Simulating the Ecological Transition

Introduction

In this tutorial, we will set up and solve a deterministic perfect foresight model of an ecological transition using the Stacked Time approach. We consider a firm that produces output using both “Brown” (carbon-heavy) and “Green” (clean) installed capital.

The firm faces an exogenous trajectory for a carbon tax \(\tau_t\) which starts at zero, ramps up linearly, and stabilizes.

1. The Model Equations

Let \(K^b_t\) and \(K^g_t\) denote Brown and Green capital.

Production: Output is given by a simple decreasing returns to scale function: \[ Y_t = A (K^b_t)^\alpha + A (K^g_t)^\alpha \quad \text{with } \alpha \in (0,1)\]

Capital Accumulation: Both capitals depreciate at rate \(\delta\). \[ K^b_{t} = (1-\delta) K^b_{t-1} + I^b_t \] \[ K^g_{t} = (1-\delta) K^g_{t-1} + I^g_t \]

Adjustment Costs: Changing the capital stock is costly due to friction. \[ \Phi(K_{t-1}, I_t) = \frac{\phi}{2} \left( \frac{I_t}{K_{t-1}} \right)^2 K_{t-1} \]

Firm’s Problem: Maximize the discounted sum of profits given a fixed interest rate \(r\) and a time-varying carbon tax \(\tau_t\): \[ \max \sum_{t=0}^\infty \left(\frac{1}{1+r}\right)^t \left[ Y_t - I^b_t - I^g_t - \tau_t K^b_t - \Phi(K^b_{t-1}, I^b_t) - \Phi(K^g_{t-1}, I^g_t) \right] \]

Question 1: Finding the Optimality Conditions

Set up the discrete-time Lagrangian and derive the first-order conditions with respect to \(I^b_t, K^b_{t}, I^g_t, K^g_{t}\).

Hint: Introduce Lagrange multipliers (Tobin’s marginal \(q_t\)) for the two capital accumulation equations. What is the interpretation of \(q_t\) in this context?

2. Setting up the \(f\) and \(g\) System

To apply our numerical solver cleanly, it is standard practice to separate the system into backward-looking transition equations and forward-looking optimality conditions.

Question 2: States and Controls

Identify the state variables (\(s_t\)) and control variables (\(x_t\)) for this model.

• Write down the transition functions \(s_t = g(s_{t-1}, x_{t-1}, z_t)\) explicitly.
• Write down the grouped optimality conditions \(f(s_t, x_t, x_{t+1}, s_{t+1}) = 0\).

3. The Steady States

Assume the tax \(\tau_t\) is constant at \(\bar{\tau}\) forever.

Question 3: The Boundary Conditions

Find the analytical expressions for the steady-state capital levels \(K^{b*}\) and \(K^{g*}\) as a function of the tax \(\bar{\tau}\).

• Compute the initial steady state when there is no climate policy (\(\tau = 0\)).
• Compute the final steady state after the policy is fully phased in (\(\tau = \tau_{\max}\)).

4. The Stacked Time Method

We want to simulate a long policy transition over \(T = 100\) periods. At \(t=0\), the government announces that starting tomorrow, the tax \(\tau_t\) will increase by \(\Delta \tau\) each period until it reaches \(\tau_{\max}\) at year \(20\), and remain there forever.

Question 4: Matrix Dimensions

Suppose we stack the model equations into a single nonlinear root-finding problem \(F(Y) = 0\) solving for the path from \(t=1\) to \(T\).

• How many endogenous variables are there per period?
• What is the total dimension of the vector \(Y\)?
• Describe the exact boundary conditions you will append to the top (initial) and bottom (terminal) of the stacked system.

Question 5: Computational Implementation

If you were to pass this massive stacked system to a standard solver in Julia like NLsolve.nlsolve, why would using a dense Jacobian matrix be computationally disastrous? How does providing a sparse, banded Jacobian (using Julia’s SparseArrays) fix this problem, and why is our system mathematically guaranteed to be banded?