Lecture 2A: Households: the Euler equation and labor supply

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The household in this chapter is one imaginary family that stands for everyone (a "representative" household). Each period it decides three things: how much to consume, how many hours to work, and how much to save in a bond that pays interest. It wants to be as happy as possible, not just now but over its whole life, and it counts the future a little less than the present (the discount factor β, about 0.99 per quarter: next quarter counts 99% as much as this one).

The result of the chapter is two rules of behaviour. The Euler equation says how the household splits consumption between today and tomorrow: if saving pays well (a high real interest rate), it consumes less today and more tomorrow. The labour supply condition says how many hours it works: more when the real wage is higher, less when it already consumes a lot. The algebra in between (a Lagrangian, first-order conditions, logarithms) is only the machinery to get from "what the household wants" to those two rules. If you understand the two rules and can plug numbers into them, you have what the exam needs; the machinery is explained step by step so you can follow it, not reproduce it.

The household problem

A representative household chooses consumption C_t, hours N_t and bonds B_t to maximise lifetime utility:

{C_{t}, N_{t}, B_{t}} max E_{0} t = 0 \sum \infty β^{t} (\frac{C _{t}^{1 - σ}}{1 - σ} - χ \frac{N _{t}^{1 + φ}}{1 + φ})

subject to the budget constraint in every period:

P_{t} C_{t} + B_{t} = (1 + i_{t - 1}) B_{t - 1} + W_{t} N_{t} + D_{t}

C_t: consumption of the final good; N_t: hours worked.
B_t: a one-period nominal bond paying the nominal rate i_t.
W_t: the nominal wage; D_t: dividends from firms (households own the firms).
σ: the inverse of the intertemporal elasticity of substitution; φ: the inverse Frisch elasticity of labor supply.

Reading the problem

Objective: the household likes consumption and dislikes work, today and in all future periods.
- β ∈ (0, 1) is the discount factor: future utility counts less than present utility. Quarterly, β ≈ 0.99.
- Utility rises with C_t at a decreasing rate; the disutility of work rises with N_t at an increasing rate.
Budget constraint, in euros: uses of funds on the left (spending P_tC_t and new savings B_t), sources of funds on the right (last period's bonds plus interest, labor income and dividends).
Two trade-offs: consume or save? (today versus tomorrow) and work or rest? (consumption versus leisure).

In plain words

Think of a student with a part-time job. Each month she decides how much to spend now, how much to put in a savings account, and how many hours to work. Spending is nice, working is tiring, and next month counts a little less than this month. The model is exactly this decision, written as a formula.

What σ and φ mean

σ: willingness to substitute over time	φ: willingness to vary hours
Curvature of utility in consumption. 1/σ is the intertemporal elasticity of substitution. From the Euler equation: E_tΔc_t+1 = (1/σ)(r_t − ρ).	Curvature of the disutility of work. 1/φ is the Frisch elasticity of labor supply. From the labor FOC: n_t = (1/φ)(w_t − σc_t).
Large σ: a strong desire to smooth consumption; spending reacts little to the real rate. σ = 1 is log utility.	Large φ: hours react little to the real wage; firms must pay much higher wages to get more hours.

What they imply for monetary policy

1/σ is the slope of the IS curve: a large σ means a weak interest rate channel.
σ + φ governs how fast marginal cost rises with output: a large σ + φ means a steeper Phillips curve.
Typical values: σ between 1 and 2 in macro models; a Frisch elasticity around 0.5 in micro studies, often higher in macro models (Chetty et al., 2011).

Solving the problem: the Lagrangian and the first-order conditions

Attach a multiplier λ_t to each period's budget constraint and maximise. λ_t is the marginal utility of one extra euro in period t. Careful with B_t: it appears in period t (as savings, −B_t) and in period t + 1 (as income, (1 + i_t)B_t, discounted by β).

What a Lagrangian does, in plain words

The household wants the highest utility, but it cannot spend more than it has: the budget constraint. A Lagrangian is a bookkeeping trick for "maximise something subject to a limit". You add the limit to the objective, multiplied by a price λ_t (lambda) that tells you how valuable one extra euro would be in that period. Then you ask, for each choice separately (consumption, hours, saving), whether one more unit would raise the total. At the best point the answer is no for all three, which gives one equation per choice: the three first-order conditions below. Each of them reads as "the benefit of one more unit equals its cost, measured in euros times λ_t".

First-order conditions

C_{t} N_{t} B_{t} : C_{t}^{- σ} = λ_{t} P_{t} : χ N_{t}^{φ} = λ_{t} W_{t} : λ_{t} = β (1 + i_{t}) E_{t} λ_{t + 1}

C_t: the utility of one more unit of consumption, C_t^−σ, equals its cost: P_t euros, each worth λ_t.
N_t: the utility cost of one more hour, χN_t^φ, equals its benefit: W_t euros, each worth λ_t.
B_t: saving one euro costs λ_t today and returns (1 + i_t) euros tomorrow, each worth βλ_t+1 in today's utility.

Eliminating λ_t gives two conditions:

Consumption-leisure (labor supply)

$\frac{χ N _{t}^{φ}}{C _{t}^{- σ}} = \frac{W _{t}}{P _{t}} \equiv w_{t}$ : the marginal rate of substitution between leisure and consumption equals the real wage.

Consumption-savings (Euler equation)

C_{t}^{- σ} = β (1 + i_{t}) E_{t} [C_{t + 1}^{- σ} \frac{P _{t}}{P _{t + 1}}]

The Euler equation: intuition

Left side: the utility lost by saving one unit today. Right side: the utility gained tomorrow.

Give up one unit of consumption today and save its value, P_t euros, in the bond.
Tomorrow you receive (1 + i_t)P_t euros, which buy (1 + i_t)P_t/P_t+1 units of goods: the gross real return.
At the optimum the household is indifferent: both sides are equal.

The Euler equation with numbers

Suppose a sandwich costs €5 today (P_t = 5), the nominal rate is 4% and sandwiches will cost €5.10 next year (2% inflation). Skip one sandwich today and put the €5 in the bond: next year you have €5.20, which buys 5.20 / 5.10 = 1.0196 sandwiches, about 2% more than the one you gave up. That 2% is the real return: 4% interest minus 2% inflation. The household keeps shifting sandwiches between today and tomorrow until the pleasure lost today exactly matches the pleasure gained tomorrow, discounted by β. If inflation were 4% as well, the €5.20 would buy exactly one sandwich: no real gain, no reason to save more.

What matters is the real rate

The nominal rate appears only in this consumption-savings margin.
Households care about (1 + i_t)P_t/P_t+1: a higher nominal rate raises saving only if it is not offset by higher expected inflation.
This is why sticky prices are needed for monetary policy to affect spending.

In plain words

A 5% interest rate with 5% inflation buys you nothing extra next year. Only if the rate rises more than prices does saving really pay off, and only then do people cut spending today.

Log-linearising the Euler equation, step by step

Why take logs, and what happens to the equation

Logarithms have one property you need: ln(a × b) = ln a + ln b. The Euler equation multiplies four things (β, 1 + i_t, C_t+1^−σ and P_t/P_t+1); taking logs turns that product into a sum, which is far easier to handle. Two approximations do the rest: ln(1 + i) ≈ i for small i (ln 1.04 = 0.039), and ln(P_t+1/P_t) is by definition the inflation rate π_t+1. The constant ln β is renamed −ρ, so that ρ is a positive number, the rate at which the household discounts the future (about 1% per quarter). After dividing by −σ you get the linear equation in the box: consumption today equals expected consumption tomorrow minus (1/σ) times how far the real rate is above ρ.

Take logs of both sides (to first order, E_t of a log ≈ log of E_t):
−σc_t = ln β + ln(1 + i_t) − σE_tc_t+1 − E_t ln(P_t+1/P_t)
Use the approximations ln(1 + i_t) ≈ i_t, ln(P_t+1/P_t) = π_t+1, and define ρ ≡ −ln β (the discount rate, about 1% per quarter):
−σc_t = −ρ + i_t − E_tπ_t+1 − σE_tc_t+1
Divide by −σ and rearrange.

Linearised Euler equation

c_{t} = E_{t} c_{t + 1} - \frac{1}{σ} (i_{t} - E_{t} π_{t + 1} - ρ)

i_t − E_tπ_t+1 is the (expected) real interest rate r_t. In steady state consumption is constant, so r = ρ: the real rate equals the discount rate.

Check ρ: with β = 0.99, ρ = −ln(0.99) = 0.01005, so about 1% per quarter (about 4% per year).

Interpreting the linearised Euler equation: c_t = E_tc_t+1 − (1/σ)(r_t − ρ)

A real rate above ρ makes saving attractive: consumption today falls relative to tomorrow.
A real rate below ρ makes borrowing attractive: consumption today rises.
1/σ measures how strongly consumption responds to the real rate.
Expectations matter: higher expected future consumption raises consumption today, because households want to smooth.
Iterating forward: c_t = −(1/σ) Σ_k≥0 E_t(r_t+k − ρ) + lim E_tc_t+k. Consumption today depends on the whole path of expected real rates, not only today's rate. This is why forward guidance (promises about future rates) works in the model.

Worked example

σ = 2, ρ = 1%, the real rate is 3%. Then E_tΔc_t+1 = (1/σ)(r_t − ρ) = (3 − 1)/2 = 1%: consumption is planned to grow 1% per quarter, so it starts lower today. With σ = 1 the same rate gap gives 2%: a stronger reaction.

This is the seed of the IS curve: once consumption equals output, it becomes the demand side of the model.

Labor supply, in logs

Take logs of χN_t^φC_t^σ = w_t: ln χ + φn_t + σc_t = ln w_t. In deviations from the steady state the constants drop out:

Labor supply

w_{t} = σ c_{t} + φ n_{t}

To work more hours (n_t up), households require a higher real wage; how much higher depends on φ.
When households consume more (c_t up), they value leisure more and also require a higher wage: a wealth effect, governed by σ.
This links wages, and so firms' costs, to output.