On a closed-form solution of the Brock–Mirman model with a finite horizon: A shadow-value interpretation

Abstract

We develop a shadow-value interpretation of the finite-horizon Brock–Mirman model and use it to obtain an explicit, time-varying closed-form solution. Viewing one unit of current output as a shadow price yields a simple linear relation over time and a short geometric adjustment: as the horizon shortens, the saving share falls while the consumption share rises. We also show that the closed form survives under proportional taxes: return- and investment-side wedges operate through the effective return and thus govern adjustment speed, whereas a pure consumption tax shifts levels only.