). Appendix A It follows from (1) that, at the steady state equilibrium, the economy is characterized by K˙ t = C˙ t = 0, thereby determining a unique pair of stationary values { K ∗ , C ∗ }. Thus, we can compute the Jacobian matrix of (1) evaluated at the steady state. The trace and determinant of the Jacobian are given by:
Tr( J ) =
(ρ + δ)[b(1 + θ2 ) − (1 + θ1 )(1 + χ )] − δ, {[b(1 + θ2 )/λ] − (1 + χ )}λ
Det( J ) = −
ψ(ρ + δ)(1 + χ )[ρ + δ(1 − a)] , a{[b(1 + θ2 )/λ] − (1 + χ )}
where ψ = [a(1 + θ1 )/λ] − 1 < 0 in order to ensure that capital exhibits a positive and diminishing marginal product. If there are two roots with negative real parts in the dynamic system, the economy will be locally indeterminate. The necessary condition for indeterminacy requires that the determinant of the Jacobian must be positive, i.e.,1 + θ2 > λ(1 + χ )/b. On the other hand, if the effect of increasing returns to variety is absent (by setting N t = 1), our condition will be reduced to that of BF, i.e., 1 + θ2 > (1 + χ )/b. References Basu, S., Fernald, J.G., 1997. Returns to scale in U.S. production estimates and implications. Journal of Political Economy 105, 249–283. Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. Journal of Economic Theory 63, 19–41. Benhabib, J., Farmer, R.E.A., 1996. Indeterminacy and sector-specific externalities. Journal of Monetary Economics 37, 421–443. Bennett, R.L., Farmer, R.E.A., 2000. Indeterminacy with non-separable utility. Journal of Economic Theory 93, 118–143.
3 4
See Hansen (1985) for the relevant discussion. The calculated values in our numerical analysis satisfy the necessary and sufficient conditions for local indeterminacy.
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