Pakistan Economic Dashboard

Model Equations

The model is a standard New Keynesian DSGE framework adapted for Pakistan, featuring households, firms, and a government. Key equations include:

1. Households

A representative household maximizes utility from consumption (C) relative to a habit stock (h), and leisure (1-L), subject to a budget constraint.

$$ \max E_0 \sum_{t=0}^{\infty} \beta^t \left( \frac{(C_t - hC_{t-1})^{1-\sigma}}{1-\sigma} - \frac{L_t^{1+\phi}}{1+\phi} \right) $$

Budget Constraint:

$$ P_t C_t + B_{t+1} \leq R_t B_t + W_t L_t + \Pi_t $$

2. Firms

Intermediate goods firms operate in a monopolistically competitive market with sticky prices (Calvo-pricing). They produce using a Cobb-Douglas technology with capital (K) and labor (L).

$$ Y_t(i) = A_t K_t(i)^{\alpha} L_t(i)^{1-\alpha} $$

Final good is a CES aggregate of intermediate goods.

3. Monetary Policy

The central bank follows a Taylor-type rule, reacting to inflation \( \pi_t \) and output gap \( \tilde{y}_t \).

$$ \frac{R_t}{R_{ss}} = \left( \frac{R_{t-1}}{R_{ss}} \right)^{\rho_R} \left[ \left( \frac{\pi_t}{\pi_{ss}} \right)^{\phi_{\pi}} \left( \frac{Y_t}{Y_{ss}} \right)^{\phi_y} \right]^{1-\rho_R} \exp(\varepsilon_{R,t}) $$

4. Climate Shocks

Agricultural Productivity Shock (\( A_{agri,t} \)):

$$ \log(A_{agri,t}) = \rho_A \log(A_{agri,t-1}) + \varepsilon_{A,t} $$

Capital Destruction Shock (\( \delta_t \)):

$$ K_t = (1 - \delta_t) K_{t-1}^{eff} + I_t $$

Where \( \delta_t \) is the depreciation rate, which spikes during an extreme weather event.

Calibrated Parameters (Long-run Values)

Parameter	Value	Description	Justification
\( \beta \)	0.99	Household discount factor	Standard value implying a 4% annual real interest rate.
\( \sigma \)	1.5	Coefficient of relative risk aversion	Common in literature for emerging economies.
\( \alpha \)	0.50	Capital share in production	Calibrated based on DSGE models for Pakistan.
\( \delta \)	0.0375	Quarterly depreciation rate	Consistent with an annual depreciation rate of 15% used in DSGE models for Pakistan.
\( \theta_p \)	0.75	Calvo price stickiness	Implies prices are fixed for an average of 4 quarters.
\( \phi_{\pi} \)	1.5	Taylor rule inflation response	Standard value for ensuring macroeconomic stability.
\( \phi_{y} \)	0.25	Taylor rule output gap response	Standard value.
\( \rho_A \)	0.85	Persistence of productivity shock	Assumed high persistence for climate-related shocks.

Variable Definitions

Endogenous Variables

• Output (Y): Total economic output (GDP).
• Consumption (C): Household spending on goods and services.
• Investment (I): Spending on new capital goods.
• Inflation (π): Rate of increase in the general price level.
• Nominal Interest Rate (R): The cost of borrowing set by the central bank.
• Poverty Rate: Percentage of households below a consumption-based poverty line (proxied by deviation in low-income household consumption).

Exogenous Shock Variables

• \(\varepsilon_{A,t}\): Shock to agricultural productivity.
• \(\varepsilon_{K,t}\): Shock to capital stock (representing destruction).
• \(\varepsilon_{R,t}\): Shock to monetary policy.