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name: econ-macro-modeling description: Macroeconomic modeling, calibration, and comparative statics. Use when working with DSGE models, growth models, general equilibrium, CES production, welfare analysis, or policy counterfactuals. allowed-tools:

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Macroeconomic Modeling

Model Selection Guide

| Model Type | Use When | Key Features | |------------|----------|--------------| | Two-period | Policy comparison, welfare | Simple, closed-form solutions | | OLG | Demographics, pensions, debt | Generational heterogeneity | | Infinite horizon | Long-run growth, BGP | Transversality conditions | | DSGE | Business cycles, monetary policy | Stochastic shocks, estimation |

Production Functions

CES (Constant Elasticity of Substitution)

$$Y = A \left[ \beta K^\rho + (1-\beta) L^\rho \right]^{1/\rho}$$

• Elasticity of substitution: $\sigma = \frac{1}{1-\rho}$
• $\rho \to 0$: Cobb-Douglas ($Y = A K^\beta L^{1-\beta}$)
• $\rho \to -\infty$: Leontief (fixed proportions)
• $\rho = 1$: Perfect substitutes

Factor Shares (CES)

$$s_L = (1-\beta) \left( \frac{AL}{Y} \right)^{1-\rho}$$ $$s_K = 1 - s_L$$

Equilibrium Concepts

1. Competitive equilibrium: Price-taking, markets clear
2. Walrasian: Auctioneer, no money
3. Monetary equilibrium: Cash-in-advance or MIU
4. Nash equilibrium: Strategic interaction (games)

Calibration Checklist

1. Targets: Identify moments to match (labor share, K/Y ratio, growth rate)
2. Parameters: List all parameters, note which are calibrated vs estimated
3. Data sources: National accounts, labor surveys, I-O tables
4. Identification: Ensure parameters are identified from chosen moments
5. Robustness: Test sensitivity to alternative calibrations

Common Calibration Targets

| Parameter | Typical Target | Data Source | |-----------|----------------|-------------| | $\beta$ (discount) | 0.96-0.99 annual | Risk-free rate | | $\alpha$ (capital share) | 0.33-0.40 | National accounts | | $\delta$ (depreciation) | 0.05-0.10 | Investment data | | $\sigma$ (IES) | 0.5-2.0 | Consumption Euler |

Welfare Analysis

Compensating Variation (CV)

Income needed to achieve reference utility at new prices: $$V(p_1, y + CV) = V(p_0, y)$$

Equivalent Variation (EV)

Income change equivalent to policy at original prices: $$V(p_0, y - EV) = V(p_1, y)$$

Consumption-Equivalent Welfare

$$\lambda: \quad U((1+\lambda)c_0) = U(c_1)$$

Standard Notation

| Symbol | Meaning | |--------|---------| | $Y, C, I, G$ | Output, consumption, investment, govt spending | | $K, L, H$ | Capital, labor, human capital | | $w, r, R$ | Wage, rental rate, gross interest rate | | $s_L, s_K$ | Labor share, capital share | | $\beta, \delta$ | Discount factor, depreciation | | $\rho, \sigma$ | CES parameter, elasticity of substitution | | $A, z$ | TFP, productivity shock |

Common Mistakes to Avoid

1. Not checking Inada conditions: Ensure interior solutions exist
2. Ignoring transitional dynamics: BGP may not be reached
3. Over-parameterization: More parameters than identifying moments
4. Units mismatch: Annual vs quarterly rates, levels vs logs
5. Forgetting budget constraints: Walras' law check