Growth, unemployment and heterogeneity

Abstract

The paper analyzes unemployment in a medium-run growth model, where aggregate demand and supply interact, using a top-down approach. The aim of the essay is the study of a nonlinear system where both aggregate demand and supply are endogenous and generate bounded unemployment, followed by a methodological effort direct to identify possible lines of convergence with the agent based models (ABM) approach. This is a by-product of the presence of heterogeneity in the model. Heterogeneity acts through two different channels and operates among class of agents: it comes into the aggregate consumption function where households are assumed employed or unemployed; it changes the learning process of pessimists and optimists. The analysis is carried on through simulations. The resulting system is fairly stable to changes in main structural parameters. On one hand, autonomous demand drives the dynamics of the system, while heterogeneity in the consumption function, due to the presence of unemployment, strengthens the links with supply aspects. On the other hand, both the rate of growth of labor productivity and labor supply are endogenous. Two major results are obtained. First, unemployment allows the so called Harrodian reconciliation between aggregate demand and supply. Second, unemployment remains bounded meaning that the interaction between aggregate demand and supply thwarts instability. These results are in keeping with those obtained by means of a bottom-up approach, typical of ABM. Possible explanations and implications of this convergence are put forward and open the venue to further deepening of complementarities among the two modeling strategies.

Mathematical appendix

The nonlinear system underlying the analysis is presented in a compact way in what follows. The first equation represents expectations, while the components of aggregate demand are formalized from (A.2) to (A.4). (A.5) represents the evolution of capital, while (A.6) sets the equilibrium in the product market.

(A.7) and (A.8) represent respectively the level and the rate of growth of supply, while (A.8), (A.9), (A.10) and (A.11) show both the levels and the rate of growth of labor supply and productivity. (A.13) defines labor demand, while (A.14) introduces the rate of unemployment. The model is closed by three definitions.

$$ E_{t - 1} g_{t} = \left( {1 - \alpha } \right)g_{t - 1} + \alpha E_{t - 2} g_{t - 1} $$

(A.1)

$$ C_{t} = \left[ {Y_{t - 1} \left( {1 + Eg_{t} } \right)} \right]\left[ {c_{1} - c_{2} \left( {u_{t - 1} - u_{0} } \right)} \right] + F_{t} $$

(A.2)

$$ F_{t} = F_{t - 1} \left( {1 + g^{*} } \right)\left[ {1 - \psi_{1} \left( {u_{t - 1} - u_{0} } \right)} \right] $$

(A.3)

$$ I_{t} = \delta K_{t} + Eg_{t} K_{t} + \beta \left[ {v^{*} \left( {1 + Eg_{t} } \right)^{2} Y_{t - 1} - \left( {1 + Eg_{t} } \right)K_{t} } \right] $$

(A.4)

$$ K_{t } = \left( {1 - \delta } \right)K_{t - 1} + I_{t - 1} $$

(A.5)

$$ Y_{t } = C_{t} + I_{t} $$

(A.6)

$$ Y_{t}^{s} = Y_{t - 1}^{s} \left( {1 + g_{t}^{s} } \right) $$

(A.7)

$$ g_{t}^{s} = \left( {1 + \tau_{t} } \right)\left( {1 + \sigma_{t} } \right) - 1 $$

(A.8)

$$ L_{t} = \left( {1 + \sigma_{t} } \right)L_{t - 1} $$

(A.9)

$$ \sigma_{t} = \rho_{0} - \rho_{1} u_{t - 1} $$

(A.10)

$$ A_{t} = \left( {1 + \tau_{t} } \right)A_{t - 1} $$

(A.11)

$$ \tau_{t} = \theta_{0} - \theta_{1} \frac{{i_{t - 1} }}{{v_{t - 1} }} $$

(A.12)

$$ i_{t} = \frac{{I_{t} }}{{Y_{t} }} $$

(A.13)

$$ v_{t} = \frac{{K_{t} }}{{Y_{t} }} $$

(A.14)

$$ N_{t} = \frac{{Y_{t} }}{{A_{t} }} $$

(A.15)

$$ u_{t} = 1 - \frac{{N_{t} }}{{L_{t} }} \ge 0 $$

(A.16)

$$ g_{t} = \frac{{Y_{t} }}{{Y_{t - 1} }} - 1 $$

(A.17)

Given an exogenous rate of autonomous demand growth g* and the expected-desired capital-output ratio v*, the system refers to 17 unknowns: I_t, K_t, C_t, F_t, Y_t, u_t, N_t, L_t, g_t, v_t, τ_t, A_t, σ_t, i_t, Eg, ${\text{Y}}_{\text{t}}^{\text{s}}$ and ${\text{g}}_{\text{t}}^{\text{s}}$.