The Association of Public Economic Theory (APET), and the University of Exeter Business School Workshop in honour of Cuong LE VAN University of Exeter, September 9-11, 2011

1. ISSUE : To develop analytical methods to study cyclical properties of stochastic endogenous Keynesian unemployment business cycles in a model with efficiency wages. Conditions for real wages and employment to be both procycliclal (“Phillips Curve ”)  Underlying Model : - Finance constrained economy with heterogeneous agents and constant returns as in Woodford (1986), Grandmont, Pintus and De Vilder (1998)

− Shirking and Efficiency Wages so as to induce workers to make effort level x requested in wage contract (w, x) set by firms (Negishi (1979), Solow (1979), Shapiro and Stiglitz (1984), Coimbra (1999) Nakajima (2006), Aloi and Lloyd-Braga (2006), …) − Unemployment Insurance (Dufourt, Lloyd-Braga and Modesto (2008))  CONCLUSIONS → Local indeterminacy (  stochastic endogenous unemployment business cycles) occurs for capital- labour substitution elasticities larger than a very small lower bound, for significant unemployment insurance (Grandmont (2008))

Raising Unemployment insurance may be good on average for employment and welfare, if relative risk aversion decreases not too fast with income (Grandmont (2008)) Phillips curve ? Real wage and employment are both procyclical when effort is mildly procyclical, which requires : − Specific heteroschedastic properties of the process of stochastic (sunspot) shocks to employed workers consumption, i.e. elasticity of but not too large − Decreasing and significantly convex relative risk aversion

2. AGENTS BEHAVIOUR  FIRMS (Continuum, Size 1) - Constant returns production function with “efficient” labour employment, - Profit maximization with respect to (real wage ), given (real rental rate of capital)

 Marginal productivities = Marginal input costs  Firms choose efficiency wage contract so as to minimize real wage per unit of effort under workers’ incentive constraint

 CAPITALISTS (continuum, size 1) Maximize discounted intertemporal (logarithmic) utility. More patient than workers  hold capital stock and no money Where gross real rate of return on capital (equal to at deterministic steady state)

 WORKERS (Continuum, size 1) - At t either employed (proba ) or unemployed (proba ) Finance constraint : Labour incomes (wage or unemployment insurance ) paid in cash at end of period, no borrowing - If more impatient that capitalists,, not too risk averse, and significant insurance (close to 1) ) hold cash only - Unemployment insurance financed by taxation of labour incomes  in equilibrium (constant money stock M > 0)

- Workers incentives (NSC) At t : if employed worker with wage contract shirks and gets caught (proba fired and gets (insurance) Non shirking condition (NSC) : Disutility of effort is out weighted by expected utility gain of not shirking

Efficiency wage contracts - At t, let future consumption of currently employed workers be where or - At t, given the distribution of the stochastic (sunspot) perturbations the choice of an efficiency wage contract by firms through profit maximization is equivalent to the choice of effort and of expected consumption so as to maximize subject to the non-shirking schedule (NSS)

3.STOCHASTIC ENDOGENOUS KEYNESIAN UNEMPLOYMENT BUSINESS CYCLES → Given stochastic process of small random (sunspot perturbations of employed workers’ consumptions with  Sequence of R.V. (effort, expected workers’ consumptions) determined by efficiency wages contracts  (stochastic) dynamics with and efficient labour, given by (1), (2) → Capitalists Where gross real rate of return on capital

- Workers

4. CYCLICAL PROPERTIES  Assumption : (local indeterminacy) Given process and history up to t-1, let hence

) Real wage and employment are both (procyclical) iff effort is mildly procyclical Given process of small (sunspot) random shocks to employed workers’ consumptions effort levels and expected consumptions are determined by firms choice of efficiency wage contracts. - Maximum of subject to non shirking schedule

- If is sequence of i.i.d. r.v., is constant over time  Real wage is anticyclical - May one expect that if future risk is reduced when Assumptions I fixed cost of effort, strictly convex strictly concave

At t, for small, reducing risk (If DDRA, must decrease fast enough, strongly convex

If hyperbolic absolute risk aversion (HARA), Above properties satisfied if small and large but small perturbation from CRRA specification, with low risk aversion CONCLUSIONS : Under above assumptions, real wage and employment both procyclical (effort mildly procyclical) if elasticity of with respect to, significant but not too large in absolute value

Qualitative conclusions likely to stand generally in models of unemployment with efficiency wages : 1. Real wage and employment both procyclical iff effort is mildly procyclical 2. Effort is procyclical if : 2.a. Specific heteroschedastic properties of risk faced by employed workers : variance of future consumption is anticyclical 2.b. Highly nonlinear attitude toward risk (convex relative risk aversion, 4 th derivative of utility function, as in finance models)

In specific model considered here, unpleasant cyclical features, as in many cash-in-advance models : Nominal price and wage both anticyclical Introducing non-constant velocity of circulation of money through variable financial (borrowing) constraint? Ex : With

Promising avenue but much more complex (raising wage at t incrases current consumption but decreases future consumption  more complex workers intertemporal incentive’s, higher dimensional dynamics ?)