Dynamic Aggregate Demand and Supply Model Notes

Dynamic Aggregate Demand and Dynamic Aggregate Supply

Overview

The lecture combines previous concepts to introduce the Dynamic Aggregate Demand (DAD) and Dynamic Aggregate Supply (DAS) model. The goals include understanding:

  • Constructing the DAD-DAS model.

  • Long-run equilibrium output and inflation.

  • Short-run equilibrium output and inflation.

  • The impact of demand and supply shocks on output and inflation over time.

Introduction

  • The traditional AD-AS model is implicitly dynamic and is used to understand economic fluctuations caused by policy changes or shocks.

  • The new model explicitly incorporates time, where variables in a period depend on variables in other periods.

  • This approach aligns with modern macroeconomic research, particularly Dynamic Stochastic General Equilibrium (DSGE) models, which are:

    1. Dynamic: Time is explicitly modeled.

    2. Stochastic: Subject to unpredictable exogenous shocks.

    3. Microfounded: Built from consumers’ and firms’ optimization (DAS-DAD model considers the first two).

  • The DAS-DAD model allows tracing the evolution of variables after shocks and policy changes.

Main Equations of the DAS-DAD Model

Demand Equations:

  • IS Curve (Goods and Services Market): Yt = Ȳ - α(rt - ρ) + ϵ_t

    • Y_t: Level of output.

    • Ȳ: Natural level of output (constant).

    • α > 0

    • ρ: Natural real interest rate (constant).

    • ϵ_t: Exogenous demand shocks (random variable).

  • Fisher Equation (Interest Rates and Inflation Expectations): it = rt + π^e_{t+1}

    • i_t: Nominal interest rate at time t.

    • r_t: Real interest rate at time t.

    • π^e_{t+1}: Inflation expectation made at time t.

  • Taylor Rule (Monetary Policy Rule): it = πt + ρ + ϕπ(πt - π^*) + ϕY(Yt - Ȳ)

    • π^*: Inflation target.

    • ϕ_π > 0: Central bank's adjustment to interest rate based on inflation deviation from the target.

    • ϕ_Y > 0: Central bank's adjustment to interest rate based on output deviation from the natural rate.

  • Adaptive Expectations: π^e{t+1} = πt

    • Agents expect next period's inflation to equal the current inflation level.

    • Results in inflation inertia.

Supply Equation:

  • Phillips Curve: πt = π^et + β(Yt - Ȳ) + vt

    • Uses the relationship between unemployment and output.

    • β > 0

    • v_t: Exogenous supply shock (random variable).

Variable Recap

  1. Endogenous Variables: Yt, πt, rt, it, π^e_{t+1}

  2. Exogenous Variables: Ȳ, ρ, π^*, ϵt, vt

  3. Parameters: α, ϕπ, ϕY, β

The model's solution expresses endogenous variables as functions of exogenous variables and parameters. The model is plotted in output-inflation space.

Dynamic Aggregate Demand (DAD)

  • To derive the DAD, the four demand equations are combined to eliminate endogenous variables other than output and inflation.

  1. Substitute the Fisher equation (solved for rt) into the IS curve:
    Y    t = Ȳ - α(it - π^e{t+1} - ρ) + ϵ_t

  2. Impose the adaptive expectations equation:
    Yt = Ȳ - α(it - πt - ρ) + ϵt

  3. Substitute the Taylor rule for it:
    Y    t = Ȳ - α(πt + ρ + ϕπ(πt - π^*) + ϕY(Yt - Ȳ) - πt - ρ) + ϵ_t

  4. Isolate Yt:
    Y    t(1 + αϕY) = Ȳ(1 + αϕY) - αϕπ(πt - π^*) + ϵ_t

  5. Solve for Yt (Aggregate Demand Curve): Yt = Ȳ - \frac{αϕπ}{(1 + αϕY)}(πt - π^*) + \frac{ϵt}{(1 + αϕ_Y)}

    • \frac{αϕπ}{(1 + αϕY)} > 0

    • \frac{1}{(1 + αϕ_Y)} > 0

This implies a negative relationship between current output and current inflation. Although it only contains subscript t, it is termed Dynamic Aggregate Demand. Shifts respond to changes in fiscal and monetary policy.

Dynamic Aggregate Supply (DAS)

The DAS derivation involves substituting the adaptive expectations equation into the Phillips Curve:

πt = π{t-1} + β(Yt - Ȳ) + vt

This implies a positive relationship between current inflation and current output. The equation is dynamic, containing variables from period t and period (t-1).

Long-Run Equilibrium

A long-run equilibrium is defined by:

  1. Output at the natural level: Y_t = Ȳ

  2. Correct expectations: π^et = πt for all t, resulting in constant inflation π{t+1} = πt.

  3. No exogenous shocks: ϵt = vt = 0

This arises due to price flexibility, ensuring output returns to the natural level and expectations are correct.

  • Y_t = Ȳ

  • r_t = ρ

  • π_t = π^*

  • i_t = ρ + π^*

Short-Run Equilibrium

The intersection of DAD and DAS determines short-run equilibrium output and inflation. Substitute DAS into DAD to solve for Yt, then substitute Yt into DAS to find π_t:

Yt = Ȳ + Aαϕπ(π^* - π{t-1}) + A(ϵt - αϕπvt)
πt = A(1 + αϕY)(π{t-1} + vt) + Aβ(αϕππ^* + ϵt)

Where A = \frac{1}{1 + αϕY + αϕπβ} > 0

These equilibrium equations hold for all t. The short-run equilibrium aligns with the long-run equilibrium when π^* = π{t-1}, ϵt = 0, and vt = 0, resulting in Yt = Ȳ and π_t = π^*.

Equilibrium output (Y_t) depends:

  • Positively on (π^* - π{t-1}). If π^* > π{t-1} = π^et, the central bank reduces the nominal interest rate, increasing Yt.

  • Positively on positive demand shocks ϵ_t > 0.

  • Negatively on negative supply shocks v_t < 0.

If π^* > π{t-1} = π^et, the central bank decreases the nominal interest rate, increasing Y_t.

Effects of an Aggregate Supply Shock

Suppose the economy starts at long-run equilibrium, and there's a one-period negative aggregate supply shock: v{t-1} = 0, vt > 0, and v_{t+1} = 0. This shock, affecting the Phillips Curve, might result from an international oil cartel increasing oil prices or a new union agreement raising wages.

  • Time t - 1: Long-run equilibrium.

  • Time t: v_t > 0 => DAS shifts up => Y ↓ and π ↑ (Stagflation).

  • Time t + 1: v{t+1} = 0, but πt > π_{t-1} => DAS shifts down a little => Y ↑ and π ↓.

  • Time t + 2: π{t+1} < πt => DAS shifts down a little => Y ↑ and π ↓…

This process continues until the economy returns to long-run equilibrium. Inflation expectations change as a result of the shocks and then gradually adjust.

Impulse Responses of an Aggregate Supply Shock

Impulse responses show the effect of a shock on endogenous variables over time. By coding the model equations in mathematical software, assigning numerical values to parameters and exogenous variables, and simulating the shock, one can observe the evolution of variables over time.

Effects of an Aggregate Demand Shock

Assume the economy starts at long-run equilibrium, and there is a positive aggregate demand shock lasting for 5 periods: ϵ{t-1} = 0, ϵt = ϵ{t+1} = … = ϵ{t+4} > 0, and ϵ_{t+5} = 0. This shock might be due to a war increasing government purchases or a stock market bubble increasing wealth.

  • Time t - 1: Long-run equilibrium.

  • Time t: ϵ_t > 0 => DAD shifts to the right => Y ↑ and π ↑.

  • Time t + 1: Inflation expectations increase because observed inflation increased in the last period => DAS shifts up => Y ↓ and π ↑.

  • Time t + 2 until t + 4: Inflation expectations continue to rise => DAS shifts up => Y ↓ and π ↑.

  • Time t + 5: ϵ_{t+5} = 0 => DAD returns to its initial position, DAS shifts up again because inflation increased in period t + 4 => Y ↓ and π ↓.

  • Time t + 6: Because inflation decreased in the last period, the DAS shifts down => Y ↑ and π ↓…

This continues until the long-run equilibrium is restored.