Our model comprises two types of economic agents: the representative firm and the trade union monopoly. In the traditional WS-PS model, the firm and the union enter into a Nash bargaining game that always results in success, i.e. an agreement where a real wage is set in such a way that the product of the net gains of each stakeholder is maximized. In our model, the negotiation is not formalized by a Nash bargaining process (which maximizes the product of the participants’ net gains). Our article does not claim to provide a new microeconomic model of bargaining; we simply represent the negotiations in terms of a confrontation between the reformulated WS and PS curves. Equilibrium, i.e., the point of intersection between WS and PS, is not necessarily reached. This is the originality of our approach, which contrasts with the standard WS-PS model where an equilibrium is always achieved in the negotiations.
The firm and the reformulated PS curve
The firm is assumed to have a Cobb–Douglas production function \(Y = AL^{\alpha }\), with Y representing the total production, L the quantity of labour units, A product per unit of labour, and \(\alpha\) the output elasticity of labour. We consider that the production function includes only one factor because the quantity of capital used in the short run is considered fixed and does not play a determining role in the causal relations.
The effective demand principle adopted here indicates that the level of production, and thus employment, is determined by the anticipated aggregate demand for goods denoted \(D_{g}\). The level of production depends on the entrepreneurs’ expectations and is assumed to be exogenous. The type of anticipation (rational, adaptive or other) is irrelevant for our purpose. However, the expectations can be considered rational.
We thus have:
From which we obtain:
$$L = \left( \frac{Y}{A} \right)^{{\frac{1}{\alpha }}} = \left( {\frac{{D_{g} }}{A}} \right)^{{\frac{1}{\alpha }}}$$
(2)
Nevertheless, Keynes adopts the first classical postulate: we should therefore check for equality between the marginal productivity of labour and real wages. However, in contrast to the traditional neoclassical approach, it is not the real wage that determines the level of employment but the opposite: since the anticipated aggregate demand is fixed, and thus also the level of employment, it follows that there is a real wage denoted here as w.
$$Y_{L}^{^{\prime}} = w\; \Leftrightarrow \alpha AL^{\alpha - 1} = w$$
We thus obtain a standard decreasing relationship between the level of employment and the real wage, the only change being that it is the level of employment that determines wages:
$$w = \alpha AL^{\alpha - 1} \Leftrightarrow L = \left( {\frac{\alpha A}{w}} \right)^{{\frac{1}{1 - \alpha }}}$$
(3)
To obtain the labour demand curve linking real wages, w, with the unemployment rate u, we simply writeFootnote 1:
$$u = 1 - \frac{L}{N} = \frac{N - L}{N}$$
From this, given (3), we can deduce that:
$$u = \frac{{N - \left( {\frac{\alpha A}{w}} \right)^{{\frac{1}{1 - \alpha }}} }}{N}$$
After rearranging the terms, we obtain:
$$w = \frac{\alpha A}{{(N(1 - u))^{1/1 - \alpha } }}\;\;\;[PS]$$
(4)
This yields the PS curve for the goods market, which reflects/?points to a positive correlation between unemployment rate and real wages.
According to Keynes (1936), the wage is the explained variable and the unemployment rate is the explanatory variable. This results from the anticipated demand for goods and so from the aggregate equilibrium product determined on the goods market as indicated by Eq. (2′).
We find an identical logic in the models of Piluso (2011) and Cartelier (1995), who highlight the following sequence:
$${\text{Anticipated demand}} \Rightarrow {\text{Level of production}} \Rightarrow {\text{level of employment}} \Rightarrow {\text{Marginal productivity of employment level}} \Rightarrow {\text{Real wage}}$$
In classical economics, the equation of the PS curve is interpreted traditionaly as follows: the real wage level explains the level of unemployment.
The trade union and the WS curve
The union utility function contains the arguments of employment level L and the real wage supplement (w-wr), where wr denotes the reservation wage.
$$U = \ln (L)^{\gamma } + \ln (w - w_{r} )$$
This utility function is derived from the WS-PS model of Piluso and Colletis (2012), itself taken from labour economics textbooks (Cahuc & Zylberberg, 1996).
The parameter \(\gamma\) represents the weight given to employment in the union’s objective. This maximizes its utility with respect to the wage claim. The problem can be written:
$$\begin{gathered} Max\,\;\ln (L)^{\gamma } + \ln (w - w_{r} )\;s.t\;L = \left( {\frac{{\alpha A^{{}} }}{w}} \right)^{{\frac{1}{1 - \alpha }}} \;\; \Leftrightarrow \;\;\;\;Max\;\gamma \ln \left[ {\left( {\frac{\alpha A}{w}} \right)^{{\frac{1}{1 - \alpha }}} \;} \right] + \ln (w - w_{r} ) \hfill \\ w \ge 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;w \ge 0 \hfill \\ \end{gathered}$$
We obtain:
$$\frac{{w - w_{r} }}{w} = \frac{1 - \alpha }{\gamma } = \mu$$
whence
$$w = \frac{{w_{r} }}{1 - \mu }$$
(5)
To plot the WS curve, let us make the standard assumption that the reserve wage of the worker (the rate below which he or she does not accept work) is equal to a weighted mean of the wage that could obtained in another firm and the unemployment benefit. Let us write:
$$w_{r} = (1 - u)w + uB$$
(6)
where u is the unemployment rate, B is the unemployment benefit, and w is the average wage rate in the economy. If the worker looks for another job in the economy, the probability of finding one is (1 − u), and that of being unemployed is u. At equilibrium, the negotiations in each employment area lead to the same real wage if \(w_{i}\) = w, so, from Eqs. (4) and (5), it can be stated that:
$$w = \frac{u}{u - \mu }B\;\;\;\;[WS]$$
(6′)
This gives a WS curve for wage demand showing that unemployment rate falls as a function of real wages.