Handling Non-Convexities: Entrepreneurship and Financial Frictions

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1 Handling Non-Convexities: Entrepreneurship and Financial Frictions Benjamin Moll Model Setup This note describes a model of entrepreneurship and financial frictions, similar to that in Buera and Shin () and Cagetti and De Nardi (6). In particular, occupational choice in combination with financial friction introduces a non-convexity in individuals optimization problems. We also introduce a second additional non-convexity, namely a choice between operating two technologies. Achdou et al. () consider a version with aggregate shocks and examine its business cycle implications s(here we focus on the model with idiosyncratic shocks only). Individuals preferences E e ρt u(c t )dt Workers earn labor income wz θ where θ Entrepreneurs choose technology, maximize profits Two technologies productive (p) and unproductive (u) y u = F u (z, k, l) = zb u k α l β y p = F p (z, k, l) = zb p ((k f k ) + ) α ((l f l ) + ) β B p > B u, but per-period overhead cost f k, f l Notation: for any scalar x, x + = max{x, } F p non-concave in k and l z: idiosyncratic shock, diffusion process Collateral constraints k λa, λ.

2 Income maximization: occupation and technology choice Individuals solve M(a, z, A; w, r) = max{wz θ, Π u (a, z, A; w, r), Π p (a, z, A; w, r)} Π j (a, z, A; w, r) = max k λa F j(z, A, k, l) (r + δ)k wl, j = p, u. max {c t} E e ρt u(c t )dt s.t. da t = [M(a t, z t, A t ; w t, r t ) + r t a t c t ]dt dz t = µ(z t )dt + σ(z t )dw t a t Optimal capital and labor choices corresponding to the productive technology k p (a, z; w, r) = min { λa, (zab p ) α β ( ) βzabp β l p (a, z; w, r) = kp (a, z; w, r) α β w ( ) β ( ) β } α α β β α β + fk r + δ w and a similar expression for optimal capital and labor choices corresponding to the unproductive technology. + fl Representative firm Y c = F c (A, K c, L c ) = AB c K η c L η c, Equilibrium Conditions Individual optimization and evolution of distribution ρv(a, z, t) = max c u(c) + a v(a, z, t)[m(a, z; w(t), r(t)) + r(t)a c] + z v(a, z, t)µ(z) + zzv(a, z, t)σ (z) + t v(a, z, t) () t g(a, z, t) = a [s(a, z, t)g(a, z, t)] z [µ(z)g(a, z, t)] + zz[σ (z)g(a, z, t)], () s(a, z, t) =M(a, z; w(t), r(t)) + r(t)a c(a, z, t) ()

3 Public firms r(t) = K F c (A, K c (t), L c (t)) δ, w(t) = L F c (A, K c (t), L c (t)) () Capital market clearing: K c (t)+ + = k u (a, z; w(t), r(t)) {Πu>max{Π p,wz θ }} k p (a, z; w(t), r(t)) {Πp>max{Π u,wz θ }} a () Labor market clearing: L c (t)+ + = l u (a, z; w(t), r(t)) {Πu>max{Π p,wz θ }} l p (a, z; w(t), r(t)) {Πp>max{Π u,wz θ }} z θ {wz θ >max{π u,π p}} (6) Given initial condition g (a, z), the two PDEs (), () together with () and the equilibrium conditions (), () and (6) fully characterize equilibrium. Numerical Solution The algorithm for solving the HJB equation () and the Kolmogorov Forward equation () are nearly identical to that used for solving the Huggett model with a diffusion process described in section here Numerical_Appendix.pdf. The algorithm is then a simple bisection algorithm on the equilibrium interest rate.. Algorithm for Steady State Use a bisection algorithm for r min r r max. Given an initial guesses r for l =,,,... follow. given r l, find ξ l = K c /L c from (). given ξ l find w from ()

4 . given r l and w l, solve the HJB equation. find K c,l from (). Compute L c,l = K c,l /ξ l and compute excess labor demand D l =L c (t) + l u (a, z; w(t), r(t)) {Πu>max{Π p,wz θ }} + l p (a, z; w(t), r(t)) {Πp>max{Π u,wz θ }} z θ {wz θ >max{π u,π p}} 6. Update r l : if D l >, choose r l+ < r l and vice versa. Results Figure plots the saving and consumption policy functions. The policy functions can be non-monotonic. Figure plots the wealth distribution. The wealth distribution has a fat right tail as in Cagetti and De Nardi (6). Saving s(a, z) 7 6 ( a) Saving, with Entrepreneurship low z medium z high z 6 8 Consumption c(a, z) ( c) Consumption, with Entrepreneurship low z medium z high z 6 8 Figure : Saving and Consumption Policy Functions References Achdou, Yves, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll.. Wealth Distribution and the Business Cycle: The Rold of Private Firms. Princeton University Working Papers. Buera, Francisco J., and Yongseok Shin.. Financial Frictions and the Persistence of History: A Quantitative Exploration. Journal of Political Economy, Forthcoming.

5 .6 ( a) with Entrepreneurship.. Density Figure : Wealth Distribution Cagetti, Marco, and Mariacristina De Nardi. 6. Entrepreneurship, Frictions, and Wealth. Journal of Political Economy, (): 8 87.