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Implementation cycles, growth and the labor market

  • Patrick Francois EMAIL logo und Huw Lloyd-Ellis
Veröffentlicht/Copyright: 2. Juli 2013
The B.E. Journal of Macroeconomics
Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 13 Heft 1

Abstract

We develop a theory of growth and cycles that endogenously relates job flows, worker flows and wages over the cycle to the processes of restructuring, innovation and implementation that drive long-run growth. Expansions are the result of clustered implementation of new ideas and recessions are the negative consequence of the restructuring that anticipates them. Due to incentive problems, production workers are employed via relational contracts and experience involuntary unemployment. Separation rates and firm turnover are counter-cyclical, but labor productivity growth and hiring rates are procyclical. Our framework also highlights the counter-cyclical forces on wages due to restructuring, and illustrates the relationship between the cyclicality of wages and long-run productivity growth.

Keywords: Endogenous cycles; endogenous growth; employment flows; wage rigidity; JEL codes: E0; E3; O3; O4
Corresponding author: Patrick Francois, Department of Economics, University of British Columbia, Vancouver, BC, Canada, CEPR and CIFAR, e-mail:
  1. 1

    As developed by Aghion and Howitt (1992), Grossman and Helpman (1991) and Segerstrom, Anant, and Dinopoulos (1990).

  2. 2

    A positive value of φ is not necessary to generate equilibrium cycles. With φ=0 there would be no growth during the first phase of the cycle that we describe below, but nothing else of qualitative difference.

  3. 3

    This view was shared by Schumpeter (1950, 132): “ … The function of entrepreneurs is to reform or revolutionize the pattern of production by exploiting an invention or, more generally, an untried technological possibility … This function does not essentially consist in either inventing anything or otherwise creating the conditions which the enterprise exploits. It consists in getting things done.”

  4. 4

    Introducing a moral hazard problem at this level as well would add unnecessary complexity without changing the qualitative implications.

  5. 5

    We assume throughout that incoming firms do not hire workers directly from the incumbents they replace. Allowing for partial transitions directly to new employers would de-couple plant level destruction from job destruction but changes nothing qualitatively.

  6. 6

    We leave unspecified the precise means by which this occurs and simply assume that the leading technology is the exclusive province of the innovator. Once the innovation has been superceded, others may enter and use it at will, since, as it can no longer generate rents, the innovator controlling it no longer has incentive to limit its use elsewhere.

  7. 7

    Note that the implied intertemporal elasticity of substitution exceeds unity. This feature can be relaxed in a model with physical capital (see Francois and Lloyd-Ellis 2008), but is essential here.

  8. 8

    Appendix B formally states this game’s information sets, timing and strategies, and shows that the posited behavior between the worker and firm are equilibrium strategies.

  9. 9

    This positive impact on incentive compatible wages of increased job loss chances has been noticed before in similar relational contract frameworks by Saint-Paul (1996), and Fella (2000), with evidence consistent with this being found in Abowd and Ashenfelter (1981) and Adams (1985).

  10. 10

    We acknowledge that this force is much less likely to be of quantitative importance in reality. Though we carry discussion of it through the analysis, the most important factor is likely to be the effect of relationship stability on wages.

  11. 11

    Throughout, we use the subscript 0 to denote the value of a variable immediately after the boom. Formally, X0(T)=limt→T+X(t).

  12. 12

    As Cohen, Nelson and Walsh (2000) document, delaying implementation to protect knowledge is a widely followed practice in reality. Graham (2004) also describes a similar critical role of secrecy in protecting value. Scaromozino, Temple, and Vulkan (2005) suggest indirect evidence of delay can be gleaned from Hobijn and Jovanovic’s (2001) evidence of major changes in US stock valuations anticipating productivity changes. Also, Pastor and Veronesi (2005) provide evidence arguing that the state of the macro economy determines decisions on IPO timing. This is also in line with delay playing a key role.

  13. 13

    The payment does not have to be a literal transfer for technology but would instead more realistically take the form of the entrant renting the incumbent’s machines, plant and production methods for the remainder of the recession. For example, this could occur at a fire-sale over the incumbent firm’s assets. After that, the entrant will implement his own methods so that the rental rate on such equipment is zero anyway. The important point is that the purchase of the assets at a price that would only be worthwhile to an entrant with a valuable innovation sends a credible signal to other innovators that profits will be higher if innovations are targetted elsewhere.

  14. 14

    To see this observe that (E1) can be expressed as which holds only if Re-arranging yields (57).

  15. 15

    During the recession, job destruction is measured as and job creation as n(t)δH(t).

  16. 16

    The wage never declines in this numerical case but can do so for other parameter values.

  17. 17

    Note that the jump in average wages upon entering the recession also contributes to overall acyclicality but is much more perverse. It arises because household employment growth, which was previously zero, becomes negative. Consequently, the marginal cost of providing effort is expected to be lower tomorrow than today which, on the margin, increases the willingness of workers to risk unemployment by shirking today. To compensate for this effect, firms must raise the wage in order to maintain incentives. This admittedly is a perverse effect of the forward looking efficiency wage framework heading into the recession. It is unlikely that such a force could be playing a significant role in reality.

Appendix

Proof of Lemma 1 Consider a firm holding an obsolete technology at time t. For there to exist a technology better than the firm’s, an innovator must have allocated h to innovation in the firm’s sector. But only firms planning to implement an innovation will find it worthwhile to undertake innovative effort, consequently, for the new technology, there exists some planned optimal date at which the technology will be implemented. Denote this date by t*. Using the large scale of production requires hiring workers who do not shirk. This is only possible if the wage is incentive compatible. Clearly, at t* the firm using the obsolete technology is not able to compete in production, so the large scale of production cannot be used. However it is also the case that incentive compatibility cannot hold, and the large scale of production not be used, if employment terminates with probability one in the next instant. Consequently at time t*–dt it is also not possible to use the large scale technology. However, the same argument applies at time t*–2dt since the worker then knows that at t*–dt employment will be terminated. The same reasoning applies for each instant until time t.

Proof of Lemma 2 From the production function we have Substituting for pi(t) using (23) re-arranges to

Proof of Lemma 3 Profit maximization implies that and so It follows that the Bellman equation associated with a worker who is employed and not shirking in sector i is given by

Similarly the Bellman equations associated with workers who are shirking in sector i and unemployed, respectively, can be expressed as

Profit maximization subject to the incentive compatibility condition implies that and so subtracting (63) from (64) we get

It follows that and that

Subtracting (65) from (63), substituting using (66) and re-writing yields (27). ■

Proof of Proposition 4 Substituting for c(t) from (29), setting and noting that when dn=0 and equation (27) rearranges to (32).

Proof of Lemma 5 Note that in any preceding no-entrepreneurship phase, r(t)=ρ+σφ. Thus, since, in a cycling equilibrium, the date of the next implementation is fixed at Tv, the expected value of entrepreneurship, δVD, also grows at the rate ρ+σφ>0. Thus, if under then the same inequality is also true the instant before, i.e., at since s(t) grows at the slower rate, φ<ρ+σφ, within the cycle. But this violates the assertion that entrepreneurship commences at Thus necessarily, at

Proof of Proposition 7 From (6) we can express the rate of job creation as

Using the fact that and substituting into (42) yields

Differentiating (35) yields

Using (83) and (35) we have that

Differentiating (12) w.r.t. time and using (34) to substitute yields

Noting that substituting into (69) using (70), (71) and (72) and re-arranging yields (42).

Proof of Proposition 8 Long-run productivity growth is given by

Substituting using (36) and integrating yields (45).

Proof of Proposition 9 Just prior to the boom, when the probability of displacement is negligible, the value of implementing immediately must equal that of delaying until the boom:

During the boom Thus the return to innovation at the boom is the value of immediate incumbency. It follows that free entry into entrepreneurship at the boom requires that

The opportunity cost to financing entrepreneurship is the rate of return on shares in incumbent firms in sectors where no innovation has occurred. Just prior to the boom, this is given by the capital gains in sectors where no innovations have occurred. Combined with (74) and (75) it follows that asset market clearing at the boom requires

Provided that R0(Tv)–R(t)>0, households will not store final output from within a cycle to the beginning of the next. However, the return on stored intermediate output in sectors with no innovations is strictly positive because its price increases at the boom. If innovative activities are to be financed at time t, households cannot be strictly better off buying claims to stored intermediate goods. Consider a sector where no innovation has occurred just prior to the boom. Since the cost of production is the same whether the good is stored or not, the rate of return on claims to stored intermediates in sector i is log pi,v+1/pi,v=Γv. It follows that the long run rate of return on claims to firm profits an instant prior to the boom must satisfy

Combining (76) and (77) yields the equality in the statement of proposition.

Proof of Proposition 10 Message space: At e→0 cost, an innovator targeting sector i, which has not yet had a signal of search success, can either send a public signal of success Zi(t)=1, or send no signal in which case Zi(t)=0. Since search occurs in continuous time we ignore the zero probability event of more than one innovation arriving simultaneously.

Description of equilibrium behavior: If, and only if, an entrepreneur succeeds he sends a signal of success immediately. If, and only if, an incumbent sees Zi(t)=1, he shuts down production. If, and only if, another entrepreneur targeting sector i sees Zi(t)=1 he subsequently targets another sector j in which Zj(t)=0.

Optimality of signal sender’s equilibrium behavior: During the phase in which search occurs, VI(t)<VD(t). Consequently, an entrepreneur would prefer delaying implementation of innovation to time T0. Suppose that at time t an incumbent sees Zi(t)=1. Then either an entrepreneur has succeeded and sent a signal, or has not succeeded and is falsely sending a signal of success. If an entrepreneur has been successful, expected returns when sending the signal are: VD(t)–e. This exceeds the return to immediate implementation, VI(t).

If the entrepreneur has not been successful, but is sending a signal falsely, then expected returns to innovation are δVD(t)–e. However, if the false sender of the signal were to follow the posited equilibrium behavior instead, and delay sending a signal until if and when a success arrives, then expected returns to search would be δ(VD(t)–e)>δVD(t)–e. Consequently, an unsuccessful entrepreneur would never falsely send a signal of success, and a successful entrepreneur strictly prefers sending a signal of success.

Optimality of recipient’s equilibrium behavior: Given equilibrium described by strategies for workers and incumbents in production, when Zi(t)=1, the incumbent firm knows that, in all subsequent periods, any positive wage offer to any worker j will lead to shirking. Consequently the firm strictly prefers to shut down production.

Optimality of unsuccessful entrepreneur’s behavior: Given that Zi(t)=1, a non-innovator conjectures that an innovation success has arrived in sector i. Consequently, if the entrepreneur were to subsequently succeed in sector i, would equal 0, since the entrepreneur would Bertrand compete with an incumbent holding an identical technology. Consequently, provided there exists another sector j, with and with Zj(t)=0, the entrepreneur will target sector j subsequently.

Proof of Proposition 11 Although, during the downturn, the realized profits of firm differ across sectors, the fact that profits are linear in the production wage implies that expected profits at time are equivalent to the profits of a firm that pays the average wage for (see Appendix B for details). This implies that the value of a firm at time Tv-1 can be expressed as

Substituting in for the endogenous variables, integrating and dividing through by Y0(Tv–1) yields

where b=ρ–(1–σ)φ. Noting that and substituting using (45) and (50) we can derive

Dividing through by and solving for Δ yields (55), where

and

Appendix B: Not-for-Publication Appendix

Formal Statement of Incumbent Firm/Worker game:

When considering the worker/firm game we proceed under the assumption that any signal of success in an incumbent firm’s sector is a credible signal of a successful innovation on the part of the signal sender. This is an equilibrium condition of the model, which we take as given for now, but the formal proof of this is shown in the proof of Proposition 5.

Consider an incumbent firm in sector i and a worker j. The labor market is anonymous in the sense that, the history of a worker’s effort choices prior to being employed at the current firm is not known to the firm. The incumbent firm only knows the history of this worker’s effort choices while employed with the current incumbent firm. The history of worker j, denoted Hj(t), summarizes the effort decision of worker j in each instant from the time he was employed in the firm tj to the present, t. That is, it summarizes the value of ej(t) ∀t∈[tj,t⌋. Equilibrium strategies will vary conditionally on this history in one of two ways. Case 1: The worker did not work for the firm in any previous period, or the worker did work for the firm and set ej(t)=1 ∀t∈[tj,t⌋. We denote this case as history Hj(t)=1. Case 2: ∃ at least one t∈[tj,t⌋ such that ej(t)=0. We denote this by Hj(t)=0.

Public Signals: Zi(t)=1, if, at time t, there has not been a signal of an innovation success in sector i subsequent to the success signal of the current incumbent firm i. Zi(t)=0 otherwise.

Actions: Incumbent Firm: Conditional on no exogenous seperation between incumbent firm, i and worker j, at time t, firm decides whether to rehire worker j, Rij(t)=1, or dismiss, Rij(t)=0. If rehired, firm decides on wage wij(t).

Worker: Conditional on no exogenous seperation between incumbent firm, i and worker j, at time t, worker observes wage and decides whether to remain with the firm Rji(t)=1, or leave Rji(t)=0, and, whether to set ej(t)=1 or 0.

Strategies: Incumbent firm: is a mapping from the worker’s history Hj and the public signals Zi received in that sector, to the firm’s actions (Rij(t), wij(t)).

Worker: is a mapping from the worker’s history Hj and the public signals Zi received in that sector, to the worker’s action set (Rji(t), ej(t)).

Equilibrium: Firm Behavior: At time t, given that worker j has not shirked in any previous period of employment with the firm and given that there has not been a signal of a new innovation success in sector i, the firm offers the worker a binding incentive compatible wage, computed in equation (30). For any other history, or if a success has been signalled in the sector, the firm does not offer worker j employment subsequently.

Worker Behavior: At time t, given that worker j has not shirked in any previous period of employment with the firm and given that there has not been a signal of a new innovation success in sector i, and given that the offered wage is at least equal to the worker accepts a position in the firm and does not shirk. For any other history, or if an innovation success has been signalled, the worker accepts the employment offer and shirks.

Specification of equilibrium strategies: Firms, i, follow in dealing with any worker j and workers j follow dealing with any firm i.

Denote wi(t) defined in equation (30) as

Incumbent firm: such that,

Worker: such that

Proof that these strategies constitute a sub-game perfect equilibrium.

From the worker’s perspective given that (Hj(t), Zi(t))=(1, 1), is the incentive compatible wage, since the incumbent firm’s strategy, specifies re-hiring if workers do not shirk. Consequently workers will not deviate to shirking along the equilibrium path of play.

Consider a deviation by the firm. Specifically, suppose that (Hj(t), Zi(t))=(1, 1) but the firm sets Clearly, given any wage below will lead the worker to shirk, and wages strictly above reduce profit, so, conditional up re-hiring the worker, is optimal for the firm. Consider a once off deviation given history (Hj(t), Zi(t))=(1, 1) in which the firm does not re-hire the worker, and hires another worker instead. The firm then draws a worker randomly from the employment pool. Such a worker also has a history Hj(t)=1, so that assuming that the firm continues with the equilibrium strategies thereafter (this will be optimal given these are a best response), the firm sets for this worker. Then, since the newly hired worker has Hj(t)=1, and is also following this worker will not shirk if and only if so that profits cannot increase under this deviation. Clearly if (Hj(t), Zi(t))=(1, 1) and the firm neither rehires nor replaces the worker, profit falls, since output falls and this is also not a worthwhile deviation. Now suppose that Hj(t)·Zi(t)≠1. According to worker j will shirk for any w. This is also a best response of workers in this sub-game because, under the worker correctly conjectures that he will not be hired subsequent to the present instant. Consequently, his best response in any subgame where he is offered a positive wage and where Hj(t)·Zi(t)≠1 is to shirk. Consequently, from the perspective of the firm, a deviation to rehiring and offering any positive wage yields strictly lower profit. Since w* has already been calculated to satisfy worker incentive compatibility, is also a best response for the worker.

Proof of Lemma 6 The Euler equation can be expressed as

Re-arranging and integrating yields

Solving for n(t) yields (35).

Comment on Proof of Proposition 9 In sectors with unimplemented innovations, entrepreneurs who hold innovations and are currently producing with the previous technology have the option of implementing the new technology before the boom, storing and then selling at the boom. Any such path of implementation will, however, affect the incentive compatible wage stream offered to production workers. Intuitively, producing and storing today for sale tomorrow implies a higher rate of layoffs tomorrow and hence a higher efficiency wage today. This upward effect on incentive compatible wages is what rules out such storage as a profitable option. Since the stream of revenue is unaffected by such storage, in order for firms’ expected discounted profits to rise it must be that the discounted wage bill for production workers falls. To see this clearly, consider the expected value of the firm with an innovation at the beginning of the cycle, VI(Tv). Since Y(·), s, P(·) and all discount rates are taken as given by the firm, VI can only rise if the value of being employed at the firm ψE(Tv) falls. But at all instants, the incentive compatible wage, wL(τ) is given by the solution to Thus any such storage which raises profits for the firm will necessarily violate incentive compatibility for unskilled workers and will lead to shirking. The same argument rules out altering the production stream to benefit from discrete jumps in the wage anywhere along the cycle.

Further details of Proposition 11 The value of an incumbent firm over a small interval dt during the cycle is

Subtracting from both sides, dividing by dt and letting dt→0 yields

Given some initial t, and Tv, the solution to this first order differential equation is

For r(τ)=φ and h(s) P(s)=H(s). Hence

Note that by partial integration

But V*(Tv)=0 and de–φ(s–t)V*(s)=–e–φ(s–t)πi(wL(s), s)ds, so that

Since profits are a linear function of the production wage we get ∀

Now if and this can be expressed as

This confirms that the expected profits at time are equivalent to the profits of a fim that pays the average wage for This implies that the value of a firm at time Tv–1 can be expressed as

Using (54) and re-arranging we get

where it is analytically convenient to let

b=ρ-(1–σ)φ.

Integrating and dividing through by Y0(Tv–1) yields

Collecting terms

Substituting using (45) and (50) we get

Collecting terms we get (79).

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Published Online: 2013-07-02
Published in Print: 2013-01-01

©2013 by Walter de Gruyter Berlin Boston

Artikel in diesem Heft

  1. Masthead
  2. Masthead
  3. Advances
  4. How have global shocks impacted the real effective exchange rates of individual euro area countries since the euro’s creation?
  5. Employment by age, education, and economic growth: effects of fiscal policy composition in general equilibrium
  6. Overeducation and skill-biased technical change
  7. Strategic wage bargaining, labor market volatility, and persistence
  8. Households’ uncertainty about Medicare policy
  9. Contributions
  10. Deconstructing shocks and persistence in OECD real exchange rates1)
  11. A contribution to the empirics of welfare growth
  12. Development accounting with wedges: the experience of six European countries
  13. Implementation cycles, growth and the labor market
  14. International technology adoption, R&D, and productivity growth
  15. Bequest taxes, donations, and house prices
  16. Business cycle accounting of the BRIC economies
  17. Privately optimal severance pay
  18. Small business loan guarantees as insurance against aggregate risks
  19. Output growth and unexpected government expenditures
  20. International business cycles and remittance flows
  21. Effects of productivity shocks on hours worked: UK evidence
  22. A prior predictive analysis of the effects of Loss Aversion/Narrow Framing in a macroeconomic model for asset pricing
  23. Exchange rate pass-through and fiscal multipliers
  24. Credit demand, credit supply, and economic activity
  25. Distortions, structural transformation and the Europe-US income gap
  26. Monetary policy shocks and real commodity prices
  27. Topics
  28. News-driven international business cycles
  29. Business cycle dynamics across the US states
  30. Required reserves as a credit policy tool
  31. The macroeconomic effects of the 35-h workweek regulation in France
  32. Productivity and resource misallocation in Latin America1)
  33. Information and communication technologies over the business cycle
  34. In search of lost time: the neoclassical synthesis
  35. Divorce laws and divorce rate in the US
  36. Is the “Great Recession” really so different from the past?
  37. Monetary business cycle accounting for Sweden
https://doi.org/10.1515/bejm-2012-0137
Schlagwörter für diesen Artikel
Endogenous cycles; endogenous growth; employment flows; wage rigidity; JEL codes: E0; E3; O3; O4

Artikel in diesem Heft

  1. Masthead
  2. Masthead
  3. Advances
  4. How have global shocks impacted the real effective exchange rates of individual euro area countries since the euro’s creation?
  5. Employment by age, education, and economic growth: effects of fiscal policy composition in general equilibrium
  6. Overeducation and skill-biased technical change
  7. Strategic wage bargaining, labor market volatility, and persistence
  8. Households’ uncertainty about Medicare policy
  9. Contributions
  10. Deconstructing shocks and persistence in OECD real exchange rates1)
  11. A contribution to the empirics of welfare growth
  12. Development accounting with wedges: the experience of six European countries
  13. Implementation cycles, growth and the labor market
  14. International technology adoption, R&D, and productivity growth
  15. Bequest taxes, donations, and house prices
  16. Business cycle accounting of the BRIC economies
  17. Privately optimal severance pay
  18. Small business loan guarantees as insurance against aggregate risks
  19. Output growth and unexpected government expenditures
  20. International business cycles and remittance flows
  21. Effects of productivity shocks on hours worked: UK evidence
  22. A prior predictive analysis of the effects of Loss Aversion/Narrow Framing in a macroeconomic model for asset pricing
  23. Exchange rate pass-through and fiscal multipliers
  24. Credit demand, credit supply, and economic activity
  25. Distortions, structural transformation and the Europe-US income gap
  26. Monetary policy shocks and real commodity prices
  27. Topics
  28. News-driven international business cycles
  29. Business cycle dynamics across the US states
  30. Required reserves as a credit policy tool
  31. The macroeconomic effects of the 35-h workweek regulation in France
  32. Productivity and resource misallocation in Latin America1)
  33. Information and communication technologies over the business cycle
  34. In search of lost time: the neoclassical synthesis
  35. Divorce laws and divorce rate in the US
  36. Is the “Great Recession” really so different from the past?
  37. Monetary business cycle accounting for Sweden
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