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An irreversible investment problem with a learning-by-doing feature

Erik Ekström, Yerkin Kitapbayev, Alessandro Milazzo, Topias Tolonen-Weckström

Abstract

We study a model of irreversible investment for a decision-maker who has the possibility to gradually invest in a project with unknown value. In this setting, we introduce and explore a feature of "learning-by-doing", where the learning rate of the unknown project value is increasing in the decision-maker's level of investment in the project. We show that, under some conditions on the functional dependence of the learning rate on the level of investment (the "signal-to-noise" ratio), the optimal strategy is to invest gradually in the project so that a two-dimensional sufficient statistic reflects below a monotone boundary. Moreover, this boundary is characterised as the solution of a differential problem. Finally, we also formulate and solve a discrete version of the problem, which mirrors and complements the continuous version.

An irreversible investment problem with a learning-by-doing feature

Abstract

We study a model of irreversible investment for a decision-maker who has the possibility to gradually invest in a project with unknown value. In this setting, we introduce and explore a feature of "learning-by-doing", where the learning rate of the unknown project value is increasing in the decision-maker's level of investment in the project. We show that, under some conditions on the functional dependence of the learning rate on the level of investment (the "signal-to-noise" ratio), the optimal strategy is to invest gradually in the project so that a two-dimensional sufficient statistic reflects below a monotone boundary. Moreover, this boundary is characterised as the solution of a differential problem. Finally, we also formulate and solve a discrete version of the problem, which mirrors and complements the continuous version.

Paper Structure

This paper contains 17 sections, 10 theorems, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Preview
  4. Some initial considerations
  5. Investment costs
  6. Signal-to-noise ratio
  7. Admissible controls
  8. Problem formulation
  9. Construction of a candidate solution
  10. Deriving an ODE for the free boundary
  11. Constant learning rates
  12. A study of the ODE for the boundary
  13. Verification
  14. Examples
  15. The discrete case
  16. ...and 2 more sections

Key Result

Proposition \oldthetheorem

Let $v$ be the value function defined as in vstopping, and let $\hat{v}$ be defined as in hatv. Then $v=\hat{v}$, and moreover, the stopping time $\tau_c:=\inf\{t\geq 0:\Pi^u_t\geq c(u)\}$ is optimal for vstopping.

Figures (5)

  • Figure 1: The trajectory of the pair $(U,\Pi^U)$ under the reflecting strategy \ref{['heur']} in the case $\rho^2(u)=\frac{1}{4(1-0.9u)}$, $k=0.5$ and $r = 0.1$.
  • Figure 2: The solution $b$ to the ODE \ref{['ode+bc']} (solid black), the curve $B$ (red) and the threshold $k$ (dashed black), in the case $\rho^2(u)=\frac{1}{4 (1-0.1u-0.8u^2)}$, $k=0.5$ and $r = 0.1$.
  • Figure 3: The optimal reflecting boundary $b$ (solid black), the curve $B$ (red) and the threshold $k$ (dashed black) in the case $\rho^2(u)=\frac{1}{4(1-0.9u)}$, $k=0.5$ and $r=0.1$.
  • Figure 4: The optimal reflecting boundary $b$ (solid black) and the threshold $k$ (dashed black) in the case $\gamma(u) = 1.25/(u + 0.2)$ and $k=0.5$.
  • Figure 5: Optimal boundaries $b_n$ (black dots), points $c_n$ (blue dots), and threshold $k$ (black dashed) with $\gamma_n = 1.25/\left(n/5 + 0.2\right)$. Remaining parameters are $k=0.5$ and $N=5$.

Theorems & Definitions (31)

  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 21 more