Abstract
It has long been known that search processes on the labour market are only another form of incompleteness on this factor market. Earlier works studied search processes and the behaviour of individuals, but seldom addressed the consequences for the whole economy.1 In recent works, the search activities were simplified so that general equilibrium models can now easily be analysed.2 However, most of the papers base on models of a closed economy. On the contrary, almost all countries take the advantages trade offers so that it is desirable to have a search and matching model of an open economy.3
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References
See e. g. Mortensen (1986), McKenna (1985) or Das (1997) for an overview.
Among the earlier works are Diamond and Maskin (1973) and Lucas and Prescott (1974). For more recent literature see e.g. Pissarides (1990), Hosios (1990a) or Moen (1997).
An exception is Davidson, Martin, and Matusz (1999) linking trade to unemployment caused by search frictions. However, the model is based on a Leontief-type production function so that a substitution between physical (or human) capital and labour cannot be studied. In addition, no adjustment processes are studied.
See e.g. Gandolfo (1994).
The shares in world exports amount to 12.4 percent (1998) for the United States and 10.0 percent (1998) for Germany. In comparison, Latin America had a share of 3.1 percent and the non OECD Asian countries 15.7 percent in all world exports in the same year (OECD, 1999).
Hamermesh and Pfann (1996) study an economy in which hiring new workers and possibly dismissing others cause costs. However, the model does not consider search processes.
See e. g. Ethier (1988) for a description of the standard specific factors model.
In the European Monetary System (EMS) e. g. the United Kingdom and Italy increased their exchange rates by more than 2.25 percent and 6 percent respectively in 1991. In the EMS, the exchange rate of the member countries where allowed to fluctuate by ±2.25 percent (±6 percent Italy) around a fixedrate. See e.g. Glismann et al. (1986) for a description of the EMS.
The assumption of infinitely living individuals is frequently justified by the argument that a finitely living head of a family takes the welfare of all present and future members into account. See e. g. Arrow and Kurz (1969) and for a critique on modelling the consumer problem in finite time with bequest motive. 10 Consequently, they not only have identical preferences, but are also equally productive in a particular sector. In addition, they cannot be distinguished with respect to their qualification.
This follows Pissarides (1990) or Wälde and Weiß (1997). Closely related are works of e. g. Davidson, Martin, and Matusz (1988) Hosios (1990a) and Hosios (1990b) or Matusz (1996).
See e. g. Pissarides (1994) or Benhabib and Bull (1983) for models where both employed and unemployed individuals may search on the labour market. In contrast, e.g. Brunello (1996) considers internal labour markets, where employees may be promoted within a firm. For the sake of simplicity however both possibilities are ignored here.
In Pissarides (1990, ch. 4) individuals choose their search intensity optimally, where a higher intensity increases the probability to be matched at any moment of search.
For and overview of the system for unemployment benefits in the OECD countries, see e. g. Goerke (1998) or Koshela and Schöb (1999). The systems for the benefits vary considerably between the countries with respect to the contribution conditions, the type of calculation, and the duration in which benefits are granted. In e. g. the United States and Germany, the benefits are proportional to the previous wage. However, in Australia and the United Kingdom, the benefits are indeed fixed. In general, the parameter b need not consist solely of benefits granted by social security systems or the government. It can also include some utility equivalent to the additional leisure available if the individual is unemployed. In this interpretation b can always be positive even if unemployed persons receive no monetary transfers. Despite this fact, the expression unemployment benefits is used henceforward.
Alternatively it could be assumed that unemployed persons spend a fixed amount for a small period of time. Then, b would denote the unemployment benefits less the search expenses.
Other specifications are possible. Hosios (1990a) considers two different labour markets, each attached to a sector. The labour markets differ with respect to their matching functions. Unemployed individuals are free to move between sectors.
Burda and Wyplosz (1994) find empirical evidence for a Cobb-Douglas specification of a matching function in Germany.
This assumption implies that every job seeker has the same probability of obtaining a job regardless of how often and how long he or she was previously unemployed.
The wage bargaining process is the subject of subsection 3.1.3.
Aghion and Howitt (1994) consider an endogenous growth model with search frictions on the labour market. Growth arises from the invention of new technologies which, at the same time, is the source of creative destruction. This framework enables Aghion and Howitt to model the separation rate s endogenously. See also Bertola and Caballero (1994) for a model with endogenous separation.
Since the consumption goods are non-durable, the only additional opportunity to transfer income across time is saving. Consequently, it is implicitly assumed that individuals have no access to the capital market. This assumption serves to keep the model as simple as possible at this stage and is dropped in chapter 6.
It is natural to assume that individuals as well as firms use the interest rate as a discount factor if this interest rate is given exogenously. However, this assumption has further consequences. With different time preference rates, an individual’s share of the total surplus in the wage bargaining will not be equal to the firms share even if a symmetric bargaining is considered. Layard, Nickell, and Jackman (1992, p. 99) therefore use a bargaining power defined as the ratio of the time preference rates. The so defined coefficients of the asymmetric Nash product however do not add up to one.
An individual is risk-neutral if the expected utility equals the utility of a lottery (cf. e. g. Mas-Colell, Whinston, and Green (1995) or other microeconomic literature). Hence 26–1.
Appendix 3.5.1 describes how the system of Bellman equations can be derived from the utility maximisation problem.
For a description of the specific factors model see e.g. Ethier (1988).
For simplicity, the costs of opening a vacancy is assumed to be fix here. This assumption, however, can be justified by assuming that firms either hire labour via head hunters or organise their personal department as a profit centre. 27 See e. g. Kantien and Schwartz (1991, ch. 12) for the problems of constraint control variables.
Note that the value of a marginal worker is not identical to the value marginal product of labour as the duration of labour relations is uncertain.
Such dismissal rules can also be given indirectly by the society in general. Then, s is the maximal rate of dismissals the society is willing to accept.
The criterion for the distinction between large and small shocks is given in section 3.3.
Specifically, the maximisation problem could still be described by equation (3.7) together with (3.6). In addition, the firms can choose a point in time at which they can dismiss an arbitrary number of workers. In the situation following a serious negative shock, firms will always choose to dismiss workers directly, if the profits are negative. However, firms will not dismiss exactly the number of workers needed to reach the new medium run levels. Even if the firms in one sector decide to reduce their labour stock instantly due to a serious negative asymmetric shock the firms of the other sector cannot instantly hire the appropriate number of unemployed persons to reach their medium-run levels due to the search processes. Therefore, the’ normal’ adjustment processes must ensue. See e. g. Kamien and Schwartz (1991, ch. 18) for an excellent description.
Pissarides (1990, p. 22) justifies this assumption by pointing out that ”this assumption is clearly the closest we can have to competitive wage determination in this market environment. In deciding how many jobs to open up the firm anticipates the wage correctly, but chooses the number of jobs by taking it as given.”
Clearly, other assumptions are reasonable. Alternatively, wages could be modelled as an outcome of a strategic bargaining problem between a firm and a worker. However, Binmore, Rubinstein, and Wolinsky (1986) show that a Nash bargaining problem may be justified by a strategic bargaining problem. A negotiation between a trade union and an employer’s association is also a reasonable assumption and for many European countries a realistic one. Even a wage equal to the value marginal product could serve a wage setting rule. However, since the cost of offering a vacancy is strictly positive, this would yield negative profits in the medium-run equilibrium and is thus incompatible with the setup of the presented model.
The decision of how many vacancies the firm wants to offer precedes matching. On the other hand, matching precedes the wage bargaining process. For this reason, the value of an additionally unfilled job will be zero per definition in the present model as the number of vacancies was chosen optimally before the matching took place.
Sufficient and necessary conditions for the existence of a solution coincide in the medium-run equilibrium. During transition periods, however, this condition is sufficient but not necessary.
Nash (1950) explicitly assumes that players are equipped with equal bargaining skills. Roth (1979, p. 17) interprets the frequently used bargaining power as a bargaining skill.
Bargaining skill and power are used interchangeably here. In general, the assumption that firms have a greater bargaining power should not be connected with the existence of high unemployment and the resulting competition among job seekers. This fact is captured in the probability of finding a job p. In addition, the wage negotiation takes place after matching has occurred.
See e. g. Osborne and Rubinstein (1990) for asymmetric Nash bargaining.
See Appendix 3.5.2 for the derivation.
Despite the fact that wages are equivalent to unemployment benefits, the individuals could decide to work because having a job may yield some nonmonetary rewards. Specifically, the social status of the worker and an unemployed person may be different.
Since product prices are given exogenously in a small open economy, a nominal wage floor is identical to a real wage floor.
Haberler (1950) discusses in some extent, that perfect mobility of labour between sectors is not necessary to equalise wages.
Buddy Levine, and Smith (1987) identify additional causes for an outward shift in the Beveridge curve: an increase in the net flow into unemployment from the labour reserve, a decrease in the search intensity of employees and a decline in the choosiness of employers.
If the assumption of equal costs for offering vacancies is changed this will not only lead to a wage gap between sectors in a steady state but alter the qualitative properties of the adjustment processes described in the following section.
Due to the fact that a two sector economy is considered, it is not possible to represent the equilibrium of the economy in the more familiar V/U-diagram normally used to illustrate the Beveridge curve.
In one sector models, the labour market equilibrium condition can be represented in a V/U-diagram and is known as Beveridge curve.
Although the labour market equilibrium condition is identical to the Beveridge curve in a V/U-diagram the shift of the LMy curve to the right side does not imply a shift of the Beveridge curve. On the contrary, the decline of a product price will not influence the Beveridge curve.
The positive employment effect refers to the derivative dLx/dpx, so that a decrease in px causes employment in X to shrink.
See e.g. Ethier (1988) for a representation of the specific factors model.
Since an exchange rate shock influences both sectors directly, the reduction of employment in sector X can be illustrated in a λ/Lx-diagram in analogy to figure 3.3.
See e.g. Goerke (1998) for an overview of the various forms of unemployment benefits in the OECD countries.
Here, a small shock is defined as a shock which enables all firms to still offer a strictly positive number of unfilled jobs. All other shocks are regarded to be large.
The rate at which each firm can dismiss workers is restricted here since jumps in the state variable, labour, were excluded per assumption.
An example for an asymmetric shock is the change of one product price. If e. g. the price px decreases the firms in sector X learn immediately that they have employed too many workers. In contrast, if the price of px increases, the value marginal product for sector X firms increases thereby raising the expected return on an additionally offered unfilled job. As a consequence, it is more profitable to offer vacancies in sector X than in Y. Therefore, sector Y firms have to realise that they have hired too many workers. Otherwise, the return on an additional vacancy would equal the one on sector X firms.
The same argument as explained for small negative symmetric shocks apply here.
Well-known examples for large asymmetric shocks are the oil price shocks. The value marginal product of labour increases considerably in the oil producing industry, e. g. by 20 percent. This effect increases the return on an unfilled job in small German oil producing firms. As a consequence, firms in other sectors offer less vacancies. Suppose that e. g. each firm of the other industries employs 100 workers and the natural fluctuation is 10 percent. Suppose further, that the oil price shock was large and the firms of the other industries need to dismiss 20 workers to increase the productivity by 20 percent. As firms are allowed to dismiss 10 workers each period, it is not possible for the firms of the other industries to instantly reach a value marginal product of labour comparable to that of the oil producing industry. In contrast, they have to close all open job opportunities and shrink independently.
The slope of the zero-motion line becomes zero if firms have no bargaining power in the wage negotiation process, i. e. if α = 1. Since the graph Dy in figure 3.1 is identical to the zero-motion line λy = 0 in figure 3.5, the total number of unemployed persons will stay constant after a drop in px has occurred, if Dy or the zero-motion line λy = 0 line is horizontally. The same result was found in subsection 3.2.2.
Although the distinction between large and small shocks is motivated by the formal approach it seems less important in continuous than in discrete time. In discrete time matching occurs e. g. at the first of each month. Suppose that a particular firm has a natural fluctuation of two workers per month and therefore has to offer two job opportunities to keep employment stable. A small negative shock would be a shock that requires the firm to reduce the labour force by one worker. Consequently, the firm needs to offer only one vacancy instead of two. All other shocks are considered to be large ones inducing the firm to close down all unfilled jobs at least for a month. In continuous time the situation is different. Matching is continuous, i. e. it takes place every minute. Therefore, the firms offer less unfilled jobs at every point in time and much smaller shocks can force the firm to close down all’ normally’ offered vacancies. Consequently, it can safely be assumed that a shock is large in continuous time.
As the event matching for the vacancy follows a Poisson process with parameter q, 1/q denotes the expected waiting time. Consequently, γ/q is the expected cost for the vacancy.
ε was defined as Vx/V in subsection 3.1.3 and denotes the probability of finding a job in sector X given the unemployed person was matched in general.
As mentioned in subsection 3.1.3 this weighted sum of the value of a marginal worker of both sectors is a compensation potential employees ask for in order to renounce from further searching activities. This compensation naturally depends on the conditions of both sectors as long as unemployed individuals can be matched into both sectors. In contrast, if one sector does not offer any vacancies the unemployed persons can only be matched into the other sector. Hence, the compensation a potential worker can call for only consists of the value of a marginal worker of the sector offering jobs.
Pontrjagin et al. (1964) develop their general maximum principle for finite discontinuities in the control variable, i. e. the number of vacancies in the maximisation problem of the firm. One of the optimality conditions is that the shadow price X has to be continuous, i. e. also at the discontinuity points of the control variable. Kamien and Schwartz (1991, ch. 12, p. 206) explicitly mention in example two, that the shadow price has to be continuous at the discontinuity point of the control variable.
According to the definition of the probability of getting a vacant job filled, q = m(U,V)/V, the fact that the number of unemployed persons is a state variable and therefore cannot jump, and that Vy= V during TP I, equation (3.21) gives a unique relation between Vy and λy.
A nonmonotonous adjustment of sectoral employment does not necessarily imply a nonmonotonous behaviour of the total number of unemployed persons.
As mentioned earlier, only one of the three possible adjustment processes is feasible for a given set of parameters. However, for given parameters, it is not possible to choose between an overshooting path of Ly and an undershooting path of Lx
An increase in the unemployment level raises the unemployment rate as the labour force is assumed to be constant.
Franz (1991) shows that there is evidence for an outward shift of the German Beveridge curve. Abraham (1991) also mentions that the Beveridge curve for the United States has shifted outwards, but that there is also evidence that it has shifted inwards again.
The vacancies and the total number of unemployed persons are measured in relation to the total labour force and hence exclude an exogenous change in the labour force as a reason for shift.
If e.g. the separation of worker-job pairs is affected by dismissal regulations, lower job security protection may lead to a higher turnover and an outward shift of the Beveridge curve.
For models of imperfect labour markets in open economies see e. g. Brecher (1974), Brecher and van Long (1989), Mezzetti and Dinopoulos (1991) Brecher (1992) or Hoon (1991).
Binmore, Rubinstein, and Wolinsky (1986) consider two reasons for bargaining: The individual’s impatience in reaching an agreement and the risk of a breakdown of the bargaining procedure. For the model presented here, the latter seems more important. Binmore and Herrero (1988) describe a model in which existing (worker-firm) pairs can be dissolved because one or both of them are matched elsewhere.
See e.g. Berthold and Fehn (1996) or Schürfeld (1998) for an overview on the German labour market.
See e. g. Goerke (1998) for an overview on the different systems in the OECD countries.
Layard, Nickell, and Jackman (1992, p. 517-524) report that in most European countries the coverage of wages collectively determined exceed 75 percent, whereas the collectively negotiated wage determines less than 25 percent in the United States.
For a comparison of the household’s saving rates of various countries, see e. g. OECD (1999).
For the Bellman equations in discrete time see e.g. Sargent (1987) or Stokey and Lucas with Edward C. Prescott (1989).
See Dixit and Pindyck (1994).
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Weiß, P. (2001). A Two-Sector Search Model of an Open Economy without Capital. In: Unemployment in Open Economies. Lecture Notes in Economics and Mathematical Systems, vol 496. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56569-4_3
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