Solve_Generic_HJB_KFE

Julia code to solve generic HJBs & KFEs

Current code: deterministic, one state, one control, infinite horizon, time/state separable
Currently, the return function only depends on the control, not the state (as in durable goods etc)
Challenge: for true generality, need to solve non-linear system of n_c focs, to get n_c policy functions (but the number of iterations w/ the implicit scheme is usually small so perhaps it's not too bad???)

Current examples:

• NGM: 1-state (k), 1-choice (c), return u(c), compare w/ closed form
• NGM (Convex-Concave tech): 1-state (k), 1-choice (c), return u(c), compare w/ Hamiltonian (multiple SS)
• CS: 1-state (a), 1-choice (c), return u(c), compare w/ closed form {To Do: extend to finite horizon}
• Investment: 1-state (k), 1-choice (i), return profit(k,i) {Not working well}
• Housing:
• Labor: non-stochastic

Next: human capital/deterministic labor supply/deterministic Lifecycle
Next: combine 2-state Poisson & Diffusion into a generic solver

Extensions:
endog := endogenous & exog := exogenous

• Multiple states (n_s):
  • (a,z): wealth a endog, income z is exog 2-state Poisson, HJB_stateconstraint_implicit
  • (a,z): wealth a endog, income z exog diffusion, HJB_diffusion_implicit
  • (a,z,t): wealth a endog, income z exog diffusion, age t exog, Lifecycle.pdf, lifecycle.m
  • (k,z): capital k endog, tech z exog diffusion, firm.pdf, firm.m
• Multiple controls (n_c):
  • (a,z): wealth a endog, income z exog 2-state Poisson, choices (c,ℓ), labor_supply.pdf, HJB_labor_supply.m
  • (a,h): wealth a endog, human capital h endog, choices (c,s), human_capital.pdf, human_capital.m
  • (a,b,z): liquid assets a endog, illiquid assets b endog, income z exog Poiss, choices (c,d), two_asset_kinked.pdf, two_asset_kinked.m
• Stochastic: see above

Optimal stopping time (HJBVI, LCP)

• Indivisible durable: saving to buy a car, car.pdf, car.m

Q: what about GE models w/ non-constant interest rates & paths of prices?