Derivation of the Continuous Equations

We start with the standard capital accumulation equation in the presence of climate damage:

$$\dot{K}_{r,s}(t) = I_{r,s}(t) - \delta_{r,s} K_{r,s}(t) - D_{r,s}(t) \tag{1}$$

where:

• $I_{r,s}(t) = I^p_{r,s}(t) + I^{\xi}_{r,s}(t)$ is total investment
• $\delta_{r,s}$ is the standard depreciation rate for region $r$, sector $s$
• $D_{r,s}(t)$ represents climate damage to capital

Summary of Continuous-Time Dynamics

We arrive at the following system of continuous-time equations:

Potential Capital Stock Equation

Starting with: $$\dot{K}^p_{r,s}(t) = I^p_{r,s}(t) - \delta_{r,s} K^p_{r,s}(t)$$

This is a first-order linear differential equation. To solve it, we rearrange: $$\dot{K}^p_{r,s}(t) + \delta_{r,s} K^p_{r,s}(t) = I^p_{r,s}(t)$$

Using the integrating factor method with $e^{\delta_{r,s} t}$: $$e^{\delta_{r,s} t}\dot{K}^p_{r,s}(t) + \delta_{r,s} e^{\delta_{r,s} t}K^p_{r,s}(t) = e^{\delta_{r,s} t}I^p_{r,s}(t)$$

The left side is the derivative of $e^{\delta_{r,s} t}K^p_{r,s}(t)$: $$\frac{d}{dt}[e^{\delta_{r,s} t}K^p_{r,s}(t)] = e^{\delta_{r,s} t}I^p_{r,s}(t)$$

Integrating both sides from $t$ to $t+\Delta t$: $$\int_t^{t+\Delta t} \frac{d}{ds}[e^{\delta_{r,s} s}K^p_{r,s}(s)]ds = \int_t^{t+\Delta t} e^{\delta_{r,s} s}I^p_{r,s}(s)ds$$

Evaluating the left side: $$e^{\delta_{r,s}(t+\Delta t)}K^p_{r,s}(t+\Delta t) - e^{\delta_{r,s} t}K^p_{r,s}(t) = \int_t^{t+\Delta t} e^{\delta_{r,s} s}I^p_{r,s}(s)ds$$

Assuming $I^p_{r,s}(s)$ is constant over $[t, t+\Delta t]$ with value $I^p_{r,s,t}$: $$e^{\delta_{r,s}(t+\Delta t)}K^p_{r,s}(t+\Delta t) - e^{\delta_{r,s} t}K^p_{r,s}(t) = I^p_{r,s,t}\int_t^{t+\Delta t} e^{\delta_{r,s} s}ds$$

$$e^{\delta_{r,s}(t+\Delta t)}K^p_{r,s}(t+\Delta t) - e^{\delta_{r,s} t}K^p_{r,s}(t) = I^p_{r,s,t}\left[\frac{e^{\delta_{r,s} s}}{\delta_{r,s}}\right]_t^{t+\Delta t}$$

$$e^{\delta_{r,s}(t+\Delta t)}K^p_{r,s}(t+\Delta t) - e^{\delta_{r,s} t}K^p_{r,s}(t) = I^p_{r,s,t}\frac{e^{\delta_{r,s}(t+\Delta t)} - e^{\delta_{r,s} t}}{\delta_{r,s}}$$

Solving for $K^p_{r,s}(t+\Delta t)$: $$K^p_{r,s}(t+\Delta t) = e^{-\delta_{r,s} \Delta t}K^p_{r,s}(t) + I^p_{r,s,t}\frac{1 - e^{-\delta_{r,s} \Delta t}}{\delta_{r,s}} \tag{7}$$

This gives us the discrete-time equation for potential capital stock with time step $\Delta t$.

This equation tracks the evolution of potential capital stock, accounting for standard depreciation (through the survival fraction $e^{-\delta_{r,s} \Delta t}$) and new capital investments. The term $\frac{1-e^{-\delta_{r,s} \Delta t}}{\delta_{r,s}}$ represents the effective multiplier for investment over the time interval, accounting for the fact that new investments also begin to depreciate immediately.

Production Capacity Factor Equation

The continuous-time equation for the capacity factor (equation Equation 6) is: $$\dot{\xi}_{r,s}(t) = \frac{I^{\xi}_{r,s}(t)}{K^p_{r,s}(t)} + \frac{I^{p}_{r,s}(t)}{K^p_{r,s}(t)} \left(1 - \xi_{r,s}(t)\right) - \delta^D_{r,s}(t) \cdot \xi_{r,s}(t)$$

Two approaches can be taken to derive the discrete-time equation: assuming or not that the potential capital stock is constant within a time step. The detailed derivations of both approaches are provided in Section 5, Section 6 and Section 7.

Complete Discrete-Time System

After integrating the continuous equations, we arrive at a five-equation system that fully characterizes capital dynamics with climate damage for any time step $\Delta t$:

• Capital Decomposition Equqtion, Equation 2: $K_{r,s}(t) = \xi_{r,s}(t) \cdot K^p_{r,s}(t)$
• Investment Split Equation, Equation 3: $I_{r,s,t} = I^p_{r,s,t} + I^{\xi}_{r,s,t}$
• Investment Allocation Equation/Inequality, Equation 4: $$I^{\xi}_{r,s,t} = \mathcal{I}_{r,s,t} \quad \text{or} \quad I^{\xi}_{r,s,t} \leq \mathcal{I}_{r,s,t}$$
• Potential Capital Stock Equation, Equation 7: $$K^p_{r,s,t+\Delta t} = e^{-\delta_{r,s} \Delta t}K^p_{r,s,t} + I^p_{r,s,t} \cdot \frac{1-e^{-\delta_{r,s} \Delta t}}{\delta_{r,s}}$$
• Production Capacity Factor Equation, Equation 9: $$\xi_{r,s,t+\Delta t} = e^{-(\delta^D_{r,s,t} + \delta_{r,s})\Delta t} \frac{K^p_{r,s,t}}{K^p_{r,s,t+\Delta t}}\xi_{r,s}(t) + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s,t+\Delta t}} \cdot \frac{1-e^{-(\delta^D_{r,s,t} + \delta_{r,s})\Delta t}}{\delta^D_{r,s,t} + \delta_{r,s}}$$

Based on the Potential Capital Stock Equation (Equation 7) and Production Capacity Factor Equation (Equation 9), we have that the dynamical equation for the capital stock is given by: $$K_{r,s,t+\Delta t} = K^p_{r,s,t+\Delta t} \cdot \xi_{r,s,t+\Delta t} = e^{-(\delta^D_{r,s,t} + \delta_{r,s})\Delta t} K_{r,s,t} + \left( I^{\xi}_{r,s,t} + I^{p}_{r,s,t} \right) \cdot \frac{1-e^{-(\delta^D_{r,s,t} + \delta_{r,s})\Delta t}}{\delta^D_{r,s,t} + \delta_{r,s}} \tag{10}$$

Note: The capital stock dynamic is independent of the investment allocation between $I^p_{r,s,t}$ and $I^{\xi}_{r,s,t}$.

Steady State

In the steady state, the capital stock and capacity factor remain constant over time, implying:

• $K^p_{r,s,t+\Delta t} = K^p_{r,s,t} = K^p_{r,s}$
• $\xi_{r,s,t+\Delta t} = \xi_{r,s,t} = \xi_{r,s}$
• $K_{r,s,t+\Delta t} = K_{r,s,t} = K_{r,s}$

Small Depreciation Rate Approximation

When the depreciation rates $\delta_{r,s}$ and $\delta^D_{r,s}$ are relatively small (which is often the case for annual rates in macroeconomic models), we can simplify our discrete-time equations using the first-order Taylor approximation of the exponential function: $e^{-x} \approx 1 - x$ for small $x$.

Applying this approximation to our equations:

Potential Capital Stock Evolution:

$$K^p_{r,s,t+\Delta t} = e^{-\delta_{r,s} \Delta t}K^p_{r,s,t} + I^p_{r,s,t} \cdot \frac{1-e^{-\delta_{r,s} \Delta t}}{\delta_{r,s}}$$

$$\approx (1-\delta_{r,s} \Delta t)K^p_{r,s,t} + I^p_{r,s,t} \cdot \frac{1-(1-\delta_{r,s} \Delta t)}{\delta_{r,s}}$$

$$= (1-\delta_{r,s} \Delta t)K^p_{r,s,t} + I^p_{r,s,t} \cdot \frac{\delta_{r,s} \Delta t}{\delta_{r,s}}$$

$$= (1-\delta_{r,s} \Delta t)K^p_{r,s,t} + I^p_{r,s,t} \cdot \Delta t$$

This recovers the familiar discrete-time capital motion equation from standard macroeconomic models when $\Delta t = 1$:

$$K^p_{r,s,t+1} = (1-\delta_{r,s})K^p_{r,s,t} + I^p_{r,s,t}$$

Production Capacity Factor Evolution (assuming constant $K^p$):

Starting from the exact discrete equation Equation 8:

$$\xi_{r,s,t+\Delta t} = e^{-\gamma_{r,s,t} \Delta t}\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s,t}} \cdot \frac{1 - e^{-\gamma_{r,s,t} \Delta t}}{\gamma_{r,s,t}}$$

where $\gamma_{r,s,t} = \delta^D_{r,s,t} + \frac{I^{p}_{r,s,t}}{K^p_{r,s,t}}$.

Using the approximation $e^{-x} \approx 1 - x$ for small $x$:

$$\approx (1-\gamma_{r,s,t} \Delta t)\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s,t}} \cdot \frac{\gamma_{r,s,t} \Delta t}{\gamma_{r,s,t}}$$

$$= (1-\gamma_{r,s,t} \Delta t)\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s,t}} \cdot \Delta t$$

For the standard annual time step ($\Delta t = 1$), this simplifies to:

$$\xi_{r,s,t+1} = (1-\gamma_{r,s,t})\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s,t}}$$

Expanding $\gamma_{r,s,t}$: $$\xi_{r,s,t+1} = (1-\delta^D_{r,s,t})\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t}}{K^p_{r,s,t}} + \frac{I^{p}_{r,s,t}}{K^p_{r,s,t}}(1 - \xi_{r,s,t})$$

Production Capacity Factor Evolution (considering time-varying $K^p$):

Starting from the discrete equation Equation 9: $$\xi_{r,s}(t+\Delta t) = e^{-(\delta^D_{r,s,t} + \delta_{r,s}) \Delta t} \frac{K^p_{r,s}(t)}{K^p_{r,s}(t+\Delta t)}\xi_{r,s}(t) + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s}(t+\Delta t)} \cdot \frac{1-e^{-(\delta^D_{r,s,t} + \delta_{r,s}) \Delta t}}{\delta^D_{r,s,t} + \delta_{r,s}}$$

For small $x$, use $e^{-x} \approx 1 - x$ and $\frac{1-e^{-x}}{x} \approx 1$:

• $e^{-(\delta^D_{r,s,t} + \delta_{r,s}) \Delta t} \approx 1-(\delta^D_{r,s,t} + \delta_{r,s}) \Delta t$
• $\frac{1-e^{-(\delta^D_{r,s,t} + \delta_{r,s}) \Delta t}}{\delta^D_{r,s,t} + \delta_{r,s}} \approx \Delta t$

This gives: $$\xi_{r,s}(t+\Delta t) \approx [1-(\delta^D_{r,s,t} + \delta_{r,s}) \Delta t] \frac{K^p_{r,s}(t)}{K^p_{r,s}(t+\Delta t)}\xi_{r,s}(t) + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s}(t+\Delta t)} \Delta t$$

For the standard annual time step ($\Delta t = 1$), this simplifies to: $$\xi_{r,s,t+1} \approx [1-(\delta^D_{r,s,t} + \delta_{r,s})] \frac{K^p_{r,s,t}}{K^p_{r,s,t+1}}\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t} + I^{p}_{r,s,t}}{K^p_{r,s,t+1}}$$

By proceeding to further simplification (see Section 8), we can find an expression with a similar pattern than for the case assuming constant $K^p$:

$$\xi_{r,s,t+1} \approx (1-\delta^D_{r,s,t})\xi_{r,s,t} + \frac{I^{\xi}_{r,s,t}}{K^p_{r,s}(t+1)} + \frac{I^{p}_{r,s,t}}{K^p_{r,s}(t+1)}(1 - \xi_{r,s,t})$$

Combined Effect on Actual Capital:

As shown in Section Section 2.3, the exact capital stock dynamics is given by: $$K_{r,s,t+\Delta t} = e^{-(\delta^D_{r,s,t} + \delta_{r,s})\Delta t} K_{r,s,t} + \left( I^{\xi}_{r,s,t} + I^{p}_{r,s,t} \right) \cdot \frac{1-e^{-(\delta^D_{r,s,t} + \delta_{r,s})\Delta t}}{\delta^D_{r,s,t} + \delta_{r,s}}$$

For small depreciation rates and annual time step ($\Delta t = 1$), using the approximations $e^{-x} \approx 1-x$ and $\frac{1-e^{-x}}{x} \approx 1$:

$$K_{r,s,t+1} \approx [1-(\delta^D_{r,s,t} + \delta_{r,s})]K_{r,s,t} + (I^{\xi}_{r,s,t} + I^{p}_{r,s,t})$$

Key insights from this approximation:

1. Additive depreciation effects: In the first-order approximation, standard depreciation ($\delta_{r,s}$) and climate damage ($\delta^D_{r,s,t}$) act additively on the capital stock.

2. Investment allocation neutrality: The capital stock evolution is independent of how total investment is split between rebuilding ($I^{\xi}_{r,s,t}$) and new capital ($I^{p}_{r,s,t}$). This split only affects the evolution of $\xi_{r,s,t}$ and $K^p_{r,s,t}$ separately.

3. Comparison with standard model: Without climate damage ($\delta^D_{r,s,t} = 0$), this reduces to the standard capital accumulation equation: $$K_{r,s,t+1} \approx (1-\delta_{r,s})K_{r,s,t} + I_{r,s,t}$$

4. Effective depreciation rate: The presence of climate damage creates an effective depreciation rate of $(\delta_{r,s} + \delta^D_{r,s,t})$ in the linearized approximation, though the exact solution shows these effects combine non-linearly through the exponential term.

Remarks

Damage on the least efficient capital vs Uniform damage

The production function specification determines how climate damage affects the marginal productivity of capital and consequently influences optimal investment allocation between new capital ($I^p$) and rebuilding ($I^{\xi}$). We consider four main specifications:

Case 1: Damage on least efficient capital (Investment allocation neutral) $$Y_{r,s,t} = A_{r,s,t} (K_{r,s,t})^{\alpha} L_{r,s,t}^{1-\alpha} = A_{r,s,t} (\xi_{r,s,t} K^p_{r,s,t})^{\alpha} L_{r,s,t}^{1-\alpha}$$

This formulation assumes damage preferentially affects the least productive capital (e.g., older, less maintained infrastructure). Since only the total effective capital stock $K = \xi K^p$ matters for production, the marginal product of capital is independent of how that stock is achieved. Therefore, $I^p$ and $I^{\xi}$ have identical effects on production, making the optimizer indifferent between investment types.

Case 2: Uniform damage (Rebuilding preferred) $$Y_{r,s,t} = A_{r,s,t} \xi_{r,s,t} (K^p_{r,s,t})^{\alpha} L_{r,s,t}^{1-\alpha}$$

Here, all capital is equally susceptible to damage, and the capacity factor directly multiplies total output. This creates a strong preference for rebuilding since:

• Marginal product of rebuilding: $\frac{\partial Y}{\partial I^{\xi}} = A \alpha K^{\alpha-1} + A (K^p)^{\alpha}$
• Marginal product of new capital: $\frac{\partial Y}{\partial I^p} = A \xi \alpha (K^p)^{\alpha-1}$

Since $\xi \leq 1$ and $\alpha < 1$, rebuilding always dominates new investment.

Case 3: Variable damage intensity $$Y_{r,s,t} = A_{r,s,t} \left[\xi_{r,s,t}\right]^{\alpha \cdot \phi(\xi_{r,s,t})} \left[K^p_{r,s,t}\right]^{\alpha} L_{r,s,t}^{1-\alpha}$$

where $\phi(\xi) \geq 1$ is a decreasing function with $\phi(1) = 1$. Since $\xi \leq 1$, this formulation implies that when capital is damaged, the remaining undamaged capital becomes more productive than the least efficient capital. This specification provides flexibility as $\phi(\cdot)$ can vary across regions and sectors to reflect different damage patterns.

Case 4: Explicit capital heterogeneity $$Y_{r,s,t} = A_{r,s,t} \left[\int_0^1 q(v) \cdot k_{r,s,t}(v) dv\right]^{\alpha} L_{r,s,t}^{1-\alpha}$$

where $k_{r,s,t}(v)$ represents capital of quality $v \in [0,1]$ and $q(v)$ is the productivity of quality-$v$ capital. Damage could destroy capital across all quality levels, allowing the optimizer to prioritize rebuilding the highest-quality capital first.

Location-specific productivity and risk

Location factors introduce additional trade-offs between rebuilding damaged capital in-place versus creating new capital in different locations. We can model this by introducing location-specific productivity and damage parameters.

Location-specific capital heterogeneity

Let capital be indexed by location $\ell \in \mathcal{L}$, with production function: $$Y_{r,s,t} = A_{r,s,t} \left[\int_{\mathcal{L}} a_{r,s}(\ell) \xi_{r,s,t}(\ell) K^p_{r,s,t}(\ell) d\ell\right]^{\alpha} L_{r,s,t}^{1-\alpha}$$

where $a_{r,s}(\ell)$ captures location-specific productivity advantages (e.g., agglomeration effects, infrastructure access).

The location-specific damage rate $\delta^D_{r,s,t}(\ell)$ affects the capacity factor dynamics: $$\frac{d}{dt} \xi_{r,s,t}(\ell) = \frac{I^{\xi}_{r,s,t}(\ell)}{K^p_{r,s,t}(\ell)} + \frac{I^{p}_{r,s,t}(\ell)}{K^p_{r,s,t}(\ell)} \left(1 - \xi_{r,s,t}(\ell)\right) - \delta^D_{r,s,t}(\ell) \cdot \xi_{r,s,t}(\ell)$$