∣Φ(ωι)Φ(ω2)(α - ɑo) - Ωι (ωι) Φ(ω2)(l - Φ(ωι))(fc - ko
Ωi (ωι) Φ(ωι) [l - Φ(ω⅛)] - Ω2 (ω2) Φ(ωι) [l - Φ(ω2)]
(A4b)
Substitution of equations (A4b) into (A3b) and the resulting expressions into (A2a)-(A2d)
yield the stable solutions for (φ, a, μ, k) in the endogenous employment case.
5.3. Fixed Employment Equilibrium
If consumer-producers supply a fixed (unitary) level of work effort l = l, instantaneous
preferences reduce to U = U(c + a, s') + V(l), where the conditions (2.2)-(2.5) obtain.
Furthermore, we can simplify production technology to y = f (k), where y and k now
represent per-capita output and physical capital, respectively, and the following standard
restrictions on f (k) hold: f'(k) > 0, f "(k) < 0, f (0) = 0, f (0) = 0, f (k) → ∞ as k → ∞.
Furthermore, we assume the usual Inada conditions are satisfied. It is then straightforward
to demonstrate that the symmetric equilibrium in the fixed employment case corresponds
to:
Uc [c + α,s (l)] + Us [C + C’S (al)] S (l) = μ - Φ, (A5a)
φ = (β + δ)φ - Uc [c + a, s (l)] - Us [C + a+ (al)] S (l), (A5b)
a = c - δa, (A5c)
μ = μ[β - f '(k)], (A5d)
k = f (k) - c.
(A5e)
Equations (A5a)-(A5c) repeat (2.ll) and (2.lb) from the model with endogenous work
effort, while (A5d, e) are the corresponding differential equations for the capital stock and
its costate variable in the fixed employment case. As in the general model with endogenous
28
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