Abstract
The models presented in the last chapters described economies, in which firms could only choose the number of vacancies in order to adjust the desired stock of labour used in production. Although additional factors of production were used, they could not be adjusted within the considered time horizon. Following this line of argumentation the preceding chapters described economies and their behaviour in the short and medium run rather than in the long run. Therefore, the models had some resemblance to the standard specific factors model. However, a direct comparison to the standard Heckscher-Ohlin-Samuelson models is not possible. As only one factor could be adjusted in the models presented in the preceding chapters, well-known results of the international trade theory, as e. g. the Stolper-Samuelson Theorem or the Rybczinsky Theorem, cannot be verified for these models.
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References
A daily observation of matching would be preferable from an analytical point of view. However, if e. g. a matching function is estimated, data availability will frequently determine a month to be the length of a period.
See e. g. Stokey and Lucas with Edward C. Prescott (1989) or Ross (1992) for Markov processes in discrete time. For stochastic problems in continuous time see e. g. Merton (1992), Dixit and Pindyck (1994) or Davis (1993).
The restriction of the asset demand may have institutional reasons. More important however, an individual will find it difficult to borrow a large amount of assets if it is highly uncertain that he or she is able to pay it back.
See e. g. Ross (1992, ch. 4) for a description of Markov chains.
The assumption that wages are determined by a trade union and an employers’ association is introduced to keep the model tractable. A more detailed discussion follows in subsection 6.1.3.
The analysis of subsection 6.1.1 parallels the work of Aiyagari (5).
This is similar to a Ponzi game. The Ponzi game is named after Charles Ponzi who financed the enormous interest debts by borrowing more and more capital. See e.g. Azariadis (1993, p.318, note 4) or Blanchard and Fischer (1989, p. 49f.) for an introduction of a no-Ponzi-game condition in a simple investment model.
In principle, individuals can accumulate assets as well as borrow capital. In the former case at+1 is positive whereas in the latter at+1 will be negative.
See e. g. Stokey and Lucas with Edward C. Prescott (1989, ch. 9) for the derivation of the stochastic Bellman equations in discrete time.
Unlike the probability of loosing a job, the probability of finding a job Pt+1 depends on time. This can be seen from the definition of Pt+1 = mt+1/Ut.
As in the previous chapters, individuals are all alike except that they can be employed or unemployed. As they can borrow or lend on the capital market, they are heterogenous not only with respect to their employment position, but also concerning their accumulated wealth. In all other aspects, e. g. the qualification, the working experience, or the productivity, they are not distinguishable. If wages were determined by individual bargaining, the heterogeneity resulting from the different amount of assets possibly accumulated by an individual would introduce the opportunity of different wages depending on the individual’s wealth. This fact is discussed in greater detail in the next subsection 6.1.3. Since individuals only differ in their accumulated assets and their employment position, the assumption that wages are the outcome of collective bargaining between a trade union and an employer’s association implies that the wages are identical for all workers employed during a certain period in time.
See e. g. Sargent (1987, p. 12ff.) for a description of the general method in discrete time.
Actually, it can be observed that different wages are paid to workers even if the union’s density is high. There are different reasons for this observation. First, individuals differ in more aspects than the accumulated assets. Most collectively negotiated contracts differentiate wages concerning qualification. A skilled worker will earn another wage than an unskilled person. However, this heterogeneity was neglected for simplicity here. Second, employers are normally free to pay higher wages than specified in the collectively negotiated contract. They may do so e. g. in order to increase the workers’ effort. This points to efficiency wages also neglected here. See e. g. Altenburg and Straub (1998) for a model combining efficiency wages and trade unions.
Generally, individuals can demand or supply assets on the capital market in order to transfer income over time. Alternatively, they could store the consumption goods. Since it was presumed in chapter 5 that consumption goods are non-storable and that individuals have no access to the capital market, they cannot transfer income over time.
As a union represents the workers in the industry-level wage bargaining process, it would be plausible to assume that unions use the individual’s time preference rate p to discount the utility stream. Doing so would introduce an asymmetry into the bargaining process as the employers’ associations use the interest rate for discounting. This asymmetry raises the possibility of endogenously modelling the union’s bargaining power. Although it may be interesting, this possibility is ignored for convenience.
The case of identical negotiation abilities is a special case of asymmetric bargaining in which ß equals 1/2. By virtue of this fact, there is no loss in generality of considering an asymmetric bargaining process.
Again, it is assumed that workers prefer to work if the wage is equivalent to the unemployment benefits. This decision may be caused by differences in the social status connected to the various employment positions.
If the wages were determined by individual bargaining between employees and firms instead of industry-level bargaining, as assumed here, a wage distribution would additionally ensue.
The assets accumulated by a particular person can in an equilibrium still vary between periods. The assumption that a stable distribution of capital exists only states, that if one person’s stock of capital increase another person’s stock of capital will decrease.
The assets A demanded by the firms is expressed in e. g. US$, whereas the capital K is measured in units. Therefore, the assets and the stock of capital are related by A = pKK, where pK denotes the price for capital. It was assumed that the price for the investment goods equals the price for the consumption goods. As the latter was normalised to one, A = K ensues.
The assumption of a depreciation rate equaling zero is responsible for the result that firms do not invest in the capital stock. If part of the capital were used up in production this share would have to be replaced so that gross investments would be positive in a long-run equilibrium.
There are several possibilities for announcing job opportunities. The firm can advertise in a newspaper, inform the labour office, or hire private labour agencies. Each possibility is connected to costs which have to be carried by the firm. However, all of these are in fact services offered by other firms or institutions. Normally, the services are also produced using labour and possibly capital. Therefore, modelling the costs for offering vacancies and services would probably reduce the equation to Y = C in an equilibrium.
Eaw does not denote the expected assets supply or demand for the particular person. Every person demands or supplies assets according to a general function A(w,p,r;). Probably, unemployed persons will reduce the stock of assets and demand assets if they have used up their own stock. Given a stable limiting distribution of the disposable income exists and is known, it is also known which amount of assets a particular person supplies or demands. Forming the expectation over the distribution of the disposable income results therefore in the economy’s mean assets supply per capita.
A clear and very intuitive explanation of martingales is given by Davis (1993, p. 21): ”The traditional picture of a martingale M is that it represents the evolution of a player’s fortune in successive plays of a fair game. […] Similarly, super-and submartingales represent unfavourable and favourable games.” For a definition of a martingale for continuous time Markov processes see e. g. the same source. For martingales in discrete time see e. g. Kushner (1967, p. 25f.) or Ethier and Kurtz (1986, p 55ff).
See e.g. Kushner (1967, p. 26). The prerequisite for this property to hold is that the supermartingale is integrable, which is given here.
Since under the assumptions imposed for the production function, the per capita demand curve of assets is negatively sloped for all available employment rates the result does not depend on specific values of λ and e.
Both equations are not only valid for an individual firm but also for the entire economy. Consequently, the subscript i vanishes.
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Weiß, P. (2001). A Simple Matching Model of a One-Sector Economy with Capital Accumulation. 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_6
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