Evolutionary learning and the stability of wage posting equilibria

Abstract

This paper presents results on the stability of the wage dispersion model presented in Mortensen (2003). Specifically, four learning processes are tested on a single parameterisation of the underlying model, and the most successful is submitted to a sensitivity analysis. The results illustrate an important problem in evolutionary dynamics first highlighted in Nelson and Winter (1982) - that to play a role in equilibrium, a strategy must be consistent with a previous disequilibrium. The results are ambiguous concerning the applicability of the equilibrium method of Mortensen (2003), as some of the learning processes are stable, whilst others are extremely unstable and exhibit complex dynamics.

Appendix

1.1 Deriving the Probability of Acceptance

Given the probability of offers received in (2), the probability of acceptance is equal to the probability that an offer w exceeds other offers received, F(w)^x, given the distribution of x. To derive (3), the following steps are required:

$$P(F(w), \lambda) = \sum\limits_{x = 0}^{\infty} \frac{e^{-\lambda}\lambda^{x}}{x!}F(w)^{x} $$

$$\Rightarrow P(F(w), \lambda) = \sum\limits_{x = 0}^{\infty} \frac{e^{-\lambda} [\lambda F(w)]^{x}}{x!} $$

$$\Rightarrow P(F(w), \lambda) = \sum\limits_{x = 0}^{\infty} \frac{e^{-\lambda [1-F(w)]}e^{-\lambda F(w)} [\lambda F(w)]^{x}}{x!} $$

$$ \Rightarrow P(F(w), \lambda) = e^{-\lambda [1-F(w)]}e^{-\lambda F(w)} \sum\limits_{x = 0}^{\infty} \frac{[\lambda F(w)]^{x}}{x!}. $$

(9)

Finally, note that the infinite sum in (9) is equal to e ^λF(w). So the probability of acceptance simplifies to P(F(w),λ) = e ^{−λ[1−F(w)]}, as in (3).

1.2 Pseudo-Code for Simulations

The following pseudo-code describes the method by which the models in Section 3 are simulated. Matlab code is available from the author on request.

• 1. Initialise parameters: T, p, b, l, α, λ.

• 2. Define $\bar {w}$ according to (6).

• 3. Define the grid of possible wage offers: $S = \left \lbrace w_{1}, . . ., w_{i}, . . ., w_{l} \right \rbrace$, with w _i+1>w _i , w ₁ = b, and w _l = p.

• 4. Compute the equilibrium distribution F:

  • (a) If $w_{i} < \bar {w}$, then $F(w_{i}) = \frac {1}{\lambda } \ln \left (\frac {p - b}{p - w_{i}} \right )$.

  • (b) If $w_{i} \geq \bar {w}$, then F(w _i )=1.

• 5. Simulate the model. For each time period 1 to T:

  • (a) Given the fitness distribution ϕ inherited from the previous period, compute the proportions of firms over offers $\tilde {f}$ following either A ₁ or A ₂.

  • (b) Cumulate $\tilde {f}$ to yield the offer distribution $\tilde {f}$.

  • (c) Given $\tilde {f}$, compute the profit distribution $\pi (w_{i}) = (p - w_{i})e^{-\lambda [1 - \tilde {F}(i)]}$.

  • (d) Given the fitness distribution ϕ inherited from the previous period, and the profit distribution π computed at step (c), compute next period’s fitness distribution $\phi ^{\prime }$ following either B ₁ or B ₂.

  • (e) Calculate this period’s KS statistic: $KS = \sup _{i} \lvert F(i) - \tilde {F}(i) \rvert$.

• 6. Step 5 results in an l by T matrix of the offer distribution $\tilde {f}$ at each time period and a vector of KS statistics of length T. These can be used to plot the figures in the main body of the paper.

Note that the equilibrium distribution and offer distribution are defined on proportions of firms playing each strategy, rather than number of firms, hence the actual number of firms does not have to be specified (but the number of strategies, l, does have to be specified). Also note that an initial fitness distribution (fitness level per strategy) has to be specified to simulate the model (these initial fitness levels comprise the model’s l state variables). Finally, note that the simulation is deterministic.