Business Cycles Theory and Policy

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Appendix A. Notation and mathematical reference

This table consolidates notation used throughout the RBC framework. Time subscripts refer to the period; a bar indicates the deterministic steady state and a hat indicates the associated logarithmic deviation.

Symbol	Meaning
Y, y	Gross domestic product / aggregate real output
C, c	Consumption
I, i	Gross investment
G	Government expenditures (in the introductory accounting model)
X, IM, Z	X denotes exports; IM or Z denotes imports in the national-accounts comparisons. In the Keynesian-cross diagrams, Z instead denotes planned aggregate expenditure; the local definition identifies the intended meaning.
kₜ₋₁	Predetermined productive capital available in period t
nₜ, lₜ	Hours worked and leisure (nₜ + lₜ = 1)
zₜ	Stochastic productivity / total factor productivity in the RBC model
wₜ, rₜ	Real wage and competitive capital rental rate
πₜ	Firm profits distributed to the household
α, θ	Capital elasticity / capital factor-income share in Cobb–Douglas production
β	Household discount factor, 0 < β < 1
δ	Capital depreciation rate
ϕ	Preference weight on leisure in the specialized utility function
ρ	Persistence of the log productivity process
uₜ	One-period technology innovation
Eₜ	Expectation conditional on information available at date t
σ, σᵤ	Standard deviation of technology innovations in the calibration; earlier σ is also used for utility curvature
x̄	Steady-state level
x̂ₜ	Log deviation ln(xₜ/x̄)

Important equilibrium equations

Production

y_{t} = z_{t} n_{t}^{1 - α} k_{t - 1}^{α}

A higher z raises output for given inputs; input elasticities sum to one.

Productivity dynamics

ln z_{t} = ρ ln z_{t - 1} + u_{t}

ρ summarizes propagation through the exogenous technology process.

Capital accumulation

k_{t} = (1 - δ) k_{t - 1} + i_{t}

Today’s installed investment makes capital available in the subsequent production period.

Wage schedule

w_{t} = (1 - α) \frac{y_{t}}{n_{t}}

Marginal product of labor equals the real wage.

Capital rental schedule

r_{t} = α \frac{y_{t}}{k_{t - 1}}

Marginal product of predetermined capital equals its rental price.

Time endowment

1 = n_{t} + l_{t}

More labor reduces leisure, holding the endowment fixed.

Euler equation

1 = β 𝔼_{t} [{(\frac{c_{t + 1}}{c_{t}})}^{- 1} (r_{t + 1} + 1 - δ)]

The stochastic return enters tomorrow’s utility-weighted consumption opportunity.

Labor optimality

ϕ c_{t} = w_{t}

Under log consumption and linear leisure, work is chosen by equating its utility opportunity cost to its wage benefit.

Resource constraint

y_{t} = c_{t} + k_{t} - (1 - δ) k_{t - 1}

Closed-economy output is allocated between consumption and gross investment.

Steady-state capital return

α \frac{\overline{y}}{\overline{k}} = \frac{1}{β} - 1 + δ

The steady-state Euler condition determines the capital rental rate. The gross return on saving adds undepreciated capital: r̄ + 1 − δ = 1/β.

Steady-state investment

\overline{i} = δ \overline{k}

Replacement investment exactly offsets depreciation.

Log deviation

{\hat{x}}_{t} = ln x_{t} - ln \overline{x}

A log deviation is approximately a proportional difference when small; multiplying by 100 expresses it in percent.

Linearized production

{\hat{y}}_{t} = {\hat{z}}_{t} + (1 - α) {\hat{n}}_{t} + α {\hat{k}}_{t - 1}

A log-linear Cobb–Douglas identity is exactly additive in log variables.

Linearized labor equation

{\hat{c}}_{t} = {\hat{y}}_{t} - {\hat{n}}_{t}

The linearized equilibrium condition after cancelling steady-state levels.

Linearized resource condition

\overline{y} {\hat{y}}_{t} = \overline{c} {\hat{c}}_{t} + \overline{k} {\hat{k}}_{t} - (1 - δ) \overline{k} {\hat{k}}_{t - 1}

Steady-state weights are necessary when adding deviations of different quantities.

Recursive solution

{\hat{k}}_{t} = λ_{k k} {\hat{k}}_{t - 1} + λ_{k z} {\hat{z}}_{t}

The policy rule separates responses to the two beginning-of-period states.

Equation consistency checks

The household Euler equation uses the gross return rₜ₊₁ + 1 − δ, not the capital rental rate rₜ₊₁ alone.
In period t, production uses kₜ₋₁ and not kₜ. Capital kₜ is selected by investment decisions at t and used in production at t+1.
The coefficient on labor is 1−α in the no-growth Cobb–Douglas production function. Factor shares add to one under perfect competition and constant returns to scale.
First-order log-linearization is a local percentage approximation. Never combine log-level equations with level resource weights without explicitly taking the approximation.
Empirical business-cycle moments are estimates associated with a specific sample and data transformation, not universal constants that must hold for every country or period.

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