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 $$ \text{SCC}t =\sum_j L $$}\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 $$ \text{Economic growth}{i,t} = \beta \cdot \text{Temperature}}+\text{Controls{i,t}+\varepsilon $$

nonlinear $$ \text{Outcome}{i,t} = \beta_1\cdot \text{Temp}}+\beta_2\cdot (\text{Temp{i,t})^2 + \text{Controls} $$ } + \varepsilon_{i,tHeterogeneous $$ \text{Outcome}{i,t} = \beta_1\cdot \text{Temp}} + \beta_2\cdot\left(\text{Temp{i,t}\cdot \text{Climate}}\right)+\text{Controls{i,t}+\epsilon $$

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.