Solutions Methods in DSGE (Open) models 1. With few exceptions, there are not exact analytical solutions for DSGE models, either open or closed economy. This is due to a combination of nonlinearity and the fact that the systems are stochastic. Almost all dynamic optimization problems give rise to non-linear equations (exceptions - a small open endownment economy with deterministic interest and income stream). But this is compounded by the fact that the stochastic features of the systems. Solutions involve taking expectations of future endogenous variables.. The standard solution methods, used in RBC and IRBC models, as well as most recent generation new open macro models, involve linear approximation around a deterministic steady state. This was discussed in the previous notes. By linear approximation, we mean the first order terms in a Taylor series approximation around a steady state. Let us quickly review the idea of orders of approximation: Say that we have a function y = g(x) where in the notation to come, y will be a non-predetermined variable in a dynamic model, x is a predetermined variable. Say that there is a deterministic steady state for x given by x,sothat: y = g(x) In general, the function g is non-linear. Using a Taylor series approximation, we can approximation this function around {x} by y g(x)+g x (x)(x x)+ 1 g xx(x)(x x) + O 3 (1) This contains 3 types of terms: zero order, first order, and second order. The notation at the end signifies that the error in the approximation is of third or higher order. When evaluated up to a first order, the expected value of y is equal to its steady state value value y. But when evaluated up to a higher order, the expectedvaluedependsonthecurvatureofthefunctionandthevarianceofx. 3. Most DSGE models just analyze the first order properties of models. For characterizing the impulse response and second moments of variables, that is fine. For instance, say we were interested in finding the variance of y, whichis a second order accurate measure. Then we could use (1) to calculate var(y) =E(y Ey) = g xvar(x) Why do we not use the higher order terms? Becuase they are second order, so if we included them, we would have terms in third or higher order in calculating the variance. But the variance is only a second order accurate measure. 4. But for other purposes, first order approximation may be inaccurate. Say that we have a welfare measure for this economy, given by EU(y),andwewishto 1
evaluate expected utility of the agents based on this welfare measure and based on the distribution of x. Then we could take a second order approximation of U(y) given by U(y) U(y)+U y (y)(y y)+ 1 U yy(y)(y y) + O 3 () It is then tempting to think of evaluating expected utility using the first order approximation of y from (1) so that () gives: EU(y) U(y)+U y (y)e(y y)+ 1 U yy(y)e(y y) + O 3 (3) = U(y)+ 1 U yy(y)g xvar(x)+o 3 (4) Except this is wrong. What we are doing here is taking into account the presence of one set of second order terms in evaluating utility, but leaving out another set of second order terms. What do we mean by this? Well, note from (1) that, evaluated up to a second order, wehave Ey y + 1 g xx(x)var(x)+o 3 (5) So, when we are evaluating up to a second order, the second line of (3) should be EU(y) U(y)+U y (y) 1 g xx(x)var(x)+ 1 U yy(y)g xvar(x)+o 3 So that the correct evaluation of welfare is given by: EU(y) U(y)+ U y (y) 1 g xx(x)+ 1 U yy(y)gx var(x)+o 3 (6) Basically, what we are saying is that when when we are evaluating expected utility, we have to take a second order approximation of utility, and a second order approximation of the economic model. If we just take a second order approximation of utility and a first order approximation of the economic model, we are leaving out some relevant terms in the evaluation of utility. These terms are second order, but they are relevant, because utility is evaluated up to second order. 5. This has very wide applicability. Woodford s book documents how to use approximations to evaluate utility in a range of sticky price models. In some cases, Woodford shows that U y (y) =0, so that we can actually validly take a first order approximation of the economic model, but in general this won t be correct. 6. Lets look at two concrete examples to see how we could be led astray. The first is from Kim and Kim, JIE 004. It is based on evaluating welfare in a country endowment economy under incomplete markets, and comparing this to an economy with complete markets. Say country i utility, i =1,, is given by: U i = C1 σ i 1 σ
and endowments are stochastic equal to Y i. Assume that Y i are log-normal and E(y) =0,wherey =log(y ) var(y) =Under autarky, we have C i = Y i. Under complete markets, we have C i = Y1+Y. Now take a log approximation to utility around a steady state c to get U i C1 σ 1 σ + C1 σ bc + 1 σ C 1 σ bc where bc =(c c). If we took a linear approximation of the model around the steady state under autarky we would get bc A i = by i = y i, and under complete markets we would get bc C i =1/(y 1 + y ). So by a linear approximation we would have Ebc A i =0and Ebc C i =0. Under autarky, we have var(c) =var(y), while under complete markets, we have var(c) =1/var(y). So we might be led to the conclusion that utility is higher under autarky when σ<1. But this is based on the assumption that E(bc) =0, which is not true. Why? Because note that C i = Y 1 + Y So that exp(c i )= exp(y 1)+exp(y ) exp(ec + 1 var (c)) = exp(1 var(y)) so that Ec = 1 var(y) 1 var(c) = 1 4 var(y), sothate(c ca ) > 0. Hence, again, we need to take into account the higher order terms which affect the means of consumption. 6. Take another example. Say we have a monetary economy with sticky prices, where each agent is a yeoman farmer, producing a differentiated output and charging a monopoly price for her output based on a constant elasticity of demand. This is a very standard model so we won t get into the details. Say that there is a utility function given by: U = C1 σ i 1 σ ηl i (7) R 1 where C = C 1 1/λ j dj 1/(1 1/λ), and production of each good is one for one o with labor supply of the yeoman farmer. The budget constraint of the household R 1 is PC = P i L i,wherep = Pj dj 1/(1 λ) 1 λ is the price level, and P i is the price o charged by the farmer. Demand for the farmer s good is given by L i = µ λ Pi C P 3
where C is aggregate consumption. Say the farmer has to set her price in advance. Then her optimal price will satisfy i " µ 1 λ E C σ (1 λ) Pi i C + ηλ 1 µ 1 λ Pi C# =0 P i P P i P Imposing symmetry and market clearing, we have the condition EC 1 σ = ηλ EC (8) λ 1 Now assume that the price level is determined by the quantity theory relationship M = PC We can evaluate utility by noting from (7) and (8) that EU = 1 EC 1 σ λ 1 σ If M is log normal then we have log normal utility. This gives us an expression EU = 1 exp((1 σ)(ec + 1 (1 σ)var(c)) λ 1 σ If we took a linear approximation of the quantity theory relationship, we might guess that E(c) =0. But again this would be incorrect, because we have to take into account the necessary condition (8), which gives us Ec = φ + 1 (σ )var(c) We can show that monetary volatility increases Ec is σ>. But it still reduces welfare. 7. So we now know that we must in general take nd order approximation of DSGE models in order to do welfare evaluation. How do we go about taking second order approximations of dynamic forward looking models? The example above was simple, since it was a static function. But when we have dynamic models with stochastic endogenous variables it is not obvious how to do this. Two papers are important. Schmitt Grohe and Uribe, and Lombardo and Sutherland. The first paper gives a general approach based on a kind of undetermined coefficients solution method. The second gives a more direct approach based on standard saddle path solution methods. Both give equivalent results for any given model however. We describe the SGU approach. 8. Take a dynamic general equilibrium rational expectations model given by: E t f(y t+1,y t,x t+1,x t )=0 (9) where f is n by 1. Here y are non-predetermined variables, and y t is an n y 1 vector, x t is an n x 1 vector, where n = n x + n y. Assume that x t = 4
[x 1 t,x t ], where x 1 are endogenous predetermined variables, and x are exogenous predetermined variables (shocks), which are described by: x t+1 = Λx t + eησε t+1 Assumptions: bounded support, stationary shocks, ε is n ε 1. Given the example of the neo-classical growth model. The general solution to this model is: y t = g(x t,σ) (10) x t+1 = h(x t,σ)+ησε t+1 (11) where η = 0 η is nx n ε, g is an n y 1, andh is n x 1. The question is, how do we construct the solutions g(x t,σ) and h(x t,σ)? Clearly, we cannot in general obtain the exact solutions for these functions. Rather, we can derive approximations around a non-stochastic steady state. This is done by the method of perturbation. First, define a non-stochastic steady state by the condition: f(y,y,x, x) =0 To get the first order approximation, we may expand (10) and (11) around the steady state as follows: y t g (x, 0) + g x (x x)+g σ σ x t h (x, 0) + h x (x x)+h σ σ Then, up to a first order, we need to obtain the coefficients g x,h x,g σ,h σ. These are obtained by noting that by the definition of f,itmustbethecasethatforany initial x and σ, wehavef (x, σ) =E t (g(h(x, σ)+ησε 0 ),g(x),h(x)+ησε 0,x)=0, so that all derivatives F x i σj(x, σ) =0. Results: a) Setting F x =0,wecanderivetheexpressionsg x and h x by solving for the roots of the first order dynamic system. b) Setting F σ =0,wefind that this gives a homogeneous equation in g σ and h σ. For all x and σ, the only solution must be that g σ = h σ =0. Simple examples: Take the small economy model described by: where β (C t )=(1+C t C) ψ. C 1 t E t β(c t )C 1 t+1 = 0 B t+1 (1 + r)b t C t + Y t = 0 Y t+1 Y ρ t exp(ε t+1 ) = 0 5
In a steady state, we have: 1 1 = 0 rb C + Y = 0 Y 1 = 0 We define C t = g(b t,y t ) and B t+1 = h (B t,y t ). Then expanding the above system around the steady state, we may solve for the coefficients of g 1,g, h 1, h, etc. For instance, in the case of g b and h b, we have: (1 ψc)g b g b h b = 0 h b (1 + r) g b = 0 It is easy to see that the solution will be g b = r + ψc and h b =1 ψc. More generally, we may use the SGU code [gx,hx] = gx_hx(nfy,nfx,nfyp,nfxp) to solve for the policy functions from the first order system. c) Second order approximation: Take a -0 Taylor series approximation of g() and h(). Then substitute this into the conditions F xx =0, F xσ =0,and F σσ =0, to determine the coefficients g xx,g xσ,g σσ, h xx,h xσ,h σσ. The algebra is quite cumbersome - see the SGU paper. However, one key feature is that the resulting system is linear in the g xx, h xx coefficients. That is, there is no need to solve an eigenvalue problem at this stage. In addition, again it turns out that g xσ = h xσ =0. So, up to the second order, the coefficients of the optimal policy rules do not depend on the volatility parameter. 9. An alternative solution method for nd order approximations of DSGE models is done in Lombardo and Sutherland. Their method is more like traditional approaches to solving RE models using a foreword solution method. The key simplification that they use is that the derivation of the second order aspects of the system may be done using the first order approximation alone. Toseethisusetheexample: E t y t+1 = ay t + byt + ce t yt+1 + v t + O 3 (1) where v t+1 = ρv t + ε t+1, ρ<1, anda>1. To solve this, we may compute yt and E t yt+1 by solving the system E t y t+1 = ay t + v t + O alone, and then substituting in the solutions to (1). This is because (1) is computed up to second order accuracy only. That means a first order solution of the model is enough to compute yt and E t yt+1. 6