ADVANCED MACROECONOMIC TECHNIQUES NOTE 1c

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1 ADVANCED MACROECONOMIC TECHNIQUES NOTE 1c Chris Edmond Linearizing a difference equation We will frequently want to linearize a difference equation of the form x t+1 = ψ(x t ) given an initial condition x 0. We can take a first order approximation to the function ψ at some candidate point x. This gives x t+1 ' ψ( x)+ψ 0 ( x)(x t x) The right hand side of this expression is a linear equation in x t with slope ψ 0 ( x) equal to the derivative of ψ evaluated at the point x. Typically, we will want to linearize around a steady state which also has the property = ψ() so that x t+1 ' + ψ 0 ()(x t ) If we treat this approximation as exact, we have an inhomogenous difference equation with constant coefficients of the form x t+1 = ax t + b where a ψ 0 () b (1 a) And we know that the solution to this difference equation is x t =(1 a t ) + a t x 0 Hence if a < 1, thena t 0 as t so that x t. We can conclude from this that the original non-linear difference equation is locally stable (since we have had to take a local approximation to the function ψ) but we cannot use this method to determine global properties of the solution 1

2 in a linear model, local and global stability are the same thing, but that is not true here. A lot of applied work in economics works by using linear approximations to non-linear models. Log-linearization, etc. A popular alternative to linearizing a model is to log-linearize it. To do this, define the log-deviation of a variable from its steady state value as ˆx t =log µ xt With this notation, a variable is at steady state when its log-deviation is zero. The popularity of this method comes from the units-free nature of the variables. Logdeviations are approximate percentage deviations from steady state and the coefficients of log-linear models are elasticities. I say approximate percentage deviation because This uses the first order approximation µ µ xt log =log 1+ x t ' x t log(1 + z) ' log(1 + z 0 )+ 1 1+z 0 (z z 0 ) around the point z 0 =0, so it is an approximation that is valid when the deviation is small. See "Problem Set Zero" for some discussion of the accuracy of this approximation. A non-linear difference equation of the form x t+1 = ψ(x t ) is log-linearized by making the change of variables x t = exp(ˆx t ) and then linearizing both sides with respect to ˆx t around the point ˆx t =0.Specifically, write exp(ˆx t+1 )=ψ( exp(ˆx t )) and now linearize both sides of this equality with respect to ˆx t (around the point ˆx t =0)to get exp(0) + exp(0)(ˆx t+1 0) ' ψ( exp(0)) + ψ 0 ( exp(0)) exp(0)(ˆx t 0) 2

3 Treating this approximation as exact and simplifying gives +ˆx t+1 = ψ()+ψ 0 ()ˆx t And recognizing that = ψ() gives ˆx t+1 = ψ 0 ()ˆx t Local stability again crucially depends on the absolute magnitude of ψ 0 (), with ψ 0 () < 1 giving a locally stable steady state. Suppose, for example, that we have the Solow difference equation k t+1 = ψ(k t ) the log-linear approximation is ˆk t+1 = ψ 0 ( k)ˆk t = s (1 δ) (1 + g)(1 + n) kα t + (1 + g)(1 + n) k t 1 (1 + g)(1 + n) [sα k α 1 +1 δ]ˆk t Calculus for log-linearizations Similar derivations allow us to do more complicated log-linearizations (this is sometimes known as the "Campbell calculus"). The following basic rule often helps immensely. Suppose that we have a differentiable function y t = f(x t,z t ) (more arguments will generalize in an obvious fashion). the log-linear approximation is ȳŷ t = f x (, z)ˆx t + f z (, z) zẑ t which is often written ŷ t = f x(, z) f(, z) ˆx t + f z(, z) z f(, z) ẑt so that the coefficients on the log-deviations ˆx t, ẑ t are elasticities. A 1% increase in ˆx t near the steady-state gives approximately a f x(, z) f(, z) 100% increase in ŷ t. Here are some further applications of this rule. 3

4 1. (Multiplication and division). Let y t = f(x t,z t )=x t z t ŷ t = f x(, z) f(, z) ˆx t + f z(, z) z f(, z) ẑt = z z ˆx t + z z ẑt = ˆx t +ẑ t And similarly, if y t = f(x t,z t )= x t z t ŷ t =ˆx t ẑ t 2. (Addition and subtraction). Let y t = f(x t,z t )=x t + z t ȳŷ t = f x (, z)ˆx t + f z (, z) zẑ t =1ˆx t +1 zẑ t = ˆx t + zẑ t 3. (Implicit functions). An important special case concerns functions of the form 0=g(x t,y t ) which may implicitly defines y t in terms of x t (or vice-versa). If g y (, ȳ) 6= 0,then 0=g x (, ȳ)ˆx t + g y (, ȳ)ȳŷ t and rearranging gives ŷ t = g x(, ȳ) g y (, ȳ)ȳ ˆx t 4

5 Examples 1. (Difference equation). Let x t+1 = f(x t ) so that ˆx t+1 = f 0 ()ˆx t The steady state values on both sides cancel and we are left with ˆx t+1 = f 0 ()ˆx t 2. (Constant elasticity function). Let z = x ε for some coefficient ε. zẑ t = ε ε 1ˆx t = ε εˆx t = ẑ t = ε ε z ˆx t = εˆx t In this case, the elasticity does not depend on the steady state values, z but only on the constant parameter ε. 3. (Consumption Euler equation an extended example). A consumer s first order condition is often 1=β U 0 (c t+1 ) U 0 (c t ) R t+1 This implies the log-linearization ˆ1 =ˆβ + U 0 (c d t+1 ) U d0 (c t )+ ˆR t+1 But since the log-deviations of constants are zero, this is just 0= U 0 (c d t+1 ) U d0 (c t )+ ˆR t+1 Moreover, if we have the constant elasticity utility function U(c) =c 1 σ with coefficient σ>0, then marginal utility is U 0 (c) =c σ and the log-deviation of marginal utility is Ud 0 (c t )= U 00 ( c) c U 0 ( c) ĉt = σĉ t 5

6 In this example, the consumption Euler equation becomes ĉ t+1 ĉ t = 1 σ ˆR t+1 The coefficient 1 σ > 0 measures the sensitivity of consumption growth to real interest rates in excess of their steady state value. Also, if we write the gross real interest rate R t+1 as one plus the net rate, R t+1 =1+r t+1,then R ˆR t+1 = rˆr t+1 But in steady state c t = c t+1 = c and so R = β 1 =1+ r (according to the original Euler equation). Hence ˆR t+1 =(1 β)ˆr t+1 4. (National income accounting). Let c t + i t = y t This implies the log-linearization cĉ t +īî t =ȳŷ t which is often written so that the coefficientsaresharesofgdp,thatis 5. (Cobb-Douglas production function). Let c ȳ ĉt + ī ȳ ît =ŷ t y t = f(k t,n t )=k α t n 1 α t, 0 <α<1 ŷ t = αˆk t +(1 α)ˆn t Chris Edmond 23 July