Lecture 2A: Why build the New Keynesian model

Start here if this is new to you

Lecture 2 looks different from Lecture 1: more symbols, fewer tables. Do not let that put you off. A model is a toy economy with a handful of actors (one typical household, a group of firms, a central bank) who each do the best they can for themselves. We write down what each wants and what limits them, work out what they will do, and then look at how the whole toy economy reacts when the central bank changes the interest rate. The payoff is that the three final equations are not assumed but explained: you can say exactly why inflation depends on expected inflation, or why a higher rate lowers output.

You need three ideas from economics and two from mathematics. From economics: utility (a number that measures how happy a household is; more consumption and less work means a higher number), marginal (the effect of one more unit: the marginal cost of a car is the cost of making one more car), and market power (a firm that sells something a little different from everyone else can set its own price instead of taking the market price). From mathematics: an optimum is where "one more unit" no longer helps, so marginal benefit equals marginal cost; and logarithms turn multiplication into addition and make small percentage changes easy to handle. Both are explained again where they are used.

Last week the Taylor rule was a description of what central banks do: the rate goes up when inflation is above target and when output is above potential. Lectures 2A and 2B ask why a central bank would behave like that, and whether it is a good way to set rates. To answer that you need a model in which (1) monetary policy actually affects output and inflation, and (2) you can ask which policy is optimal, meaning best for households. That model is the New Keynesian (NK) model.

Today (2A): build the NK model from the decisions of households and firms and derive its three equations.
Next (2B): optimal monetary policy in that model, and why it looks like a Taylor rule.

In plain words

A model is a simplified economy written in equations. Instead of assuming how people behave, the NK model starts from what households and firms want (utility, profit) and lets the equations follow from their choices. That way every number in the final equations has an economic meaning.

Why sticky prices matter

Benchmark: flexible prices	New Keynesian
Firms take prices as given (perfect competition).	Firms have market power (monopolistic competition).
Prices adjust immediately to any shock.	Firms set their own nominal prices.
A change in i_t is matched by a change in expected inflation: the real rate does not move.	Changing prices is costly: prices are sticky.
Money is neutral: monetary policy does not move output.	Monetary policy moves the real interest rate and output in the short run.

The lecture builds the model in three steps: (1) firms become price setters, (2) add costs of changing prices, (3) study optimal monetary policy (next lecture).

In plain words

If every price jumped instantly, a higher interest rate would just come with higher expected inflation and nothing real would change. Because many prices change only now and then, the central bank can move the real rate for a while, and that is what changes spending.

The economy in the model

Households consume, work and save in a nominal bond. The economy is "cashless": there is no money in utility or in the budget constraint.
Firms, in two layers:
- Retail firms are perfectly competitive. They bundle many different goods into one final good that households buy.
- Wholesale firms form a continuum indexed by j ∈ [0, 1]. Each makes a differentiated good, sets its own price and pays a cost when it changes that price.
The central bank sets the nominal interest rate i_t.

Why cashless?

In modern economies the central bank sets an interest rate, not the money supply. Money exists in the background (the central bank's balance sheet adjusts so the rate holds), but policy works through i_t.

Notation warning: ε is the elasticity of demand between differentiated goods, and the markup is ε/(ε − 1). Some textbooks (and older course notes) use ε for the markup itself.

Roadmap of the derivation

Households: solve their problem, which gives two conditions: how much to consume today versus tomorrow (the Euler equation) and how much to work (labor supply).
Firms: solve their problem, which gives the demand for each good and the markup under flexible prices, and, with costs of changing prices, a Phillips curve linking inflation to marginal cost.
Combine: express marginal cost in terms of the output gap.
Result: three equations: the IS curve, the Phillips curve and the policy rule.

Every step follows the same recipe: write the problem → take first-order conditions → interpret → approximate around the steady state. The lecturer stresses that the intuition matters more than the algebra.

Toolkit: notation and approximations

Upper case for levels, lower case for logs: c_t ≡ ln C_t, y_t ≡ ln Y_t, n_t ≡ ln N_t.
A hat is a percentage deviation from the steady state: ẑ_t ≡ ln Z_t − ln Z̄ ≈ (Z_t − Z̄)/Z̄. Example: if Z̄ = 100 and Z_t = 102, then ẑ_t ≈ 0.02, a 2% deviation.
Steady state: the point where all variables are constant and there are no shocks.
For small z: ln(1 + z) ≈ z. So ln(1 + i_t) ≈ i_t and ln(P_t+1/P_t) ≈ π_t+1, the inflation rate. Check: ln(1.02) = 0.0198 ≈ 0.02.
E_t: the expectation using the information available in period t.

Why log-linearise?

The exact conditions are nonlinear and hard to solve.
Close to the steady state a linear approximation is accurate.
Linear equations are simple to interpret, to solve and to estimate: this is what is taken to the data.

Reading the notation of Lecture 2

E_t ("expectation at time t"): the best forecast you can make with what is known in period t. E_tπ_t+1 is expected inflation for next period. It is not a different variable, just "the forecast of".
Σ (sigma, capital) means "add up". Σ_t=0^∞ β^tu_t is u₀ + βu₁ + β²u₂ + …: utility in every future period, each one counting a little less because β is below 1 (0.99, then 0.98, then 0.97, …).
Capital versus small letters: C_t is consumption in euros or units; c_t is its logarithm, and in the final equations it means "percent above or below normal". c_t = 0.01 or "1%" means consumption is 1% above its steady-state level.
Steady state: the resting position of the model, where nothing changes from period to period and there are no surprises. All the small-letter equations describe movements around that resting point.
A hat (m̂c_t) is the same idea for a variable whose log we did not give a separate letter: percent deviation from steady state.
∫₀¹ … dj: an integral over all firms j between 0 and 1. Read it as "add up over all firms"; it is a sum with infinitely many tiny firms, and you never need to compute it.
First-order condition (FOC): the equation you get by asking "would one more unit make me better off?" and setting the answer to "no, I am already at the best point". It always says marginal benefit = marginal cost.

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