New Keynesian Model Study Notes

The New Keynesian model modifies the Ramsey model with two main changes: - Introduction of multiple monopolistic firms: Each firm is responsible for producing unique goods, which alters the market structure significantly. This change allows firms to act as price setters rather than price takers, impacting their pricing strategies and market competition. - Price Stickiness: In the short run, some firms cannot adjust prices according to current economic conditions, leading to a situation where prices remain fixed despite changes in demand or costs.

Lifetime Utility Function: The representative household's utility is given by: $U(C_1) - V(L_1) + \frac{1}{1 + \rho} \bigg[ U(C_2) - V(L_2) \bigg]$
Definitions: - $U$: neoclassical utility function representing satisfaction derived from consumption. - $V$: disutility of labor function, where $V'(L) > 0$ (indicating the more labor, the more disutility) and $V''(L) > 0$ (indicating increasing marginal disutility of labor).
Budget Constraints: - Flow budget constraints are defined for each period: - Period 1: $P_1 C_1 + B_2 \leq P_1 w_1 L_1 - T_1 + \text{profit}_1$ - Period 2: $P_2 C_2 + B_3 \leq (1 + i_1) B_2 + P_2 w_2 L_2 - T_2 + \text{profit}_2$ - Initial savings are given as $B_1 = 0$ , and the No-Ponzi Game Constraint is characterized by $B_3 \geq 0$ , ensuring that households do not accumulate negative savings over their lifetime.
Labor Supply Constraints: - Without loss of generality, assume optimal labor supply is constrained to $L_t \leq N$, where households choose to work less due to the disutility experienced from labor.
Role of Money: - Money serves as a unit of account and is assumed not to facilitate transactions more effectively than other assets, following the cash-in-advance model.
Equity Ownership: - Households are considered equity owners of firms, receiving profits in a lump-sum manner, which modifies the budget constraints by including profits in their income.

The optimality conditions for the households include: - Consumption Euler Equation: $U'(C_1) = \frac{(1 + i_1)}{(1 + \rho)} \frac{P_1}{P_2} U'(C_2)$ - Labor Supply Conditions: - $V'(L_1) = w_1 U'(C_1)$ - $V'(L_2) = w_2 U'(C_2)$

The model incorporates many monopolistic firms instead of a single representative firm, with each firm employing the production function: $Y_{it} = A_t L_{dit}$ - Where: - $Y_{it}$: Represents output produced by firm $i$. - $A_t$: Technological factor that influences productivity. - $L_{dit}$: Labor demand associated with firm $i$.
Demand Function: Firms set prices based on demand: $Y_{it} = \left( \frac{P_{it}}{P_t} \right)^{-\eta} Y_t$ - Where: - $\eta > 1$ indicates the elasticity of demand.
Profit Maximization: Firms aim to maximize after-tax profit defined by: $\text{profit}<em>{it} = (1 - \tau_t) P</em>{it} Y_{it} - P_t w_t L_{dit}$
Optimal Pricing Condition: - Through derivation, we arrive at the pricing formula: $P_{it} = \frac{\theta}{(\theta - 1)(1 - \tau_t)} P_t w_t A_t$ - This reflects a markup over nominal marginal costs, known as the Lerner rule.

Pricing behavior shifts between period 1 and period 2: - In period 1, a fraction $0 \leq \alpha \leq 1$ of firms cannot optimally set prices, leading them to set a fixed price $P_0$. - Firms that can adjust prices will do so according to the Lerner rule, leading to a pricing structure defined as: $P_{i1}=\begin{cases} P_0 & \text{if firm cannot adjust} \\frac{\theta(\theta - 1)(1 - \tau_{1}) P_1 w_1 A_1}{\text{if firm can adjust}} & \text{for firms that can adjust} \end{cases}$ - This results in firm-specific price settings within the same time period.

The nominal GDP is defined as: $P_t Y_t = \text{sum of } P_{it} Y_{it} \text{ across firms via integration: } P_t Y_t = \int P_{it} Y_{it} di$
Expression Relating Price Index: - From derivations, we derive: $P^{1-\theta}<em>{t} = \left( \int P</em>{it}^{1-\theta} di \right)$
Analysis in period 2 indicates all firms set the same price, impacting nominal equilibrium: $P_{i2} = P_2$
The market clearing conditions ensure that labor demand aligns with output, which must equal aggregate output.

Monetary Policy: - The central bank controls nominal interest rates without directly influencing transactions, maintaining the target for $i_1$ to achieve price stability.
Fiscal Policy: - The government operates a balanced budget and rebats any sales taxes in a lump-sum manner.

Equilibrium is defined by a set of prices and allocations satisfying the following system of equations: - $U'(C_1) = \frac{(1 + i_1)}{(1 + \rho)} \frac{P_1}{P_2} U'(C_2)$ - $V'(L_1) = w_1 U'(C_1)$
Notes on Equilibrium Variables: The variables $Y$, $C$, and $L$ represent output, consumption, and labor for each period, respectively.

The equilibrium conditions can be consolidated into two essential equations governing output and inflation in period 1: - First Fundamental Equation (Aggregate Demand Curve): $1 + \pi_1 = … \text{(aggregate demand relation) }$
- Second Fundamental Equation (New Keynesian Phillips Curve): $1 + \pi_1 = … \text{(supply relationship) }$

Increasing $\rho$ shifts demand higher, affecting equilibrium levels, resulting in increased output and inflation in period 1. The real interest rate reflects changing prices.

Enhancing technology directly impacts firms, decreasing real marginal costs, thereby influencing price adjustments.
The aggregate supply curve shifts to reflect reductions in inflation in period 1, guiding overall economic changes.

Using the Pareto planner framework, we evaluate the allocation of resources to maximize lifetime utility under constraints, revealing inefficiencies in decentralized solutions.
Aligning resource allocation with social optimum underscores discrepancies in output levels, highlighting the need for strategic policies to rectify market failures due to monopolistic pricing.

Reflects the planner's optimal response through interest rate adjustments to stabilize prices, emphasizing the central bank's role in achieving price stability conducive to socially efficient output levels amidst economic fluctuations.
This analytical approach illustrates how monetary policy effectively optimizes inflation control.

Comprehensive citations that underpin theoretical perspectives generated from numerical outputs include: - Gal´ı, J. and P. Rabanal (2004). - Lerner, A. P. (1934). - Phillips, A. W. (1958).