Lecture 2A: From marginal cost to the three equations

Start here if this is new to you

This chapter puts the household rules and the firm rules together. The link is marginal cost, the cost of producing one more unit. When the economy produces more than normal, firms need more hours, workers demand higher wages for them, so marginal cost rises and firms want to raise prices: that is how a booming economy creates inflation. The output gap x_t measures how far output is above or below the level that would prevail if prices were fully flexible (natural output). Positive gap: overheating, rising costs, upward pressure on prices. Negative gap: slack, falling costs, downward pressure.

The destination is three short equations. The Phillips curve: inflation today depends on expected inflation and on the output gap. The IS curve: the output gap depends on how far the real interest rate is above the natural rate. The policy rule: how the central bank sets the rate. If you can state what each one says in a sentence, read the direction of every term (does it push up or down?), and plug in numbers, you are in good shape. Every derivation step below is a substitution: replace one variable by what another equation says it equals.

Marginal cost and output

Marginal cost is not observed in the data. So express it in terms of output, using three conditions (logs, deviations from steady state):

Marginal cost: mc_t = w_t/A_t ⇒ m̂c_t = w_t − a_t
Labor supply (households): w_t = σc_t + φn_t
Market clearing: goods, c_t = y_t (adjustment costs are second order around zero inflation); labor, y_t = a_t + n_t ⇒ n_t = y_t − a_t

Substitute the second and third into the first:

Step by step: the substitution

Marginal cost is the wage divided by productivity: m̂c_t = w_t − a_t (in logs, division becomes subtraction).
Replace the wage by what households demand: w_t = σc_t + φn_t. Now m̂c_t = σc_t + φn_t − a_t.
Everything produced is consumed (no investment or government in this small model), so c_t = y_t.
Output is productivity times hours, so hours are output minus productivity: n_t = y_t − a_t.
Substitute both: m̂c_t = σy_t + φ(y_t − a_t) − a_t = (σ + φ)y_t − (1 + φ)a_t.

Read the result: output pushes costs up through two channels (σ: richer households want higher wages; φ: more hours need higher wages), productivity pushes them down (fewer hours per unit, and the same wage buys more output).

Marginal cost in terms of output

m c_{t} = σ y_{t} + φ (y_{t} - a_{t}) - a_{t} = (σ + φ) y_{t} - (1 + φ) a_{t}

Higher output means more hours (requiring higher wages, via φ) and more consumption (requiring higher wages, via σ), so higher marginal cost.
Higher productivity means fewer hours per unit, so lower marginal cost.

Worked example

σ = 1, φ = 1. Output is 2% above steady state, productivity 1% above. m̂c_t = (1 + 1) × 2 − (1 + 1) × 1 = 2%: firms' costs are 2% above normal.

Natural output and the output gap

Natural output yⁿ_t is output if prices were flexible. Then firms always charge their desired markup, so m̂c_t = 0:

0 = (σ + φ) y_{t}^{n} - (1 + φ) a_{t} ⟹ y_{t}^{n} = \frac{1 + φ}{σ + φ} a_{t}

Subtract this from the expression for m̂c_t:

Marginal cost is proportional to the output gap

$m c_{t} = (σ + φ) (y_{t} - y_{t}^{n}) = (σ + φ) x_{t}$ , with x_t ≡ y_t − yⁿ_t

The output gap x_t: how far output is from where it would be with flexible prices.
Natural output moves with productivity: not every fall in output is a policy problem. If a bad harvest lowers productivity, natural output falls too and the gap may be zero.
x_t > 0: the economy is "overheating"; firms' costs are above normal and they want to raise prices.

Worked example

σ = 2, φ = 1, productivity 3% above steady state: yⁿ_t = (1 + 1)/(2 + 1) × 3 = 2%. If actual output is 3% above steady state, the output gap is x_t = 3 − 2 = 1%.

Equation 1: the New Keynesian Phillips curve (NKPC)

Substitute m̂c_t = (σ + φ)x_t into the linear Phillips curve:

NKPC

$π_{t} = β E_{t} π_{t + 1} + κ x_{t} + u_{t}$ , with $κ \equiv \frac{( ε - 1 ) ( σ + φ )}{ψ}$

κ is the slope of the Phillips curve. Sticky prices (large ψ) give a flat Phillips curve.
u_t is a cost-push shock. It appears if desired markups vary over time (time-varying ε_t, wage markups, energy costs that do not move natural output).
Iterating forward: π_t = κ Σ_k≥0 β^kE_tx_t+k + Σ_k≥0 β^kE_tu_t+k. Inflation today is driven by expected future gaps and cost pressures.

What the Phillips curve says, in one breath

Firms set prices for a while, so when they do set them they look ahead. If they expect inflation next period, they raise prices now to keep up (the βE_tπ_t+1 term). If the economy is running hot, their costs are high and they raise prices for that reason too (the κx_t term). If an oil shock pushes up costs without changing how much the economy can produce, that shows up as u_t. The slope κ is small because changing prices is costly: a 1% output gap raises quarterly inflation by only a fraction of a percentage point. That is why expectations do most of the work, and why a central bank that is believed has an easier job.

Worked examples

κ: ε = 6, σ = 1, φ = 1, ψ = 100 gives κ = 5 × 2 / 100 = 0.1.
NKPC: β = 0.99, expected inflation 2%, κ = 0.1, gap 1%, no shock: π_t = 0.99 × 2 + 0.1 × 1 = 2.08%.
A permanent gap of 1% with κ = 0.1, β = 0.99: π_t = κ × 1/(1 − β) = 0.1/0.01 = 10. Small gaps that are expected to last a long time add up to a lot of inflation, which is why expectations matter so much.

Traditional versus New Keynesian Phillips curve

Traditional (backward-looking)	New Keynesian (forward-looking)
π_t = π_t−1 + κx_t	π_t = βE_tπ_t+1 + κx_t
Inflation has inertia: it depends on past inflation.	Inflation depends on expected inflation.
Disinflation requires a period of low output.	A credible central bank can lower inflation by changing expectations.
An empirical regularity.	Derived from firms' decisions.

In the data inflation is more persistent than the pure NK version implies, so empirical work often adds lagged inflation: a hybrid Phillips curve (Galí & Gertler, 1999).

Equation 2: the dynamic IS curve

Start from the linearised Euler equation and impose goods market clearing, c_t = y_t: y_t = E_ty_t+1 − (1/σ)(i_t − E_tπ_t+1 − ρ). The same equation holds under flexible prices, with natural output and the natural real rate rⁿ_t: yⁿ_t = E_tyⁿ_t+1 − (1/σ)(rⁿ_t − ρ), so rⁿ_t ≡ ρ + σE_tΔyⁿ_t+1. Subtract the second from the first:

Dynamic IS curve

x_{t} = E_{t} x_{t + 1} - \frac{1}{σ} (i_{t} - E_{t} π_{t + 1} - r_{t}^{n})

The output gap depends on the gap between the actual real rate and the natural rate.

rⁿ_t, the natural rate of interest: the real rate that would keep output at its natural level.
Policy is tight when the real rate is above rⁿ_t (the gap falls), and loose when it is below.
With yⁿ_t = ((1 + φ)/(σ + φ))a_t: rⁿ_t = ρ + σ((1 + φ)/(σ + φ))E_tΔa_t+1. Expected productivity growth raises the natural rate: households expect to be richer and want to borrow, so a higher rate is needed to keep demand in line.
Demand shocks (preferences, fiscal spending) also move rⁿ_t.

What the IS curve says, in one breath

Households compare the real interest rate with the natural rate rⁿ_t, the rate at which they would be happy to consume exactly natural output. If the real rate is higher, saving is attractive, spending falls and output drops below natural: a negative gap. If it is lower, spending rises and the gap is positive. The 1/σ in front says how sensitive spending is. Expected future output matters as well: if households expect a strong economy next year, they spend more now. The central bank controls the nominal rate i_t, and because prices are sticky it thereby controls the real rate for a while; that is the whole transmission mechanism in one equation.

Link to the Taylor rule

The intercept r* in last week's Taylor rule is an estimate of the natural rate: the neutral real rate.

Worked examples

IS: E_tx_t+1 = 0, σ = 1, i_t = 5%, E_tπ_t+1 = 2%, rⁿ_t = 2%: x_t = 0 − (5 − 2 − 2) = −1%. The real rate (3%) is above the natural rate (2%), so output falls below natural.
Natural rate: ρ = 1%, σ = 1, φ = 1, expected productivity growth 1%: rⁿ_t = 1 + 1 × (2/2) × 1 = 2%.

Equation 3: monetary policy

Two equations, three unknowns (x_t, π_t, i_t): the model is closed by a description of monetary policy, for example a Taylor-type rule:

i_{t} = ρ + ϕ_{π} π_{t} + ϕ_{x} x_{t}

Or by optimal policy: the central bank chooses i_t to maximise welfare (next lecture).
Money in the background: the central bank's balance sheet adjusts so that i_t is achieved. In the long run the quantity-theory link holds: money growth equals inflation.

Worked example

ρ = 1%, ϕ_π = 1.5, ϕ_x = 0.5, inflation 2%, output gap −1%: i_t = 1 + 1.5 × 2 + 0.5 × (−1) = 3.5%.

The three-equation New Keynesian model

Block	Equation
Demand: dynamic IS curve	x_t = E_tx_t+1 − (1/σ)(i_t − E_tπ_t+1 − rⁿ_t)
Supply: New Keynesian Phillips curve	π_t = βE_tπ_t+1 + κx_t + u_t
Policy	i_t = rule, or optimal choice

The parameters and where they appear

Parameter	Meaning	Typical value	Appears in
β	discount factor (patience)	0.99 (quarterly)	NKPC; ρ = −ln β
σ	inverse intertemporal elasticity	1 to 2	IS slope 1/σ; κ
φ	inverse Frisch elasticity	1 to 2	κ; natural output
ε	elasticity of demand (competition)	6 (20% markup)	κ
ψ	cost of changing prices	large	κ
κ	slope of the Phillips curve	0.01 to 0.1	NKPC

κ = (ε − 1)(σ + φ)/ψ combines preferences, competition and price stickiness.
Estimates of κ are small: a 1% output gap moves quarterly inflation only a little. A flat Phillips curve is a recurrent finding, especially before 2021.
Every parameter has an economic interpretation: the advantage of deriving the model instead of assuming it.

Andere hoofdstukken van International Monetary Relations and Policy

Test jezelf met oefenvragen