A Guide on Solving Non-convex Consumption-Saving Models

Abstract

Consumption-saving models with adjustment costs or discrete choices are typically hard to solve numerically due to the presence of non-convexities. This paper provides a number of tools to speed up the solution of such models. Firstly, I use that many consumption models have a nesting structure implying that the continuation value can be efficiently pre-computed and the consumption choice solved separately before the remaining choices. Secondly, I use that an endogenous grid method extended with an upper envelope step can be used to solve efficiently for the consumption choice. Thirdly, I use that the required pre-computations can be optimized by a novel loop reordering when interpolating the next-period value function. As an illustrative example, I solve a model with non-durable consumption and durable consumption subject to adjustment costs. Combining the provided tools, the model is solved almost 50 times faster than with standard value function iteration for a given level of accuracy. Software is provided in both Python and C++.

Appendix: Model in Sect. 3.1

1.1 Bellman Equation

Defining the vector of state variables by $s_{t}=(p_{t},n_{t}^{1},n_{t}^{2},m_{t})$. The recursive Bellman equation for the extended benchmark model with two durable stocks can be written

$$\begin{aligned} v_{t}(s_{t})= & {} \max \left\{ \begin{array}{c} v_{t}^{keep}(p_{t},n_{t}^{1},n_{t}^{2},m_{t})\\ v_{t}^{adj.}(p_{t},x_{t})\\ v_{t}^{adj.,1}(p_{t},n_{t}^{2},x_{t}^{1})\\ v_{t}^{adj.,2}(p_{t},n_{t}^{2},x_{t}^{2}) \end{array}\right\} \nonumber \\&{\text {s.t.}}\nonumber \\ x_{t}= & {} m_{t}+(1-\tau _{1})n_{t}^{1}+(1-\tau _{2})n_{t}^{2}\nonumber \\ x_{t}^{1}= & {} m_{t}+(1-\tau _{1})n_{t}^{1},\nonumber \\ x_{t}^{2}= & {} m_{t}+(1-\tau _{2})n_{t}^{2}, \end{aligned}$$

(6.1)

where

$$\begin{aligned} v_{t}^{keep}(p_{t},n_{t}^{1},n_{t}^{2},m_{t})= & {} \max _{c_{t}}u(c_{t},n_{t}^{1},n_{t}^{2})+\beta {{\mathbb {E}}}_{t}[v_{t+1}(s_{t+1})]\nonumber \\&{\text {s.t.}}\nonumber \\ a_{t}= & {} m_{t}-c_{t}\nonumber \\ m_{t+1}= & {} (1+r)a_{t}+y_{t+1}\nonumber \\ n_{t+1}^{1}= & {} (1-\delta _{1})n_{t}^{1}\nonumber \\ n_{t+1}^{2}= & {} (1-\delta _{2})n_{t}^{2}\nonumber \\ a_{t}\ge & {} 0, \end{aligned}$$

(6.2)

and

$$\begin{aligned} v_{t}^{adj.}(p_{t},x_{t})= & {} \max _{c_{t},d_{t}^{1},d_{t}^{2}}u(c_{t},d_{t}^{1},n_{t}^{2})+\beta {{\mathbb {E}}}_{t}[v_{t+1}(s_{t+1})]\nonumber \\&{\text {s.t.}}\nonumber \\ a_{t}= & {} x_{t}-c_{t}-d_{t}^{1}-d_{t}^{2}\nonumber \\ m_{t+1}= & {} (1+r)a_{t}+y_{t+1}\nonumber \\ n_{t+1}^{1}= & {} (1-\delta _{1})d_{t}^{1}\nonumber \\ n_{t+1}^{2}= & {} (1-\delta _{2})d_{t}^{2}\nonumber \\ a_{t}\ge & {} 0, \end{aligned}$$

(6.3)

and

$$\begin{aligned} v_{t}^{adj.,1}(p_{t},n_{1}^{2},x_{t}^{1})= & {} \max _{c_{t},d_{t}^{1}}u(c_{t},d_{t}^{1},n_{t}^{2})+\beta {{\mathbb {E}}}_{t}[v_{t+1}(s_{t+1})]\nonumber \\&{\text {s.t.}}\nonumber \\ a_{t}= & {} x_{t}^{1}-c_{t}-d_{t}^{1}\nonumber \\ m_{t+1}= & {} (1+r)a_{t}+y_{t+1}\nonumber \\ n_{t+1}^{1}= & {} (1-\delta _{1})d_{t}^{1}\nonumber \\ n_{t+1}^{2}= & {} (1-\delta _{2})n_{t}^{2}\nonumber \\ a_{t}\ge & {} 0, \end{aligned}$$

(6.4)

and

$$\begin{aligned} v_{t}^{adj.,2}(p_{t},n_{1}^{1},x_{t}^{2})= & {} \max _{c_{t},d_{t}^{2}}u(c_{t},n_{t}^{1},d_{t}^{2})+\beta {{\mathbb {E}}}_{t}[v_{t+1}(s_{t+1})]\nonumber \\&{\text {s.t.}}\nonumber \\ a_{t}= & {} x_{t}^{2}-c_{t}-d_{t}^{2}\nonumber \\ m_{t+1}= & {} (1+r)a_{t}+y_{t+1}\nonumber \\ n_{t+1}^{1}= & {} (1-\delta _{1})n_{t}^{1}\nonumber \\ n_{t+1}^{2}= & {} (1-\delta _{2})d_{t}^{2}\nonumber \\ a_{t}\ge & {} 0. \end{aligned}$$

(6.5)

1.2 Nesting

Defining the post-decision value function

$$\begin{aligned} w_{t}(p_{t},d_{t}^{1},d_{t}^{2},a_{t})=\beta {{\mathbb {E}}}_{t}[v_{t+1}(p_{t+1},n_{t+1}^{1},n_{t+1}^{1},m_{t+1})], \end{aligned}$$

the keeper and adjuster value functions can be re-formulated as

$$\begin{aligned} v_{t}^{keep}(p_{t},n_{t}^{1},n_{t}^{2},m_{t})= & {} \max _{c_{t}}u(c_{t},n_{t}^{1},n_{t}^{2})+w_{t}(p_{t},d_{t}^{1},d_{t}^{2},a_{t})\nonumber \\&{\text {s.t.}}\nonumber \\ a_{t}= & {} m_{t}-c_{t}\nonumber \\ m_{t+1}= & {} (1+r)a_{t}+y_{t+1}\nonumber \\ n_{t+1}^{1}= & {} (1-\delta _{1})n_{t}^{1}\nonumber \\ n_{t+1}^{2}= & {} (1-\delta _{1})n_{t}^{2}\nonumber \\ a_{t}\ge & {} 0, \end{aligned}$$

(6.6)

and

$$\begin{aligned} v_{t}^{adj.}(p_{t},x_{t})= & {} \max _{d_{t}^{1},d_{t}^{2}}v_{t}^{keep}(p_{t},d_{t}^{1},d_{t}^{2},m_{t})\nonumber \\&{\text {s.t.}}\nonumber \\ m_{t}= & {} x_{t}-d_{t}^{1}-d_{t}^{2}\nonumber \\ d_{t}^{1}+d_{t}^{2}\in & {} [0,x_{t}], \end{aligned}$$

(6.7)

and

$$\begin{aligned} v_{t}^{adj.,1}(p_{t},n_{1}^{2},x_{t}^{1})= & {} \max _{d_{t}^{1},}v_{t}^{keep}(p_{t},d_{t}^{1},n_{t}^{2},m_{t})\nonumber \\&{\text {s.t.}}\nonumber \\ m_{t}= & {} x_{t}^{1}-m_{t}-d_{t}^{1}\nonumber \\ d_{t}^{1}\in & {} [0,x_{t}^{1}], \end{aligned}$$

(6.8)

and

$$\begin{aligned} v_{t}^{adj.,2}(p_{t},n_{1}^{1},x_{t}^{2})= & {} \max _{d_{t}^{2}}v_{t}^{keep}(p_{t},n_{t}^{1},d_{t}^{2},m_{t})\nonumber \\&{\text {s.t.}}\nonumber \\ m_{t}= & {} x_{t}^{2}-d_{t}^{2}\nonumber \\ d_{t}^{2}\in & {} [0,x_{t}^{2}]. \end{aligned}$$

(6.9)

1.3 EGM

Define the post-decision marginal value of cash by

$$\begin{aligned} q(p_{t},d_{t},a_{t}) =\beta R{{\mathbb {E}}}_{t}\left[ \alpha c_{t+1}^{\alpha (1-\rho )-1}((d_{t+1}^{1}+\underline{d}^{1})^{\gamma }(d_{t+1}^{2}+\underline{d}^{2})^{1-\gamma })^{(1-\alpha )(1-\rho )}\right] . \end{aligned}$$

(6.10)

The Euler-equation for $c_{t}$ is then given by

$$\begin{aligned} u_{c}(c_{t},d_{t}^{1},d_{t}^{2})=q(p_{t},d_{t}^{1},d_{t}^{2},a_{t}). \end{aligned}$$

(6.11)

This implies that EGM can be performed as follows

$$\begin{aligned} c_{t} =z(a_{t},d_{t}^{1},d_{t}^{2},p_{t}) \end{aligned}$$

(6.12)

$$\begin{aligned}&=\left( \frac{q_{t}(p_{t},d_{t},a_{t})}{\alpha ((d_{t}^{1}+\underline{d}_{1})^{\gamma }(d_{t}^{2}+\underline{d}_{2})^{1-\gamma })^{(1-\alpha )(1-\rho )}}\right) ^{\frac{1}{\alpha (1-\rho )-1}}\nonumber \\ m_{t}&=a_{t}+c_{t}. \end{aligned}$$

(6.13)

The upper envelope algorithm is unchanged.

1.4 Implementation

Parametrization The chosen parameters are $\beta =0.965$, $\rho =2$, $\alpha =0.9$ $\underline{d}_{1}=\underline{d}_{2}=10^{-2}$, $R=1.03$, $\tau _{1}=0.08$, $\tau _{1}=0.12$, $\delta _{1}=0.10$, $\delta _{2}=0.20$, $\gamma =0.5$, $\sigma _{\psi }=\sigma _{\xi }=0.1$, and $\lambda =1$.

Grids Constructed as in the benchmark model, but with the following changes $\#_{p}=50$, ${\overline{n}}=2$, $\#_{n}=50$, ${\overline{m}}=10$, $\#_{m}=100,$ ${\overline{x}}=12$, $\#_{x}=100$, $\#_{a}=100$.

Interpolation Done as in the benchmark model.

Simulation The initial values in the simulation are drawn as:

$$\begin{aligned} \log p_{0}&\sim {\mathcal {N}}(\log (1),0.2)\\ \log d_{0}^{1}&\sim {\mathcal {N}}(\log (0.4),0.2)\\ \log d_{0}^{1}&\sim {\mathcal {N}}(\log (0.4),0.2)\\ \log a_{0}&\sim {\mathcal {N}}(\log (0.2),0.1). \end{aligned}$$