[IAM]Climate Economics and Finance

Nordhaus Dynamic Integrated model of Climate and Economy (DICE) Model

$$F_t = \eta\cdot \log_2\left(\frac{M_{AT,t}}{M_{AT,1750}}\right) + F_t^{\text{Abate}} + F_t^{\text{Ex}}$$

• $F_t$ : increased radiative forcing
• $M_{AT,t}$ : increased atmospheric carbon concentrations
• $F_t^{\text{Abate}}$ : non-CO2 forcing net of abatement
• $F_t^{\text{Ex}}$​ : exogenous forcing
• $D(T_t)$​ : damage function

$$Y_t^{\text{Net}} = \underbrace{(1-\Lambda_t(\mu_t))}_{\text{left after abatement}}\cdot [\underbrace{(1-D(T_t))}_{\text{left after damage}}\cdot \underbrace{Y_t^{\text{Gross}}}_{\text{gross output}}]$$

Implications :

• reduce emission $\mu_t\downarrow$ costs money today, but reduce climate damages in the future $T_t\downarrow$​

$$r = \rho + \phi g^* +\beta^{\text{CLIM}}\pi - \sigma^2_c\phi^2(0.5)$$

• $\rho$ : utility discount (patiency)
• $g^*$ : consumption growth
• $\phi$ : utility concavity

Social Cost of Carbon

$$\text{SCC}_t =\sum_j L_{t+j}\left(\frac{1}{1+\rho}\right)^j \frac{\Delta u(c_{t+j})}{\Delta u(c_t)} \frac{\Delta Y_{t+j}}{\Delta T_{t+j}}\frac{\Delta T_{t+j}}{\Delta E_t}$$

• $L_t$ : population
• $\rho$ : utility discount factor

Panel

linear

$$\text{Economic growth}_{i,t} = \beta \cdot \text{Temperature}_{i,t}+\text{Controls}_{i,t}+\varepsilon_{i,t}$$

nonlinear

$$\text{Outcome}_{i,t} = \beta_1\cdot \text{Temp}_{i,t}+\beta_2\cdot (\text{Temp}_{i,t})^2 + \text{Controls}_{i,t} + \varepsilon_{i,t}$$

Heterogeneous

$$\text{Outcome}_{i,t} = \beta_1\cdot \text{Temp}_{i,t} + \beta_2\cdot\left(\text{Temp}_{i,t}\cdot \text{Climate}_{i}\right)+\text{Controls}_{i,t}+\epsilon_{i,t}$$

Cyclones and Growth

Marginal Abatement cost function

Social Cost of Carbon & Excel-based IAM

• temperature change: $T_t = \eta\cdot \log_2\left(\frac{M_{AT,t}}{M_{AT,1750}}\right)$
  • $\eta$ : temperature sensitivity
  • $M_{AT,t}$ : total amount of CO2 at time $t$
  • $M_{AT,1750}$ : total amount of CO2 before industry
• damage function : $\frac{\text{Damage}_t}{\text{GDP}_t} = \psi_1 T_t^{\psi_2}$
  • $\psi_1,\psi_2$ : parameters
  • $T_t$ : temperature change
• Present Value(PV) of damage : $\frac{5\cdot \text{Damage}_t}{(1+r)^{t-t_0}}$​
• Present Value Total(PVT) of damage : $\sum_{t}\frac{5\cdot \text{Damage}_t}{(1+r)^{t-t_0}}$
• Total abatement cost (TAC) : $\frac{\text{TAC}_t}{\text{GDP}_t}=\theta_1\cdot (1+g)^{t-t_0}\mu^{\theta_2}$
  • $\theta_1,\theta_2,g$ : abatement parameters
  • $\mu$ : percentage emission reduction
• PV of TAC : $\frac{5\cdot \text{TAC}_t}{(1+r)^{t-t_0}}$​
• utility function: $u(c_t) = \frac{c_t^{1-\phi}}{1-\phi}$
  • $c_t = \frac{\text{GDP}_t^{\text{Net}}}{\text{Population}_t}$
  • $\phi$ : utility parameters
• PV of social welfare : $\text{PVSW}_t = L_t\cdot u(c_t)\cdot \left(\frac{1}{1+\rho}\right)^{t-t_0}$
  • $\rho$ : utility discount factor
  • $L_t$ : population
• Social Cost of Carbon (SCC) : $\text{SCC} = \frac{\text{PVT of damage}}{\Delta S}$

Lucas Tree Asset Pricing Model

$$P_t = \mathbb E_t\left[\sum_{j=1}^\infin \beta^j\left(\frac{u'(d_{t+j})}{u'(d_t)}\right)\cdot d_{t+j}\right]$$

• $\beta$ : impatience
• $u'(\cdot)$ : marginal utility of income
• $P_t$​ : price of the stock
• $d_{t+j}$ : dividends（股息)

Consumption Capital Asset Pricing Model(CCAPM)

$$\underbrace{u'(c_t)}_{\text{Marginal Cost(MC)}} = \underbrace{\beta\cdot \mathbb E_t\left[u'(c_{t+1})(1+r_{f,t+1})\right]}_{\text{Marginal Benefit(MB)}}$$

$$\mathbb E_t(r_{j,t+1}) - r_{f,t+1} = -(1+r_{f,t+1})\cdot \text{Cov}\left[\frac{u'(c_{t+1})}{u'(c_t)},r_{j,t+1}\right]$$

• $r_{f,t+1}$​ : risk free rate
• $u'(c_{t+1})$​ : marginal utility of consumption
• $\mathbb E_t(r_{j,t+1})$​ : expected return value
• $c_t$ : consumption at time $t$

Implications :

• $\text{Cov}\left[\frac{u'(c_{t+1})}{u'(c_t)},r_{j,t+1}\right]<0 \Leftrightarrow r\propto c\propto \frac{1}{u}\Leftrightarrow \text{risk}\uparrow$
• $\text{Cov}\left[\frac{u'(c_{t+1})}{u'(c_t)},r_{j,t+1}\right]>0 \Leftrightarrow r\propto u \propto \frac{1}{c}\Leftrightarrow \text{risk}\downarrow$​
• CCAPM implies that we should value carbon abatement relatively more

Efficient Market Hypothesis (EMH)

key idea : Asset prices reflect all available information about their value

Implications :

• Stock price movements random (walks)
• Trade-off between risk and expected return
• Known climate risks should already be priced into asset value

Balance Sheet

$$\text{Total Assets} = \text{Total Liabilities} + \text{Stockholders' equity}$$

Concept:

• Asset: something owned by the bank
  • Examples: bank reserves, cash equivalents, long-term investment
• Liability: something owed to another institution or person
  • demanded deposits(活期存款), short-term borrowing, long-term debts
• Stockholders’ equity

Example

Assets	Amount	Liabilities and Stockholders’ Equity	Amount
Reserves	$74	Demand deposits	$935
Cash equivalents	$274	Short-term borrowing	$429
Long-term investments	$1,453	Long-term debt	$208
Total assets	$1,801	Total liabilities	$1,572
		Stockholders’ equity	$229
		Total liabilities + Stockholders’ equity	$1,801

Bank

• Identify profitable lending opportunities : savers & borrowers
• Maturity Transformation : short-term liabilities into long-term investments
• Risk Management :
  • insolvent(资不抵债) : Stockholders’ equity > 0, loss of value in long-term investment
  • fire sale: too many depositors withdraw deposits at the same time, banks sell illiquid

Implications :

• $\text{risk}\uparrow\Leftrightarrow \text{long-term investments}\downarrow$​ : As long as stockholders’ equity is positive, this loss “comes out of” stockholders’ equity
• If The Efficient Markets Hypothesis holds and climate change turns out to be as expected, we would NOT expect physical climate impacts to pose a risk to bank solvency in the future.