Business Cycles Theory and Policy

Come studiareCerca un argomento oppure apri un capitolo dall’indice. I titoli ti portano direttamente alla spiegazione corretta.

6. Steady state, calibration and log-linear approximation

Introduction

A nonlinear RBC model is an internally consistent system of forward-looking difference equations, but it is not usually convenient to study directly by hand. The solution follows a standard sequence: identify the deterministic steady state, assign parameter values, approximate equilibrium conditions in log deviations and solve the resulting linear system.

The steady state gives a reference point. Calibration links structural parameters to empirical moments and equilibrium identities rather than selecting values without justification. Log-linearization then measures small percentage departures from this point; its quality must be judged relative to the size of the shocks being analyzed.

A first-order approximation preserves local effects but loses genuinely nonlinear features of large excursions, occasionally binding constraints and precautionary behavior. Thus the model's first-order impulse responses are most informative for small disturbances around the calibrated steady state.

Integrated theoretical explanation

The complete benchmark RBC system contains five central relationships. The Euler condition connects consumption today with expected consumption and capital returns tomorrow. The labor condition links consumption and hours to the marginal product of work. The resource constraint allocates output between consumption and capital accumulation. The production function determines goods available from productivity, labor and inherited capital. Finally, the autoregressive technology process describes how new innovations alter the exogenous state. Solving the model means finding paths that satisfy all five equations simultaneously, not solving each in isolation.

A deterministic steady state is a constant configuration of the endogenous variables in an environment without further technology innovations. Productivity is normalized to one. Because capital no longer changes from period to period, the investment required merely replaces the amount depreciated, so ī = δk̄. The steady-state resource restriction is ȳ = c̄ + δk̄. The Euler equation yields 1 = β[αȳ/k̄ + 1 − δ], which identifies the required steady-state marginal product of capital as αȳ/k̄ = β⁻¹ − 1 + δ. These are equilibrium restrictions rather than independent numerical assumptions.

The capital–output ratio follows immediately from the required return: k̄/ȳ = α/(β⁻¹ − 1 + δ). Because investment just replaces depreciation, the steady-state investment share satisfies ī/ȳ = δ(k̄/ȳ). Consumption's output share is then c̄/ȳ = 1 − δ(k̄/ȳ). Given a normalization for hours n̄ and the Cobb–Douglas technology, the level of steady-state output is pinned down by these ratios. The leisure-preference parameter ϕ can be chosen so that the labor first-order condition holds at the target steady-state hours. Working through these steps in order avoids attempting to solve simultaneously for every unknown without using the identifying equilibrium relationships.

Calibration means using the structure of the model and observed economic magnitudes to choose plausible parameter values. The capital share α is associated with the observed share of income paid to capital when factor prices equal marginal products. Depreciation is informed by the pace at which productive assets wear out; β is linked to the equilibrium capital return; ϕ helps match average hours; ρ characterizes technology persistence; and the innovation standard deviation controls shock size. A calibrated value does not become a universal economic constant: it is a choice connected to a dataset, frequency, production structure and target moments.

The Solow residual is a measure of total factor productivity inferred from output and measured factor inputs. Taking logs of Cobb–Douglas production produces ln zₜ = ln yₜ − α ln kₜ₋₁ − (1 − α)ln nₜ. Its construction assumes that the production elasticity and input series are appropriately measured. The residual may also contain variation caused by utilization, composition, omitted inputs or measurement error. Treating every movement in the measured residual as a pure exogenous technology shock is a strong interpretation, and the empirical implications of the RBC framework should be assessed with that limitation in mind.

Linearizing around a steady state is useful because the original equilibrium conditions contain nonlinear products, powers, ratios and expectations. For a strictly positive variable, define x̂ₜ = ln(xₜ/x̄). A small x̂ₜ is approximately the proportional deviation (xₜ − x̄)/x̄. The approximation takes advantage of exp(x̂ₜ) ≈ 1 + x̂ₜ near zero. In logarithms, Cobb–Douglas production has an exact linear representation, ŷₜ = ẑₜ + (1 − α)n̂ₜ + αk̂ₜ₋₁. Other equations require a first-order Taylor expansion because their level expressions contain sums or more complicated nonlinearities.

The goods-market resource identity provides a particularly important example. Begin with yₜ = cₜ + kₜ − (1 − δ)kₜ₋₁ and express every positive variable as its steady-state level multiplied by exp of its log deviation. Keeping first-order terms and cancelling the steady-state identity gives ȳŷₜ = c̄ĉₜ + k̄k̂ₜ − (1 − δ)k̄k̂ₜ₋₁. The weights are steady-state levels; they cannot be omitted simply because the model has been written in percentage deviations. Dividing by ȳ converts the equation into ratios that can be interpreted as resource shares.

For the labor condition, ϕcₜ = (1 − α)yₜ/nₜ, take logarithms and subtract the steady-state equation. All fixed constants cancel, leaving ĉₜ = ŷₜ − n̂ₜ. The familiar condition that real wages equal the marginal product of labor produces the same percentage logic. For the technology process, the log law of motion is already linear. The Euler equation is more delicate because it contains a conditional expectation of a future product and must be expanded jointly around steady-state consumption and returns. The resulting system is linear in state and control deviations but retains forward-looking expectations.

A first-order approximation is reliable only locally. It describes the immediate marginal effect of small changes around the calibrated reference point, not necessarily the response to very large crises, irreversible investment constraints or nonlinear regime changes. Its main advantage is that a complicated dynamic optimization model can be studied using a system of linear rational-expectations difference equations. Once solved, that linear system yields policy functions, impulse responses and model-generated statistics that can be compared systematically with business-cycle data.

Formal framework and essential relationships

1 = β [α \frac{\overline{y}}{\overline{k}} + 1 - δ], \overline{y} = \overline{c} + δ \overline{k}

With no growth and no innovations, the Euler equation equates discounting to the steady-state gross capital return; investment merely replaces depreciation.

ϕ \overline{c} = (1 - α) \frac{\overline{y}}{\overline{n}}, \overline{y} = {\overline{n}}^{1 - α} {\overline{k}}^{α}

The steady-state labor equation and production function jointly determine hours, output, consumption and capital once parameters are chosen.

Parameter	Illustrative calibration	Economic role
ρ	0.95	Persistence of log technology
σᵤ	0.007	Standard deviation of productivity innovations
α	0.36	Capital income share
ϕ	2.56	Marginal utility weight on leisure
δ	0.0225	Capital depreciation per quarter
β	≈ 0.99	Household discount factor

The calibration uses σ for the standard deviation of innovations, whereas σ denotes utility curvature in the earlier general CRRA specification. The meaning must therefore be inferred from the equation or parameter table under discussion.

{\hat{x}}_{t} = ln (\frac{x_{t}}{\overline{x}}), x_{t} = \overline{x} exp ({\hat{x}}_{t}) \approx \overline{x} (1 + {\hat{x}}_{t})

A hat denotes a log deviation from steady state, which is approximately a proportional change for small shocks. The approximation exp(h) ≈ 1+h is the local first-order Taylor formula.

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

The linearized labor and production equations turn nonlinear ratios and products into combinations of deviations with economically interpretable coefficients.

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

The linearized resource constraint is expressed in units of steady-state quantities, not a simple unweighted sum of percent changes.

Explanations, derivations and empirical applications

Approximating The Solution

The deterministic steady state, structural calibration and approximation are separate stages of the quantitative solution. Together they transform a nonlinear general-equilibrium system into a tractable local description of dynamic responses.

The quantitative procedure proceeds in a fixed logical order: calculate the deterministic steady state, choose calibrated or estimated parameters, log-linearize the nonlinear equilibrium conditions, solve the linear rational-expectations system, and examine the resulting impulse responses. Starting with the steady state is necessary because all first-order coefficients are evaluated at that reference allocation.

Nonlinear System Of Equations

The equilibrium conditions to be calibrated and approximated are:

1 = β E_{t} [\frac{c_{t}}{c_{t + 1}} (α \frac{y_{t + 1}}{k_{t}} + 1 - δ)]

ϕ c_{t} = w_{t} = (1 - α) \frac{y_{t}}{n_{t}}

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

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

log z_{t} = ρ log z_{t - 1} + u_{t}, E (u_{t}) = 0, Var (u_{t}) = σ_{u}^{2}, | ρ | < 1

Steady State

A deterministic steady state removes innovations and holds stationary endogenous variables constant:

c_{t} = \overline{c}, y_{t} = \overline{y}, k_{t} = \overline{k}, n_{t} = \overline{n}, log z_{t} = 0, \overline{z} = 1

1 = β E_{t} [\frac{c_{t}}{c_{t + 1}} (α \frac{y_{t + 1}}{k_{t}} + 1 - δ)]

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

Constant consumption removes the intertemporal consumption ratio; the steady-state return offsets discounting.

ϕ c_{t} = w_{t} = (1 - α) \frac{y_{t}}{n_{t}}

ϕ \overline{c} = (1 - α) \frac{\overline{y}}{\overline{n}}

Investment And Output

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

\overline{y} = \overline{c} + δ \overline{k}, \overline{i} = δ \overline{k}

With constant capital, investment replaces depreciation.

Steady State Factor Shares

\overline{w} = (1 - α) \frac{\overline{y}}{\overline{n}}, \frac{\overline{w} \overline{n}}{\overline{y}} = 1 - α

\overline{r} = α \frac{\overline{y}}{\overline{k}}, \frac{\overline{r} \overline{k}}{\overline{y}} = α

Calibrating The Model

There are 6 parameters in the model to set β, δ, α, ϕ, ρ, σ; Now we will calibrate the model to match several stylized facts. We need a calibration target for each parameter; directly from the data; or the steady state equations.

Calibrating Directly From The Data

Some parameters can be set directly from the data.

The log of the cyclical productivity component can be fitted to an AR(1) process. The illustrative quarterly calibration uses persistence ρ = 0.95 and innovation standard deviation σᵤ = 0.007; these values are calibration choices, not estimates for every economy.

What Are Solow Residuals?

Calibration assigns parameter values using directly observable persistence and shock measures, factor shares, and steady-state accounting identities. For example, competitive Cobb–Douglas production makes the capital exponent equal to capital's income share. Solow residuals attribute the part of output not accounted for by measured input quantities to total factor productivity, which may also include measurement error or omitted factors.

With Cobb–Douglas production, the level of measured productivity is the residual after accounting for capital and labor:

log X_{t} = log y_{t} - (1 - α) log n_{t} - α log k_{t - 1}

Allowing the measured capital share to vary over time replaces α by αₜ. Interpreting this residual as technology requires assumptions about input measurement, utilization and production structure.

Solow Residuals And Productivity Shocks

Kaldor-style facts are long-run restrictions that help determine whether an aggregate production model is plausible. Balanced growth links rising labor productivity and capital intensity with relatively stable factor proportions, capital returns and income shares. They do not imply a constant output level or that short-run fluctuations are unimportant.

A stochastic productivity process can be approximated by fitting an autoregressive model to the measured productivity residual. The estimated persistence determines how long a typical innovation affects productivity, while the residual innovation dispersion determines shock size. Interpreting the statistical residual as pure exogenous technology remains an identifying assumption of the benchmark RBC model.

Calibration

α = \frac{\overline{r} \overline{k}}{\overline{y}} = 0.36

The capital elasticity is calibrated to the observed capital-income share under the competitive Cobb–Douglas assumptions.

\overline{n} = \frac{1}{3}, \frac{\overline{i}}{\overline{y}} = \frac{1}{4}, \frac{\overline{c}}{\overline{y}} = \frac{3}{4}, \frac{\overline{y}}{\overline{k}} = 0.09

ϕ = \frac{1 - α}{\overline{n} (\overline{c} / \overline{y})} = \frac{0.64}{(1 / 3) \times 0.75} = 2.56

\overline{n} = \frac{1}{3}, \frac{\overline{i}}{\overline{y}} = \frac{1}{4}, \frac{\overline{c}}{\overline{y}} = \frac{3}{4}, \frac{\overline{y}}{\overline{k}} = 0.09

δ = \frac{\overline{i}}{\overline{y}} \frac{\overline{y}}{\overline{k}} = 0.25 \times 0.09 = 0.0225

\overline{n} = \frac{1}{3}, \frac{\overline{i}}{\overline{y}} = \frac{1}{4}, \frac{\overline{c}}{\overline{y}} = \frac{3}{4}, \frac{\overline{y}}{\overline{k}} = 0.09

β = \frac{1}{α (\overline{y} / \overline{k}) + 1 - δ} = \frac{1}{0.36 \times 0.09 + 1 - 0.0225} \approx 0.990197

Calibration: Summary

Parameter	Illustrative value	Calibration basis
ρ	0.95	Persistence of log productivity
σᵤ	0.007	Innovation standard deviation
α	0.36	Capital-income share
ϕ	2.56	Hours and consumption share
δ	0.0225	Investment share and output–capital ratio
β	≈0.99	Steady-state Euler condition

Log Deviations

A log deviation is ln(xₜ/x̄), approximately the percentage difference from the steady-state level when the deviation is small. First-order Taylor expansion turns products and powers into weighted sums of log deviations, greatly simplifying the equilibrium system. It is a local method: when departures become large, approximation errors can be economically material.

x_{t} = \overline{x} e^{{\hat{x}}_{t}}, {\hat{x}}_{t} = log x_{t} - log \overline{x} = log (\frac{x_{t}}{\overline{x}})

For small changes, the log deviation approximates a proportional deviation. Multiplying by 100 gives percentage units.

Example: Log Deviations

\overline{c} = 5.43, c_{t} = 5.55, {\hat{c}}_{t} = log (5.55 / 5.43) \approx 0.02186

Consumption is approximately 2.2% above its reference level. With cₜ = 5.35, the log deviation is approximately −0.01484, or −1.48%.

Loglinearization Around The Steady State

e^{{\hat{x}}_{t}} \approx 1 + {\hat{x}}_{t}, x_{t} = \overline{x} e^{{\hat{x}}_{t}} \approx \overline{x} (1 + {\hat{x}}_{t})

This is a local first-order approximation around zero log deviation.

Example: Loglinearization

f (x_{t}) = x_{t}^{2} = {\overline{x}}^{2} e^{2 {\hat{x}}_{t}} \approx {\overline{x}}^{2} (1 + 2 {\hat{x}}_{t})

How Good Is The Approximation?

\overline{x} = 2, x_{t} = 2.05, {\hat{x}}_{t} = log (2.05 / 2) \approx 0.0246926

f (2.05) = 4.2025, f (2.05) \approx 4 [1 + 2 log (2.05 / 2)] \approx 4.19754

The approximation error is about 0.00496, small relative to the change in the function.

\overline{x} = 2, x_{t} = 2.5, {\hat{x}}_{t} = log (2.5 / 2) \approx 0.223144

f (2.5) = 6.25, f (2.5) \approx 4 [1 + 2 log (2.5 / 2)] \approx 5.78515

The error increases to about 0.46485. Larger departures make the local approximation less accurate.

Loglinearizing

ϕ \overline{c} e^{{\hat{c}}_{t}} = (1 - α) \frac{\overline{y}}{\overline{n}} e^{{\hat{y}}_{t} - {\hat{n}}_{t}}

ϕ \overline{c} = (1 - α) \frac{\overline{y}}{\overline{n}} \Rightarrow {\hat{c}}_{t} = {\hat{y}}_{t} - {\hat{n}}_{t}

The labor condition is exactly linear in log deviations.

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

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

Canceling the steady-state production identity and taking logs gives an exact log relation.

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

Expand each exponential to first order and subtract the steady-state resource identity:

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

Unlike the Cobb–Douglas equation, this relation is an approximation.

1 = β E_{t} [\frac{c_{t}}{c_{t + 1}} (α \frac{y_{t + 1}}{k_{t}} + 1 - δ)]

1 = E_{t} [β α \frac{\overline{y}}{\overline{k}} e^{{\hat{c}}_{t} - {\hat{c}}_{t + 1} + {\hat{y}}_{t + 1} - {\hat{k}}_{t}} + β (1 - δ) e^{{\hat{c}}_{t} - {\hat{c}}_{t + 1}}]

Expand both exponentials and use the steady-state Euler condition:

0 = E_{t} [β α \frac{\overline{y}}{\overline{k}} ({\hat{y}}_{t + 1} - {\hat{k}}_{t}) + {\hat{c}}_{t} - {\hat{c}}_{t + 1}]

Linear System Of Equations

0 = E_{t} [β α \frac{\overline{y}}{\overline{k}} ({\hat{y}}_{t + 1} - {\hat{k}}_{t}) + {\hat{c}}_{t} - {\hat{c}}_{t + 1}]

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

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

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

{\hat{z}}_{t} = ρ {\hat{z}}_{t - 1} + u_{t}

Synthesis

Calibration fixes economically interpretable parameters; first-order approximation translates nonlinear dynamics into a tractable local system.

Torna all’inizio ↑