Solving General Equilibrium Models. with Incomplete Markets and Many Assets
|
|
- Jason Greer
- 5 years ago
- Views:
Transcription
1 Solving General Equilibrium Models with Incomplete Markets and Many Assets June 23, 2006 Martin D. D. Evans Viktoria Hnatkovska Georgetown University and NBER Georgetown University Department of Economics Department of Economics Washington DC Washington DC Tel: (202) Abstract This paper presents a new numerical method for solving general equilibrium models with many assets. The method can be applied to models where there are heterogeneous agents, time-varying investment opportunity sets, and incomplete markets. It also can be used to study models where time-varying risk premia and optimal portfolio choices arise endogenously. We illustrate how the method is used by solving one and two-sector versions of a two country general equilibrium model with production and dynamic portfolio choice. We check the accuracy of our method by comparing the numerical solution to the one-sector model against its known analytic properties. We then apply the method to the two-sector model where no analytic solution is available. In both models the standard accuracy tests con rm the e ectiveness of our method. JEL Classi cation: C68; D52; G. Keywords: Portfolio Choice; Incomplete Markets; Time-Varying Risk Premia; Perturbation Methods; Projection Methods. Financial support from the National Science Foundation is gratefully acknowledged.
2 Introduction This paper presents a new numerical method for solving general equilibrium models with many assets. The method can be applied to models where there are heterogeneous agents, time-varying investment opportunity sets, and incomplete markets. In this paper we illustrate how the method is used by solving one and twosector versions of a two country general equilibrium model with production. Formal accuracy tests con rm that our technique provides a very accurate solution to both models. Equilibrium in the one-sector model we study is characterized by complete risk-sharing, constant portfolio rules and constant risk premia. This model can be solved analytically, so we use it to provide a simple check of the accuracy of our numerical procedure. Solving for the equilibrium in the two-sector model is much more complex because markets are incomplete. In this environment portfolio decisions a ect real allocation decisions, and vice-versa. This complex interaction between the real and nancial sides of the economy cannot be addressed by existing solution methods. Our method allows for these interactions. Furthermore, it allows us to nd the dynamic portfolio rules and time-varying risk premia that arise in equilibrium as a consequence of market incompleteness. Our ability to compute an accurate numerical solution to a model with these complex features means that our method has many important applications. Our technique combines a hybrid projection-perturbation method with continuous-time approximations. In so doing, we contribute to the literature along several dimensions. First, relative to the nance literature, our method delivers optimal portfolios in a discrete-time general equilibrium setting in which returns are endogenously determined. It also enables us to characterize the dynamics of returns and the stochastic investment opportunity set as functions of macroeconomic state variables. 2 Second, relative to the macroeconomics literature, portfolio decisions are derived without assuming complete asset markets or constant returns to scale in production. 3 Our solution method also relates to the literature on numerical methods. First, it builds on the perturbation methods developed and applied in Judd and Guu (993, 997), Judd (998) and further discussed in Collard and Juillard (200), Jin and Judd (2002), Schmitt-Grohe and Uribe (2004) among others. These methods extend solution techniques relying on linearizations by allowing for second- and higher-order terms in the approximation of the policy functions. Second, our method builds upon the projection technique introduced by Judd (992). This method parametrizes the decision rules using basis functions and chooses the optimal rules that minimize some residual function. Unfortunately, these methods can only be used in applications that omit a key feature of models with portfolio choice: namely, the conditional heteroskedasticity of the state vector that captures the time-varying nature of the investment opportunity set. Continuous time approximations of Campbell, et. al. (2003) can handle such endogenous heteroskedasticity, however 2 A number of approximate solution methods have been developed in partial equilibrium frameworks. Kogan and Uppal (2000) approximate portfolio and consumption allocations around the solution for a log-investor. Barberis (2000), Brennan, Schwartz, and Lagnado (997) use discrete-state approximations. Brandt, Goyal, and Santa-Clara (200) solve for portfolio policies by applying dynamic programming to an approximated simulated model. Brandt and Santa-Clara (2004) expand the asset space to include asset portfolios and then solve for the optimal portfolio choice in the resulting static model. 3 Solutions to portfolio problems with complete markets are developed in Heathcote and Perri (2004), Serrat (200), Kollmann (2005), Baxter, Jermann and King (998), Uppal (993), Engel and Matsumoto (2004). Pesenti and van Wincoop (996) analyze equilibrium portfolios in a partial equilibrium setting with incomplete markets.
3 they are developed only in partial equilibrium. We extend their approach to a general equilibrium setting. The paper is structured as follows. Section presents the one sector version of the model we use to illustrate our solution method. Section 2 describes the method in detail. Section 3 presents and compares the numerical solution of the model to its analytic counterpart. A formal assessment of the method s accuracy is provided in Section 4. Section 5 presents the two sector version of the model and examines its equilibrium properties. Section 6 concludes. The One-Sector Model This section describes the one sector version of the model we employ to illustrate our solution method. It is a standard international asset pricing model with portfolio choice and builds upon Danthine and Donaldson s (994) formulation of an asset pricing model with production. We consider a frictionless production world economy consisting of two symmetric countries, called home (h) and foreign (f). Each country is populated by a continuum of identical households who consume and invest in di erent assets, and rms who produce a single good that is freely traded between the two countries. Firms are perfectly competitive and issue equity that is traded on the world stock market.. Firms Our rms are in nitely lived. They issue equity claims to the stream of their dividends, and households can use this equity for their saving needs. Each rm owns capital and undertakes independent investment decisions. A representative rm in the h country starts period t with the stock of rm-speci c capital K t and an equity liability A t equal to. Period t production is Z t K t ; with > 0: Z t denotes the state of productivity. The output produced by rms in the f country is given by an identical production function using rm-speci c foreign capital, ^Kt ; and productivity, ^Zt : (Hereafter we use ^ to denote foreign variables.) The goods produced by h and f rms are identical and can be costlessly transported between countries. Under these conditions, the law of one price must prevail to eliminate arbitrage opportunities. At the beginning of period t, each rm observes the productivity realization, produces output, and uses the proceeds to nance investment I t and to pay dividends to the shareholders. We assume that rms allocate output to maximize the value of the rm to its shareholders every period. Let P t denote the ex-dividend price of a share in the representative h rm at the start of period t; and let D t be the dividend per share paid at t: With A t = ; the value of the rm at the start of period t is P t + D t ; and the optimization problem it faces can be summarized as subject to max I t (D t + P t ) ; () K t+ = ( )K t + I t ; (2) D t = Z t K t I t ; (3) where > 0 is the depreciation rate on physical capital. The representative rm in the f country solves an 2
4 analogous problem; that is to say they choose investment ^I t to maximize ^D t + ^P t, where ^P t is the ex-dividend price of a share and ^D t is the dividend per share paid at t. As before, we normalize the equity liability ^A t to : Let z t [ln Z t ; ln ^Z t ] 0 denote the state of productivity in period t: We assume that z t follows an AR() process: z t = az t + S =2 e e t ; where a is a 2 2 matrix and e t is a 2 vector of i.i.d. mean zero, unit variance shocks. S =2 e is a 2 2 matrix of scaling parameters..2 Households Each country is populated by a continuum of households who have identical preferences. The preferences of households in the h country are de ned in terms of h consumption C t ; and are given by X E t i=0 i ln C t+i ; (4) where 0 < < is the discount factor. E t denotes expectations conditioned on information at the start of period t: Preferences for households in country f are similarly de ned in terms of foreign consumption, ^Ct : Households in our economy can save by holding domestic equity shares, international bonds and equity issued by foreign rms. The budget constraint of the representative h household can be written as W t+ = R w t+ (W t C t ) ; (5) where W t is nancial wealth, and R w t+ is the (gross) return on wealth between period t and t +. This return depends on how the household allocates wealth across the available array of nancial assets, and on the realized return on those assets. In particular, R w t+ = R t + h t (R h t+ R t ) + f t(r f t+ R t ); (6) where R t is the return on bonds, and R h t+ and R f t+ are the returns on h and f equity. The fraction of wealth that h country households hold in h and f equity are h t and f t respectively: The budget constraint for f households is similarly de ned as ^W t+ = ^R w t+( ^W t ^Ct ); with ^Rw t+ = R t + ^ h t (R h t+ R t ) + ^ f t(r f t+ R t ); where ^ h t and ^ f t denote the shares of wealth allocated by f households into h and f country equities. Households in country h choose how much to consume and how much wealth to allocate into the equity 3
5 of h and f rms to maximize expected utility (4) subject to (5) and (6) given current equity prices and the interest rate on bonds: This problem can be recursively expressed as: Q (W t ) = max ln Ct + E t Q R w t+ (W t C t ) ; (7) fc t; h t ;f t g with C t 0 and W t > 0: Q(:) denotes the household s value function. The optimization problem facing f households is analogous..3 Equilibrium This section summarizes the conditions characterizing the equilibrium in our model. The rst order conditions for the representative h household s problem in (7) are = E t Mt+ R h t+ ; (8a) = E t [M t+ R t ] ; (8b) = E t Mt+ R f t+ ; (8c) where M t+ (@U=@C t+ ) = (@U=@C t ) is the discounted intertemporal marginal rate of substitution (IMRS) between consumption in period t and period t + : The returns on equity issued by h and f rms are de ned as R h t+ = (P t+ + D t+ ) =P t and R f t+ = ^Pt+ + ^D t+ = ^P t : With these de nitions, the Euler equation in (8a) can be rewritten as P t = E t [M t+ (P t+ + D t+ )] : Using this expression to substitute for P t in the h rm s investment problem ()-(3) gives the following recursive formulation: V(K t ; Z t ) = max I t (D t + P t ) ; = max I t (D t + E t [M t+ (D t+ + P t+ )]) ; = max I t Z t K t I t + E t [M t+ V(K t+ ; Z t+ )] ; (9) where V(:) denotes the value of the rm. The rst order condition associated with this optimization problem is = E t Mt+ R k t+ ; where R k t+ Z t+ (K t+ ) + ( ) is the return on capital. This condition determines the optimal investment of h rms and thus implicitly identi es the level of dividends in period t; D t ; via equation (3). The rst order conditions for rms in country f take an analogous form. It is worth noting that our model has equity home bias built in as rms use the IMRS of domestic agents, (e.g. M t+ in the case of h rms) to value the dividend steam in (9). Although the array of assets available to households is su cient for complete risk-sharing in this version of the model, in the two sector 4
6 version we present below markets are incomplete. As a result, the IMRS for h and f households will di er and households in the two countries will generally prefer di erent dividend streams. In principle, this formulation of how rms choose investment/dividends can induce home bias in household equity holdings. Solving for the equilibrium in this economy requires nding equity prices fp t ; ^Pt g; and the interest rate R t ; such that markets clear when households follow optimal consumption, savings and portfolio strategies, and rms make optimal investment decisions. Under the assumption that bonds are in zero net supply, market clearing in the bond market requires that 0 = B t + ^B t : (0) The goods market clears globally. In particular, since h and f rms produce a single good that can be costlessly transported between countries, the market clearing condition is C t + ^C t = Y t I t + ^Y t ^It = D t + ^D t : () The market clearing conditions in the h and f equity markets are = A h t + ^A h t and = A f t + ^A f t: (2) where A i t ( ^A i t) denotes the number of shares of equity issued by i = {h, f} rms held by h (f) households. These share holdings are related to the portfolio shares by the identities, P t A h t h t (W t C t ) and ^P t A f t f t (W t C t ). The share holdings of f households are ^A h t and ^A f t with P t ^Ah t ^ h t ( ^W t ^Ct ) and ^P t ^Af t ^ f t( ^W t ^Ct ): 2 Solution Method 2. Overview Our solution method relies on several existing numerical methods for functional problems. In particular, it borrows some of the features of perturbation and projection methods, and amends them with the continuous time approximations. We next provide a brief overview of these techniques and highlight the novelty of our approach. The goal of both the perturbation and projection methods consists of solving a system of equations that characterize the equilibrium of the model. This system has the general form 0 = E t f (Y t+ ; Y t ; X t+ ; X t ; " t+ ) ; (3) where f is a known function. X t is a vector of variables that describe the state of the economy at time t: In our illustrative model, X t contains the state of productivity, capital stocks and households wealth. Y t is a vector of non-predetermined variables at time t: It includes consumption, dividends, asset prices, etc. " t 5
7 are innovations with the scaling parameter (matrix) : In our model the disturbance terms in " t are i.i.d. productivity shocks with mean zero and unitary variance; " t = S =2 e e t : Both methods start by conjecturing the form of the solution to the problem summarized in (3). In particular, the optimal policy functions characterizing competitive equilibria of the model can be presented as Y t = G(X t ; ) and X t+ = H (X t ; " t+ ) : Substituting these conjectures into (3) we get 0 = E t f (G(H (X t ; " t+ ) ; ); G(X t ; ); H (X t ; " t+ ) ; X t ; " t+ ) F (X t ; " t+ ) : (4) Even though the true form of G(:) and H(:) are unknown, we can write their approximations as bg(x t ; ) = X i' i (X t ; ) ; (5) i bh(x t ; " t+ ) = X i' i (X t ; ) ; (6) i where { i } and { i } are parameter vectors, and ' i (X t ; ) are some elementary functions. The form of the approximations varies across the methods which we discuss next. Perturbation Methods The perturbation method has become a popular tool in solving stochastic general equilibrium models because it balances accuracy and computational time. The method, as developed by Collard and Juillard (200), Jin and Judd (2002), and Schmitt-Grohe and Uribe (2004), consists of approximating G(X t ; ) and H(X t ; " t+ ) around a non-stochastic steady state X and a set of perturbation parameters. The steady state values for X are obtained from the steady state equation 0 = f (Y ; Y ; X ; X ; 0) : The standard deviations of the innovations to productivity in are usually used as perturbation parameters. Policy functions are approximated using n-th. order Taylor series expansions, which implies that the elementary functions ' i (X t ; ) in (5) and (6) are n-th. order polynomials of X t and : To obtain the unknown coe cients and ; the perturbation method relies on the Implicit Function Theorem. In particular, rst the system of equilibrium conditions F (X t ; " t+ ) = 0 is approximated up to the n-th. order around the steady state and = 0: Since the equilibrium conditions in F (:) must be equal to zero for any X or ; the system of derivatives of F (:) up to any order must be equal to zero as well. Namely, F x nx n (X t; ) = 0 8X t ; ; n x ; n (7) where F x nx n (X t; ) represents the n x -th. derivative of F (:) with respect to X t ; and the n -th. derivative with respect to : The task of nding the coe cients in (5) and (6) now reduces to solving the system of restrictions in (7) evaluated at the steady state and = 0. 6
8 Projection Methods The projection method was introduced in Economics by Judd (992). In its general formulation, the technique consists of choosing basis functions over the space of continuous functions and using them to approximate G(X t ; ) and H(X t ; " t+ ): In most applications, families of orthogonal polynomials, like Chebyshev s polynomials, are used to form ' i (X t ; ) : 4 Given the chosen order of approximation, n; the problem of solving the model translates into nding the coe cient vectors and that minimize a residual function. This function is formed by replacing the true policy functions in (4) by their approximate counterparts, b G(X t ; ) and b H(X t ; " t+ ) : R(X t ; " t+ ; b G; b H; ; ) = E t f bg( b H (Xt ; " t+ ) ; ); b G(X t ; ); b H (X t ; " t+ ) ; X t ; " t+ : The optimal values of and minimize this residual function, given some weighting function which determines the size of approximation error. This procedure creates the projection, whose name varies depending on the choice of the weight. The least squares projection chooses coe cients and to minimize the norm of the residual function. Other alternatives include the Galerkin method and the Collocation method (Judd 992, 998). To obtain the conditional expectations that appear in the residual function as well as the integrals necessary to calculate the projections, the method relies on numerical integration techniques. Continuous Time Approximations The use of continuous time approximations is advocated by Campbell (994), while their application to portfolio choice problems is described in Campbell, Chan and Viceira (2003), CCV hereafter. The technique uses a continuous time approximation for portfolio returns which preserves the multiplicative nature of portfolio weighting. CCV approximations hold exactly in continuous time when asset prices follow di usions and remain very accurate in discrete time for short time intervals. The method is applicable to partial equilibrium models in which the process for returns is exogenous. It consists of taking Taylor series expansions of the model equilibrium conditions, which in combination with approximate budget constraints and exogenous processes for returns give a system of linear-quadratic equations for portfolio shares and consumption in terms of state variables. An iterative numerical procedure can then be used to solve this system for optimal consumption and portfolio rules. Our Method Our method draws on all three approaches outlined above. It combines a hybrid projection-perturbation method with the continuous time approximations in a way that supplements the strengths of each approach. Our reasoning follows that of Judd (998). Each method has important advantages. In solving the problem 4 Chebyshev s polynomial method belongs to the family of the spectral projection methods for which the basis functions are nonzero almost everywhere. Finite element projection methods use basis functions that are nonzero within a small support. For a comparison of the two see Aruoba et.al. (2005). 7
9 summarized in (4), the perturbation method xes the and coe cients in (5)-(6) at the values implied by the derivatives in (7). At the same time, the ' i (X t ; ) functions are exible, chosen to re ect the problem at hand. Projection methods, on the other hand, choose values for { ; g optimally, given a choice for the ' i (X t ; ) functions. Thus a researcher who uses either method faces a trade-o between having ' i (X t ; ) or { ; g set a priori. Our method allows for exibility in both the ' i (X t ; ) functions and the { ; g coe cients. Speci cally, we start by solving a perturbation problem, which provides us with approximate policy functions for some vector of coe cients. Given the ' i (X t ; ) functions derived in the rst step, we then solve for the coe cients that minimize a residual function. The remainder of this section presents our procedure in general terms. The set of equations characterizing the equilibrium of our model can be written in a general form as 0 = E t f Y t+ ; Y t ; X t+ ; X t ; S =2 (X t ) " t+ ; (8) X t+ = H X t ; S =2 (X t ) " t+ ; where S =2 (X t ) is a lower triangular matrix or Cholesky factor. As before, X t is a vector of variables that describe the state of the economy at time t; while Y t is a vector of non-predetermined variables at time t: The function f(:) denotes the equations characterizing the equilibrium, while H (:; :) determines how past states a ect the current state. " t+ is a vector of mean zero i.i.d. innovations with unit variances. The vector of shocks driving the equilibrium dynamics of the model is U t+ S =2 (X t ) " t+ : This vector includes exogenous shocks, like the productivity shocks, and innovations to endogenous variables, like the shocks to households wealth. The shocks have a conditional mean of zero and a conditional covariance equal to S (X t ) ; a function of the current state vector X t : E (U t+ jx t ) = 0; (9) E U t+ Ut+jX 0 t = S =2 (X t ) S =2 (X t ) 0 = S (X t ) : An important aspect of our formulation is that it explicitly allows for the possibility that shocks driving the equilibrium dynamics are conditionally heteroskedastic. By contrast, standard perturbation methods assume that U t+ follows an i.i.d. process, in which case S (X t ) would be a constant matrix. As we shall see, it is not possible to characterize the equilibrium of a model with portfolio choice and incomplete markets in this way. Conditional heteroskedasticity arises as an inherent feature of such models, and must be accounted for in any solution technique. Given our formulation in (8) and (9), a solution to the model is characterized by a decision rule for the non-predetermined variables Y t = G (X t ; S (X t )) ; (20) 8
10 that satis es the equilibrium conditions in (8): 0 = E t f G H X t ; S =2 (X t ) " t+ ; S H X t ; S =2 (X t ) " t+ ; G (X t ; S (X t )) ; H X t ; S =2 (X t ) " t+ ; X t ; S =2 (X t ) " t+ : Or in a more compact notation, 0 = F(X t ): as The rst step in our method follows the perturbation procedure by approximating the policy functions bg = X i' i (X t ) ; i bh = X i' i (X t ) ; i bs = X s i' i (X t ) ; i for some unknown coe cient sequences { i }, { i }, and {s i }. ' i (X t ) are ordinary polynomials in X t : Next we approximate f; as b f: The equations associated with the real side of the economy are approximated using Taylor series expansions, while those pertinent to the portfolio side are approximated using continuous time expansions of CCV. We denote the derivatives in these expansions as {& i }. Next we apply the projection method that uses ' i (X t ) as basis functions. Substituting b G; b H, and b S into bf and taking expectations analytically gives us an approximation for F : bf X t ; G; b H; b S; b ; ; ; s = X i' i (X t ) ; i where { i } are functions of {& i }, { i }, { i }, and {s i }. This is our residual function R(X t ; G; b H; b S; b ; ; ; s): The coe cient vectors ; ; ; and s are found my minimizing this function in conjunction with a weighting matrix. Two features of our method deserve special note. The rst concerns our treatment of expectations. In particular, the system of equilibrium conditions F (:) often involves computing conditional expectations of highly nonlinear functions (e.g., Euler equations). In most cases, such calculations require integration and are challenging. The projection method addresses the problem by using numerical integration based on quadrature methods. Our method instead approximates F (:) by combining Taylor series expansions with continuous time approximations, and then calculates the conditional expectations of the approximated functions analytically. This is a great computational saving, and makes solving relatively complex models feasible [see, for example, Evans and Hnatkovska (2005) and Hnatkovska (2005)]. The second feature concerns the function S (X t ) ; which identi es the covariance of the shocks driving the state vector X t : We need to accommodate conditional heteroskedasticy in the dynamics of X t because it arises from the structure of models that incorporate portfolio choice with incomplete markets. This is 9
11 true even when the exogenous shocks to the economy (e.g. productivity shocks) are homoskedastic. When markets are incomplete we need to track the distribution of wealth across the economy to identify the IMRS for each household and hence compute their portfolio choices. 5 This means that X t must include the wealth of individual households; elements that will be conditionally heteroskedastic when optimal portfolio shares vary with the state of the economy (see below). The S (X t ) function is therefore necessary to represent the general equilibrium implications of time-varying portfolio choice when markets are incomplete. 6 An accurate characterization of the portfolio choice problem facing households also requires the S (X t ) function because it identi es how the second and higher-order conditional moments of all the variables vary with X t. These moments allow us to compute the equilibrium risk premia and optimal portfolio shares as functions of X t : The ability to solve an incomplete markets model with endogenous time-varying risk premia and portfolios means that our method has many important applications. The subsections that follow describe each step of our solution method in detail. 2.2 Log-Approximations To further clarify why our formulation in (8) and (9) allows for conditional heteroskedasticity in the dynamics of the state vector, we return to the model. In particular, let us focus on the log-approximated equations arising from the households rst order conditions and budget constraint. Hereafter we use lowercase letters to denote the log transformation of the corresponding variable, measured as a deviation from its steady state level or initial value. Following CCV we use a rst-order log-approximation to households budget constraints. In the case of h households it is given by w t+ = ln ( C t =W t ) + r w t+; = (c t w t ) + r w t+; (2) where is the steady state consumption-wealth ratio and ln( ): In our model, households have log preferences so the optimal consumption-wealth ratio is a constant equal to : In this case c t w t = 0 and = ln : rt+ w is the log return on optimally invested wealth which CCV approximate as r w t+ = r t + 0 ter t t (diag ( t ) t t ) ; (22) where 0 t [ h t f t ] is the vector of portfolio shares, er 0 t+ [ r h t+ r t r f t+ r t ] is a vector of excess equity returns, and t is the conditional covariance of er t+ : The approximation error associated with this 5 When markets are complete we can solve the portfolio problems using the common IMRS that can be identi ed from real allocations alone (i.e. without the distribution of wealth). See, for example, Kollmann (2005). 6 The S (X t) function also impacts on real allocation decisions, like consumption and investment. So in general there will be a feedback between the portfolio decisions of households and the real economy. By contrast, when markets are complete equilibrium allocations can be computed separately from household s nancial decisions (see, for example, Obstfeld and Rogo 996, pp. 302). 0
12 expression disappears in the limit where asset prices follow continuous time di usion processes. obtain Next, we turn to the rst-order conditions in (8). E t r t+ r t + 2 V t r t+ Using the standard log-normal approximation, we = CV t m t+ ; r t+ ; (23a) r t = E t m t+ 2 V t(m t+ ); (23b) where r t+ is the log return for equity = fh, fg ; and m t+ ln M t+ is the log IMRS. V t (:) and CV t (:; :) denote the variance and covariance conditioned on period t information. With log utility m t+ = ln c t+ = ln w t+ ; so (23a) can be rewritten in vector form as Combining this expression with (2) and (22) gives E t er t+ = t t 2 diag ( t) : (24) w t+ = (c t w t ) + r t t t t + 0 t (er t+ E t er t+ ) : (25) Equation (25) provides us with a log-approximate version of the h household s budget constraint. It shows that the growth in household wealth between t and t + depends upon the consumption/wealth ratio in period t (a constant in the case of log utility), the period-t risk free rate, r t ; portfolio shares, t ; the variance-covariance matrix of excess returns, t ; and the unexpected return on assets held between t and t + ; 0 t (er t+ E t er t+ ) : Notice that the susceptibility of wealth in t + to unexpected returns depends on the period-t portfolio choices, t. Consequently, the volatility of wealth depends endogenously on the portfolio choices made by households and the equilibrium behavior of returns. In an equilibrium where returns have an i.i.d. distribution, t will be constant, 7 and wealth will be conditionally homoskedastic. Of course in a general equilibrium setting the properties of returns are themselves determined endogenously, so there is no guarantee that optimally chosen portfolio shares or the second moments of returns will be constant. Consequently, if wealth is an element in the state vector X t ; our solution method needs to allow for the presence of conditional heteroskedasticity in X t : Our (approximate) solution for the S (X t ) function reveals the extent to which heteroskedasticity arises in the dynamics of wealth given the optimal choice of t for a general distribution of equilibrium returns. 8 Of course, standard perturbation methods can still be used to solve models where the equilibrium dynamics of the wealth must be homoskedastic, or where wealth can be excluded from X t : The latter case occurs when the array of assets available for trading allows for perfect risk-sharing so that markets are complete. When markets are incomplete, by contrast, it is not possible to characterize the equilibrium dynamics of the 7 This can be seen in our case of log utility by inspecting (24). Clearly, if E ter t+ and t are constant because returns are i.i.d., t must also be constant. 8 It is worth emphasizing that heteroskedasticity does not arise here because we are dealing with a log-approximated version of the household s budget constraint. It is an inherent feature of the household s budget constraint because portfolio choices a ect the susceptibility of future wealth to the unexpected returns on individual assets (see equations 5 and 6 above). The log-approximation in (25) simply illustrates the point in a particularly clear way.
13 economy without including household wealth in the state vector X t : 9 As a consequence, in this setting it is necessary to allow for conditional heteroskedasticity in the dynamics of the state variables as our formulation in (8) and (9) does. Equation (24) implicitly identi es the optimal choice of the h households portfolio shares, t : This equation was derived from the households rst-order conditions under the assumption that the joint conditional distribution of log returns is approximately normal. Notice that the approximation method does not require an assumption about the portfolio shares chosen in the steady state. By contrast, standard perturbation methods consider Taylor series approximations to the model s equilibrium conditions with respect to decision variables around the value they take in the non-stochastic steady state. As Judd and Guu (2000) point out, this method is inapplicable when the steady-state value of the decision variable is indeterminate. This is an important observation when solving a model involving portfolio choice. In the non-stochastic steady state assets are perfect substitutes in household portfolios because returns are identical, so the optimal choice of portfolio is indeterminate. While the steady state portfolio shares are absent from equation (24), the problem of indeterminacy still arises in our one-sector model. In particular, we have to take a stand on the steady state distribution of asset holdings when log-approximating the market clearing conditions: Consider, for example, the market clearing condition for h equity in (2). Combining this condition with the portfolio share de nitions, and the fact that the consumption-wealth ratio for all households is equal to ; we obtain P t ^W = h t + ^ h t t : W t W t We consider a second-order Taylor series approximation to this expression around the steady state values for P t =W t and a particular value for ^W t =W t : Speci cally, we parameterize the value of ^W =W and then work out its implications for the value of P=W: 0 This is particularly simple in the case where wealth is assumed to be equally distributed (i.e. ^W =W = ). Here symmetry and market clearing in the goods market require that D = C = ( )W: It follows that P=W = [( )=] (P=D) = because the Euler equation for stock returns implies that the steady state value of P=D equals =( log-approximation embedding goods market clearing becomes + p t w t + 2 (p t w t ) 2 = h t + ^ h t + ^w t w t + 2 ( ^w t w t ) 2 : ): In this case, the second-order Log-approximations implied by the other market clearing conditions are similarly obtained. Speci cally, when initial wealth is assumed to be equally distributed, market clearing in f equity, bonds and goods imply, 9 Speci cally, we need to track m t+ and ^m t+ ; which for standard utility speci cations are functions of c t+ and ^c t+ respectively. We therefore need to solve the consumption/savings problem of each household to identify c t+ and ^c t+ ; a task that necessitates the inclusion of household wealth in the state vector. 0 Our approach of parametrizing the initial wealth distribution across agents is an alternative to the Judd and Guu (2000) bifurcation procedure for dealing with portfolio indeterminacy. 2
14 correspondingly, that + ^p t ^w t + 2 (^p t ^w t ) 2 = ^ f t + f t + w t ^w t + 2 (w t ^w t ) 2 ; p t + ^p t = d t + ^d t ; (26) c t + ^c t = d t + ^d t : This approach to the indeterminacy problem has another important advantage. In cases where wealth must be an element of X t ; its presence introduces a nonstationary unit root component in the X t process: shocks to returns will generally have permanent e ects on wealth. As we show below, our procedure accommodates the presence of a unit root by characterizing the equilibrium dynamics of the model in a neighborhood of the initial state, X 0 : To study the equilibrium properties of the model we must therefore specify the elements of X 0 : Thus, choosing the initial distribution of wealth not only provides a way to resolve indeterminacy concerning portfolio shares in the non-stochastic steady state, it also allows us to analyze the equilibrium dynamics of a model that is inherently nonstationary. Of course, we do have to keep in mind that the accuracy of the equilibrium dynamics provided by our solution will deteriorate as X t moves further from X 0 : The remaining equations characterizing the model s equilibrium are log-approximated in a standard way. Optimal investment by h and f rms requires that E t rt+ k r t + 2 V t rt+ k E t^r t+ k r t + 2 V t ^r t+ k = CV t r k t+; w t+ ; (27a) = CV t ^r k t+; ^w t+ ; (27b) where r k t+ and ^r k t+ are the log returns on capital approximated by r k t+ = z t+ ( ) k t+ and ^r k t+ = ^z t+ ( ) ^k t+ ; (28) with ( ) < : The dynamics of the h and f capital stock are approximated by k t+ = k t + z t d t and ^kt+ = ^k t + ^z t ^d t : (29) Finally, we turn to the relationship between the price of equity, dividends and returns. As in Campbell and Shiller (989), we relate the log return on equity to log dividends and the log price of equity by r h t+ = p t+ + ( )d t+ p t and r f t+ = ^^p t+ + ( ^) ^d t+ ^p t ; (30) with =( + exp(d p)) and ^ =( + exp( ^d ^p)) where d p and ^d ^p are the average log For example, when households have log preferences the rst two terms on the right in (25) are constant. Under these circumstances, a positive unexpected return will permenantly raise wealth unless the household nds it optimal to adjust their future portfolio shares so that 0 t+i t+i t+i falls and/or r t+i falls by a compensating amount. 3
15 dividend-price ratios in the h and f countries. In the non-stochastic steady state = ^ = : Making this substitution, iterating forward, taking conditional expectations, and imposing lim j! E t j p t+j = 0 and lim j! E t j ^p t+j = 0; we obtain p t = ^p t = X i ( )E t d t++i E t rt++i h ; (3a) i=0 X n o i ( )E t ^dt++i E t rt++i f : (3b) i=0 These approximations show how log equity prices are related to expected future dividends and returns. 2.3 State Variables Dynamics The key step in our solution procedure is deriving a general yet tractable set of equations that describe the equilibrium dynamics of the state variables. dimensionality of the state vector. One problem we immediately face in this regard is the As we noted above, the distribution of wealth plays an integral role in determining equilibrium prices and returns when markets are incomplete, so household wealth needs to be included in the state vector. In models with a continuum of heterogenous households it is obviously impossible to track the wealth of individuals, so moments of the wealth distribution need to be included in the state vector. The question of how many moments to include is not easily answered. Dimensionality is still a problem when heterogeneity across households is limited. In our model there are only two types of households, so it su ces to keep track of h and f households wealth. The dimensionality problem occurs under these circumstances because uncertainty enters multiplicatively into the dynamics of wealth. (Recall that portfolio shares determine the susceptibility of wealth to unexpected return shocks.) If wealth is part of the state vector, X t, and both portfolio shares and realized returns depend on X t ; the level of wealth will depend on the elements in X t X 0 t: This means that the equilibrium dynamics of wealth will in general depend on the behavior of the levels, squares and cross-products of the individual state variables. This dependence between the lower and higher moments of the state variables remains even after log-approximation. In equation (25) we see that h household wealth depends on the quadratic form for portfolio shares, which are themselves functions of the state vector, including wealth. As a result, the state vector needs to be expanded to include squares and cross-products. Of course a similar logic applies to the equilibrium behavior of squares and cross-products involving wealth. So by induction, a complete characterization of the equilibrium wealth dynamics could easily require an in nite number of elements in X. Our solution procedure uses a nite subset of state variables X X that provides a good approximation to the equilibrium dynamics. We will use the model presented in Section to illustrate our procedure. Let x t [z t ; k t ; ^k t ; w t ; ^w t ] 0 where k t ln (K t =K), ^k t ln ^Kt =K ; w t ln(w t =W 0 ) and ^w t ln( ^W t = ^W 0 ): More generally, x t will be an l vector that contains the variables that make up the state vector. We will approximate the equilibrium 4
16 dynamics of the model with the vectors X t = x t ~x t and U t = 6 4 where ~x t vec (x t x 0 t) : The shock vector U t is partitioned conformably with X t and both vectors contain L = + l + l 2 elements. To determine the dynamics of X t ; we rst conjecture that x t follows 0 u t ~u t ; x t+ = 0 + (I )x t + 2 ~x t + u t+ ; (32) where 0 is the l vector of constants, is the l l matrix of autoregressive coe cients and 2 is the l l 2 matrix of coe cients on the second-order terms. u t+ is a vector of innovations with zero conditional mean, and conditional covariance that is a function of X t : E (u t+ jx t ) = 0; E u t+ u 0 t+jx t = (X t ) = 0 + x t x 0 t 0 : Below we shall use the vectorized conditional variance of u t which we write as 2 3 h i vec ((X t )) = x t 7 5 = X t: (33) ~x t The next step is to derive an equation describing the dynamics of ~x t consistent with (32) and (33). For this purpose we consider the continuous time analogue to (32) and derive the dynamics of ~x t+ via Ito s lemma. As the Appendix shows, the resulting process can be approximated in discrete time by ~x t+ = 2 D 0 + ( 0 I) + (I 0 ) x t + I ( I) (I ) + 2 D ~xt + ~u t+ (34) where ~u t+ = [(I x t ) + (x t I)] u t+ ; 0 I 0 I ; and U = X r X E rs Er;s: 0 s E r;s is the elementary matrix which has a unity at the (r; s) th position and zero elsewhere. Equation (34) approximates the dynamics of ~x t+ because it ignores the role played by cubic and higher order terms involving the elements of x t : In this sense, (34) represents a second order approximation to the dynamics of 5
17 the second order terms in the state vector. 2 Notice that the variance of u t+ a ects the dynamics of ~x t+ via the D matrix and that ~u t+ will generally be conditionally heteroskedastic. We can now combine (32) and (34) into a single equation: x t+ ~x t = I 2 2 D 0 ( 0 I) + (I 0 ) I ( I) (I ) + 2 D x t ~x t 0 u t+ ~u t ; or more compactly X t+ = AX t + U t+ ; (35) with E (U t+ jx t ) = 0: We also need to determine the conditional covariance of the U t+ vector. Appendix we show that E U t+ U 0 t+jx t S (Xt ) = (X t ) (X t ) 0 (X t ) 0 (X t ) In the C A ; (36) where vec ( (X t )) = 0 + x t + 2 ~x t ; vec (X t ) 0 = 0 + x t + 2 ~x t ; vec ( (X t )) = 0 + x t + 2~x t : The i; i and i matrices are complicated functions of the parameters in (32) and (33); their precise form is given in the Appendix. To this point we have shown how to approximate the dynamics of X t given a conjecture concerning 0 ; ; 2 ; 0 and : We now turn to the issue of how these matrices are determined. For this purpose we make use of two further results. Let a t and b t be two generic endogenous variables related to the state vector by a t = a X t and b t = b X t ; where a and b are L vectors. Our second-order approximation for the dynamics of X t implies that CV t (a t+ ; b t+ ) = A ( a ; b ) X t ; (R) and a t b t = B ( a ; b ) X t : (R2) A (:; :) and B (:; :) are L vectors with elements that depend on a ; b and the parameters of the X t process. The precise form of these vectors is also shown in the Appendix. 2 One way to check the accuracy of this approximation is to derive a generalization of (34) involving third order terms and then compute the contribution of these terms to the dynamics of x t and ~x t: Since the elements of x t are measured in terms of percentage deviations from steady state or initial values, third order terms are unlikely to be signi cant. Nevertheless, as we note below, we are cognizant of the approximation error in (34) when examining the solution to a model. 6
18 To see how these results are used, we return to the model. The dynamics of the state vector depend upon households portfolio choices, f h t ; f t; ^ h t ; ^ f tg ; rms dividend choices, {d t ; ^d t g; equilibrium equity prices, fp t ; ^p t g ; and the risk free rate, r t : Let us assume, for the present, that each of these non-predetermined variables is linearly related to the state. (We shall verify that this is indeed the case below.) In particular, let i be the L row vector that relates variable i to the state X t and let h i be the L vector that selects the i th element out of X t : We can now easily derive the restrictions on the dynamics of productivity, capital and wealth. Recall that the rst two rows of x t comprise the vector of productivities that follow an exogenous AR() process. The corresponding elements of 0 ; ; 2 ; 0 and are therefore entirely determined by the parameters of this process. The next elements in x t are the log capital stocks. If equilibrium dividends satisfy d t = d X t and ^d t = ^dx t ; we can rewrite the log-approximated dynamics for k t and ^k t shown in (29) as h k X t+ = h^kx t+ = h k + h z d X t ; h^k + h^z ^d X t : Notice that these equations must hold for all realizations of X t : So substituting for X t+ with (35) and equating coe cients we obtain h k A = h k + h z d and h^ka = h^k + h^z ^d: These equations place restrictions on the elements of 0 ; ; and 2 : Furthermore, because k t+ and ^k t+ are solely functions of the period t state, the corresponding element rows and columns of E(u t+ u 0 t+jx t ) (X t ) are vectors of zeros. This observation puts restrictions on the elements of 0 and : Deriving the equilibrium restrictions on the dynamics of wealth in (25) is a little more complicated and requires the use of R and R2. Our starting point is the approximation for log equity returns in (30) which we now write in terms of the state vector: r h t+ = h X t+ p X t and r f t+ = f X t+ ^p X t ; where h p +( ) d and f ^p +( ) ^d: Notice that unexpected log returns are r t+ E t r t+ = (X t+ E t X t+ ) for = fh, f}, so applying R we obtain " # A(h ; h )X t A( h ; f )X t V t (er t+ ) t = : A( h ; f )X t A( f ; f )X t Now recall that our log-approximated version of the h household budget constraint contains a quadratic function of the portfolio shares and t : To evaluate this component, let us assume that the portfolio shares 7
19 satisfy h t = h X t and f t = f X t, so that h 0 t t t = i " A( h X t f h ; h )X t A( h ; f )X t X t A( h ; f )X t A( f ; f )X t # " h X t f X t # : Applying R2 to the right hand side gives 0 t t t = B ( h ; B(A( h ; h ); h ) + B(A( h ; f ); f )) X t +B ( f ; B(A( h ; f ); h ) + B(A( f ; f ); f )) X t ; = X t : According to (25), E t w t+ = w t + ln + r t t t t, while the dynamics of the state vector in (35) imply that E t w t+ = h w AX t : Equating these moments for all possible values of X t requires that h w A = h w + ln h + r + 2 : This expression provides us with another set of restrictions on the elements of 0 ; ; and 2. The model also places restrictions on the second moments of the state variables. For example, the restrictions on the variances and covariances of the productivity vector z t are by assumption homoskedastic: h z vec((x t )) = vec (CV t (z t+ ; z t+ )) = vec(s e ): Second moments involving the capital stock are also straightforward: they are all zero because capital is chosen one period in advance (see equation (2) above). Next, consider the moments involving wealth. First note that for any variable a t = a X t ; CV t (w t+ ; a t+ ) = h t CV t (r h t+; a t+ ) + f tcv t (r f t+; a t+ ): Applying R and R2 to the right hand side, gives CV t (w t+ ; a t+ ) = h X t A( h ; a )X t + f X t A( f ; a )X t = (B( h ; A( h ; a )) + B( f ; A( f ; a ))) X t : Our conjecture for the conditional covariance of x t in (33) implies that the second moments of wealth depend only on the constant and second order terms in X t : This conjecture requires that h i h a vec((x t )) h a 0 0 = B( h ; A( h ; a )) + B( f ; A( f ; a )); where h a vec((x t )) = CV t (w t+ ; a t+ ): For a we use the elements of x t [z t ; w t ; ^w t ] 0 : An analogous set of 8
20 restrictions applies to the dynamics of f household wealth. 2.4 Non-Predetermined Variable Dynamics To this point we have shown how the equilibrium conditions of the model impose restrictions on the dynamics of the state variables under the assumption that the vector of non-predetermined variables Y t (i.e., h t ; f t; ^ h t ; ^ f t; d t ; ^d t ; p t ; ^p t and r t ) satis es Y t = X t ; for some matrix with rows i : We now turn to the question of how the elements of are determined from the equilibrium conditions and the dynamics of the state vector. We begin with the restrictions on h equity prices. In particular, our aim is to derive a set of restrictions that will enable us to identify the elements of p where p t = p X t in equilibrium. Our derivation starts with expected returns. Speci cally, we note from the log approximated rst order conditions in (23a) with = fhg that E t r h t+ = r t + CV t (w t+ ; r h t+) 2 V t(r h t+); = r t + B( h ; A( h ; h )) + B( f ; A( f ; h )) 2 A( h; h ) X t ; = ( r + h er)x t : Combining this expression for expected returns with the assumed form for equilibrium dividends, the dynamics of the state vector, and (3a) gives X p t = i f( ) d E t X t++i ( r + h er) E t X t+i g ; i=0 = [( ) d A ( r + h er)] (I A) X t : Thus, given our assumption about dividends, the risk free rate, and the optimality of portfolio choices we nd that log equity prices satisfy p t = p X t where p = [( ) d A ( r + h er)] (I A) : (37) A similar exercise con rms that ^p t = ^p X t where ^p = ( ) ^da ( r + f er) (I A) : (38) The restrictions in (37) and (38) depend on the form of the dividend policies via the d and ^d vectors. These vectors are determined by the rms rst order conditions. In particular, using the fact that V t r k t+ = 2 V t (z t+ ) and CV t r k t+; w t+ = CVt (z t+ ; w t+ ) from (28); we can use R and R2 to write the log- 9
TECHNICAL WORKING PAPER SERIES SOLVING GENERAL EQUILIBRIUM MODELS WITH INCOMPLETE MARKETS AND MANY ASSETS. Martin D. D. Evans Viktoria Hnatkovska
TECHNICAL WORKING PAPER SERIES SOLVING GENERAL EQUILIBRIUM MODELS WITH INCOMPLETE MARKETS AND MANY ASSETS Martin D. D. Evans Viktoria Hnatkovska Technical Working Paper 318 http://www.nber.org/papers/t0318
Capital Structure and Investment Dynamics with Fire Sales
Capital Structure and Investment Dynamics with Fire Sales Douglas Gale Piero Gottardi NYU April 23, 2013 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 23, 2013 1 / 55 Introduction Corporate
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Comprehensive Examination: Macroeconomics Fall, 202 Answer Key to Section 2 Questions Section. (Suggested Time: 45 Minutes) For 3 of
In the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now
PERMANENT INCOME AND OPTIMAL CONSUMPTION On the previous notes we saw how permanent income hypothesis can solve the Consumption Puzzle. Now we use this hypothesis, together with assumption of rational
Chapter 1. GMM: Basic Concepts
Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating
Small Open Economy RBC Model Uribe, Chapter 4
Small Open Economy RBC Model Uribe, Chapter 4 1 Basic Model 1.1 Uzawa Utility E 0 t=0 θ t U (c t, h t ) θ 0 = 1 θ t+1 = β (c t, h t ) θ t ; β c < 0; β h > 0. Time-varying discount factor With a constant
ECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2
ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2 The due date for this assignment is Tuesday, October 2. ( Total points = 50). (Two-sector growth model) Consider the
1 Bewley Economies with Aggregate Uncertainty
1 Bewley Economies with Aggregate Uncertainty Sofarwehaveassumedawayaggregatefluctuations (i.e., business cycles) in our description of the incomplete-markets economies with uninsurable idiosyncratic risk
Session 4: Money. Jean Imbs. November 2010
Session 4: Jean November 2010 I So far, focused on real economy. Real quantities consumed, produced, invested. No money, no nominal in uences. I Now, introduce nominal dimension in the economy. First and
Lecture 3, November 30: The Basic New Keynesian Model (Galí, Chapter 3)
MakØk3, Fall 2 (blok 2) Business cycles and monetary stabilization policies Henrik Jensen Department of Economics University of Copenhagen Lecture 3, November 3: The Basic New Keynesian Model (Galí, Chapter
The Basic New Keynesian Model. Jordi Galí. June 2008
The Basic New Keynesian Model by Jordi Galí June 28 Motivation and Outline Evidence on Money, Output, and Prices: Short Run E ects of Monetary Policy Shocks (i) persistent e ects on real variables (ii)
Incomplete Markets, Heterogeneity and Macroeconomic Dynamics
Incomplete Markets, Heterogeneity and Macroeconomic Dynamics Bruce Preston and Mauro Roca Presented by Yuki Ikeda February 2009 Preston and Roca (presenter: Yuki Ikeda) 02/03 1 / 20 Introduction Stochastic
Speculation and the Bond Market: An Empirical No-arbitrage Framework
Online Appendix to the paper Speculation and the Bond Market: An Empirical No-arbitrage Framework October 5, 2015 Part I: Maturity specific shocks in affine and equilibrium models This Appendix present
The Basic New Keynesian Model. Jordi Galí. November 2010
The Basic New Keynesian Model by Jordi Galí November 2 Motivation and Outline Evidence on Money, Output, and Prices: Short Run E ects of Monetary Policy Shocks (i) persistent e ects on real variables (ii)
Problem 1 (30 points)
Problem (30 points) Prof. Robert King Consider an economy in which there is one period and there are many, identical households. Each household derives utility from consumption (c), leisure (l) and a public
Lecture 3: Dynamics of small open economies
Lecture 3: Dynamics of small open economies Open economy macroeconomics, Fall 2006 Ida Wolden Bache September 5, 2006 Dynamics of small open economies Required readings: OR chapter 2. 2.3 Supplementary
ECON0702: Mathematical Methods in Economics
ECON0702: Mathematical Methods in Economics Yulei Luo SEF of HKU January 14, 2009 Luo, Y. (SEF of HKU) MME January 14, 2009 1 / 44 Comparative Statics and The Concept of Derivative Comparative Statics
Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models
Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Fall 22 Contents Introduction 2. An illustrative example........................... 2.2 Discussion...................................
Competitive Equilibrium and the Welfare Theorems
Competitive Equilibrium and the Welfare Theorems Craig Burnside Duke University September 2010 Craig Burnside (Duke University) Competitive Equilibrium September 2010 1 / 32 Competitive Equilibrium and
Advanced Economic Growth: Lecture 8, Technology Di usion, Trade and Interdependencies: Di usion of Technology
Advanced Economic Growth: Lecture 8, Technology Di usion, Trade and Interdependencies: Di usion of Technology Daron Acemoglu MIT October 3, 2007 Daron Acemoglu (MIT) Advanced Growth Lecture 8 October 3,
Fluctuations. Shocks, Uncertainty, and the Consumption/Saving Choice
Fluctuations. Shocks, Uncertainty, and the Consumption/Saving Choice Olivier Blanchard April 2002 14.452. Spring 2002. Topic 2. 14.452. Spring, 2002 2 Want to start with a model with two ingredients: ²
1 The Basic RBC Model
IHS 2016, Macroeconomics III Michael Reiter Ch. 1: Notes on RBC Model 1 1 The Basic RBC Model 1.1 Description of Model Variables y z k L c I w r output level of technology (exogenous) capital at end of
Asset pricing in DSGE models comparison of different approximation methods
Asset pricing in DSGE models comparison of different approximation methods 1 Introduction Jan Acedański 1 Abstract. There are many numerical methods suitable for approximating solutions of DSGE models.
High-dimensional Problems in Finance and Economics. Thomas M. Mertens
High-dimensional Problems in Finance and Economics Thomas M. Mertens NYU Stern Risk Economics Lab April 17, 2012 1 / 78 Motivation Many problems in finance and economics are high dimensional. Dynamic Optimization:
Lecture 4 The Centralized Economy: Extensions
Lecture 4 The Centralized Economy: Extensions Leopold von Thadden University of Mainz and ECB (on leave) Advanced Macroeconomics, Winter Term 2013 1 / 36 I Motivation This Lecture considers some applications
FEDERAL RESERVE BANK of ATLANTA
FEDERAL RESERVE BANK of ATLANTA On the Solution of the Growth Model with Investment-Specific Technological Change Jesús Fernández-Villaverde and Juan Francisco Rubio-Ramírez Working Paper 2004-39 December
R = µ + Bf Arbitrage Pricing Model, APM
4.2 Arbitrage Pricing Model, APM Empirical evidence indicates that the CAPM beta does not completely explain the cross section of expected asset returns. This suggests that additional factors may be required.
4- Current Method of Explaining Business Cycles: DSGE Models. Basic Economic Models
4- Current Method of Explaining Business Cycles: DSGE Models Basic Economic Models In Economics, we use theoretical models to explain the economic processes in the real world. These models de ne a relation
ECON 5118 Macroeconomic Theory
ECON 5118 Macroeconomic Theory Winter 013 Test 1 February 1, 013 Answer ALL Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages
Dynamic stochastic general equilibrium models. December 4, 2007
Dynamic stochastic general equilibrium models December 4, 2007 Dynamic stochastic general equilibrium models Random shocks to generate trajectories that look like the observed national accounts. Rational
Appendix for "O shoring in a Ricardian World"
Appendix for "O shoring in a Ricardian World" This Appendix presents the proofs of Propositions - 6 and the derivations of the results in Section IV. Proof of Proposition We want to show that Tm L m T
Addendum to: International Trade, Technology, and the Skill Premium
Addendum to: International Trade, Technology, and the Skill remium Ariel Burstein UCLA and NBER Jonathan Vogel Columbia and NBER April 22 Abstract In this Addendum we set up a perfectly competitive version
Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path
Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path Ryoji Ohdoi Dept. of Industrial Engineering and Economics, Tokyo Tech This lecture note is mainly based on Ch. 8 of Acemoglu
The Ramsey Model. Alessandra Pelloni. October TEI Lecture. Alessandra Pelloni (TEI Lecture) Economic Growth October / 61
The Ramsey Model Alessandra Pelloni TEI Lecture October 2015 Alessandra Pelloni (TEI Lecture) Economic Growth October 2015 1 / 61 Introduction Introduction Introduction Ramsey-Cass-Koopmans model: di ers
Monetary Economics: Solutions Problem Set 1
Monetary Economics: Solutions Problem Set 1 December 14, 2006 Exercise 1 A Households Households maximise their intertemporal utility function by optimally choosing consumption, savings, and the mix of
Economics 701 Advanced Macroeconomics I Project 1 Professor Sanjay Chugh Fall 2011
Department of Economics University of Maryland Economics 701 Advanced Macroeconomics I Project 1 Professor Sanjay Chugh Fall 2011 Objective As a stepping stone to learning how to work with and computationally
Topic 8: Optimal Investment
Topic 8: Optimal Investment Yulei Luo SEF of HKU November 22, 2013 Luo, Y. SEF of HKU) Macro Theory November 22, 2013 1 / 22 Demand for Investment The importance of investment. First, the combination of
Macroeconomics II Money in the Utility function
McCless]Prof. McCless UCEMA Prof. McCless UCEMA Macroeconomics II Money in the Utility function [ October, 00 Money in the Utility function Money in the Utility function Alternative way to add money to
Getting to page 31 in Galí (2008)
Getting to page 31 in Galí 2008) H J Department of Economics University of Copenhagen December 4 2012 Abstract This note shows in detail how to compute the solutions for output inflation and the nominal
GMM-based inference in the AR(1) panel data model for parameter values where local identi cation fails
GMM-based inference in the AR() panel data model for parameter values where local identi cation fails Edith Madsen entre for Applied Microeconometrics (AM) Department of Economics, University of openhagen,
Approximation around the risky steady state
Approximation around the risky steady state Centre for International Macroeconomic Studies Conference University of Surrey Michel Juillard, Bank of France September 14, 2012 The views expressed herein
Solving and Simulating Models with Heterogeneous Agents and Aggregate Uncertainty Yann ALGAN, Olivier ALLAIS, Wouter J. DEN HAAN, and Pontus RENDAHL
Solving and Simulating Models with Heterogeneous Agents and Aggregate Uncertainty Yann ALGAN, Olivier ALLAIS, Wouter J. DEN HAAN, and Pontus RENDAHL August 24, 2010 Abstract Although almost nonexistent
Dynare Class on Heathcote-Perri JME 2002
Dynare Class on Heathcote-Perri JME 2002 Tim Uy University of Cambridge March 10, 2015 Introduction Solving DSGE models used to be very time consuming due to log-linearization required Dynare is a collection
problem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves
Ross (1976) introduced the Arbitrage Pricing Theory (APT) as an alternative to the CAPM.
4.2 Arbitrage Pricing Model, APM Empirical evidence indicates that the CAPM beta does not completely explain the cross section of expected asset returns. This suggests that additional factors may be required.
Chapter 4. Applications/Variations
Chapter 4 Applications/Variations 149 4.1 Consumption Smoothing 4.1.1 The Intertemporal Budget Economic Growth: Lecture Notes For any given sequence of interest rates {R t } t=0, pick an arbitrary q 0
Advanced Macroeconomics II. Real Business Cycle Models. Jordi Galí. Universitat Pompeu Fabra Spring 2018
Advanced Macroeconomics II Real Business Cycle Models Jordi Galí Universitat Pompeu Fabra Spring 2018 Assumptions Optimization by consumers and rms Perfect competition General equilibrium Absence of a
The Real Business Cycle Model
The Real Business Cycle Model Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations of aggregate economic variables around their trend.
Lecture 2 The Centralized Economy
Lecture 2 The Centralized Economy Economics 5118 Macroeconomic Theory Kam Yu Winter 2013 Outline 1 Introduction 2 The Basic DGE Closed Economy 3 Golden Rule Solution 4 Optimal Solution The Euler Equation
1 Two elementary results on aggregation of technologies and preferences
1 Two elementary results on aggregation of technologies and preferences In what follows we ll discuss aggregation. What do we mean with this term? We say that an economy admits aggregation if the behavior
Another Look at the Boom and Bust of Financial Bubbles
ANNALS OF ECONOMICS AND FINANCE 16-2, 417 423 (2015) Another Look at the Boom and Bust of Financial Bubbles Andrea Beccarini University of Münster, Department of Economics, Am Stadtgraben 9, 48143, Münster,
RBC Model with Indivisible Labor. Advanced Macroeconomic Theory
RBC Model with Indivisible Labor Advanced Macroeconomic Theory 1 Last Class What are business cycles? Using HP- lter to decompose data into trend and cyclical components Business cycle facts Standard RBC
Markov Perfect Equilibria in the Ramsey Model
Markov Perfect Equilibria in the Ramsey Model Paul Pichler and Gerhard Sorger This Version: February 2006 Abstract We study the Ramsey (1928) model under the assumption that households act strategically.
ECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko
ECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko Indirect Utility Recall: static consumer theory; J goods, p j is the price of good j (j = 1; : : : ; J), c j is consumption
Macroeconomics IV Problem Set I
14.454 - Macroeconomics IV Problem Set I 04/02/2011 Due: Monday 4/11/2011 1 Question 1 - Kocherlakota (2000) Take an economy with a representative, in nitely-lived consumer. The consumer owns a technology
Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6
1 Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6 1 A Two-Period Example Suppose the economy lasts only two periods, t =0, 1. The uncertainty arises in the income (wage) of period 1. Not that
What Drives US Foreign Borrowing? Evidence on External Adjustment to Transitory and Permanent Shocks
What Drives US Foreign Borrowing? Evidence on External Adjustment to Transitory and Permanent Shocks Giancarlo Corsetti Panagiotis Th. Konstantinou y European University nstitute, University of Rome, CEPR
Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley
Time Series Models and Inference James L. Powell Department of Economics University of California, Berkeley Overview In contrast to the classical linear regression model, in which the components of the
Testing for Regime Switching: A Comment
Testing for Regime Switching: A Comment Andrew V. Carter Department of Statistics University of California, Santa Barbara Douglas G. Steigerwald Department of Economics University of California Santa Barbara
Monetary Economics Notes
Monetary Economics Notes Nicola Viegi 2 University of Pretoria - School of Economics Contents New Keynesian Models. Readings...............................2 Basic New Keynesian Model...................
Stochastic simulations with DYNARE. A practical guide.
Stochastic simulations with DYNARE. A practical guide. Fabrice Collard (GREMAQ, University of Toulouse) Adapted for Dynare 4.1 by Michel Juillard and Sébastien Villemot (CEPREMAP) First draft: February
1. Using the model and notations covered in class, the expected returns are:
Econ 510a second half Yale University Fall 2006 Prof. Tony Smith HOMEWORK #5 This homework assignment is due at 5PM on Friday, December 8 in Marnix Amand s mailbox. Solution 1. a In the Mehra-Prescott
Technical Appendix for Financial Business Cycles
Technical Appendix for Financial usiness Cycles Matteo Iacoviello Federal Reserve oard December 4, 05 This appendix contains some of the derivations for the paper Financial usiness Cycles.. The Extended
Government The government faces an exogenous sequence {g t } t=0
Part 6 1. Borrowing Constraints II 1.1. Borrowing Constraints and the Ricardian Equivalence Equivalence between current taxes and current deficits? Basic paper on the Ricardian Equivalence: Barro, JPE,
Housing and the Business Cycle
Housing and the Business Cycle Morris Davis and Jonathan Heathcote Winter 2009 Huw Lloyd-Ellis () ECON917 Winter 2009 1 / 21 Motivation Need to distinguish between housing and non housing investment,!
Online Appendix for Precautionary Saving of Chinese and US Households
Online Appendix for Precautionary Saving of Chinese and US Households Horag Choi a Steven Lugauer b Nelson C. Mark c May 06 Abstract This online appendix presents the analytical derivations and estimation
"A Theory of Financing Constraints and Firm Dynamics"
1/21 "A Theory of Financing Constraints and Firm Dynamics" G.L. Clementi and H.A. Hopenhayn (QJE, 2006) Cesar E. Tamayo Econ612- Economics - Rutgers April 30, 2012 2/21 Program I Summary I Physical environment
Daily Welfare Gains from Trade
Daily Welfare Gains from Trade Hasan Toprak Hakan Yilmazkuday y INCOMPLETE Abstract Using daily price quantity data on imported locally produced agricultural products, this paper estimates the elasticity
Topic 2. Consumption/Saving and Productivity shocks
14.452. Topic 2. Consumption/Saving and Productivity shocks Olivier Blanchard April 2006 Nr. 1 1. What starting point? Want to start with a model with at least two ingredients: Shocks, so uncertainty.
Lecture 2. (1) Aggregation (2) Permanent Income Hypothesis. Erick Sager. September 14, 2015
Lecture 2 (1) Aggregation (2) Permanent Income Hypothesis Erick Sager September 14, 2015 Econ 605: Adv. Topics in Macroeconomics Johns Hopkins University, Fall 2015 Erick Sager Lecture 2 (9/14/15) 1 /
ECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal
Macroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models
Macroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models Prof. George McCandless UCEMA Spring 2008 1 Class 1: introduction and rst models What we will do today 1. Organization of course
ECOM 009 Macroeconomics B. Lecture 2
ECOM 009 Macroeconomics B Lecture 2 Giulio Fella c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 2 40/197 Aim of consumption theory Consumption theory aims at explaining consumption/saving decisions
Lecture 2 The Centralized Economy: Basic features
Lecture 2 The Centralized Economy: Basic features Leopold von Thadden University of Mainz and ECB (on leave) Advanced Macroeconomics, Winter Term 2013 1 / 41 I Motivation This Lecture introduces the basic
New Notes on the Solow Growth Model
New Notes on the Solow Growth Model Roberto Chang September 2009 1 The Model The firstingredientofadynamicmodelisthedescriptionofthetimehorizon. In the original Solow model, time is continuous and the
A Method for Solving DSGE Models with Dispersed Private Information 1
A Method for Solving DSGE Models with Dispersed Private Information Cedric Tille Geneva Graduate Institute HEID and CEPR Eric van Wincoop University of Virginia and NBER March 7, 20 Cedric Tille gratefully
Volume 29, Issue 4. Stability under learning: the neo-classical growth problem
Volume 29, Issue 4 Stability under learning: the neo-classical growth problem Orlando Gomes ISCAL - IPL; Economics Research Center [UNIDE/ISCTE - ERC] Abstract A local stability condition for the standard
Discrete State Space Methods for Dynamic Economies
Discrete State Space Methods for Dynamic Economies A Brief Introduction Craig Burnside Duke University September 2006 Craig Burnside (Duke University) Discrete State Space Methods September 2006 1 / 42
Macroeconomics Theory II
Macroeconomics Theory II Francesco Franco FEUNL February 2016 Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 1 / 18 Road Map Research question: we want to understand businesses cycles.
Slides II - Dynamic Programming
Slides II - Dynamic Programming Julio Garín University of Georgia Macroeconomic Theory II (Ph.D.) Spring 2017 Macroeconomic Theory II Slides II - Dynamic Programming Spring 2017 1 / 32 Outline 1. Lagrangian
ECON0702: Mathematical Methods in Economics
ECON0702: Mathematical Methods in Economics Yulei Luo SEF of HKU January 12, 2009 Luo, Y. (SEF of HKU) MME January 12, 2009 1 / 35 Course Outline Economics: The study of the choices people (consumers,
Chapter 2. GMM: Estimating Rational Expectations Models
Chapter 2. GMM: Estimating Rational Expectations Models Contents 1 Introduction 1 2 Step 1: Solve the model and obtain Euler equations 2 3 Step 2: Formulate moment restrictions 3 4 Step 3: Estimation and
slides chapter 3 an open economy with capital
slides chapter 3 an open economy with capital Princeton University Press, 2017 Motivation In this chaper we introduce production and physical capital accumulation. Doing so will allow us to address two
Economics 241B Estimation with Instruments
Economics 241B Estimation with Instruments Measurement Error Measurement error is de ned as the error resulting from the measurement of a variable. At some level, every variable is measured with error.
Lecture 9: The monetary theory of the exchange rate
Lecture 9: The monetary theory of the exchange rate Open Economy Macroeconomics, Fall 2006 Ida Wolden Bache October 24, 2006 Macroeconomic models of exchange rate determination Useful reference: Chapter
Sophisticated Monetary Policies
Federal Reserve Bank of Minneapolis Research Department Sta Report 419 January 2008 Sophisticated Monetary Policies Andrew Atkeson University of California, Los Angeles, Federal Reserve Bank of Minneapolis,
Lecture 6: Recursive Preferences
Lecture 6: Recursive Preferences Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013 Basics Epstein and Zin (1989 JPE, 1991 Ecta) following work by Kreps and Porteus introduced a class of preferences
Global Value Chain Participation and Current Account Imbalances
Global Value Chain Participation and Current Account Imbalances Johannes Brumm University of Zurich Georgios Georgiadis European Central Bank Johannes Gräb European Central Bank Fabian Trottner Princeton
Comprehensive Exam. Macro Spring 2014 Retake. August 22, 2014
Comprehensive Exam Macro Spring 2014 Retake August 22, 2014 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question.
Solution Methods. Jesús Fernández-Villaverde. University of Pennsylvania. March 16, 2016
Solution Methods Jesús Fernández-Villaverde University of Pennsylvania March 16, 2016 Jesús Fernández-Villaverde (PENN) Solution Methods March 16, 2016 1 / 36 Functional equations A large class of problems
Lecture 6: Discrete-Time Dynamic Optimization
Lecture 6: Discrete-Time Dynamic Optimization Yulei Luo Economics, HKU November 13, 2017 Luo, Y. (Economics, HKU) ECON0703: ME November 13, 2017 1 / 43 The Nature of Optimal Control In static optimization,
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:
International Macro Finance
International Macro Finance Economies as Dynamic Systems Francesco Franco Nova SBE February 21, 2013 Francesco Franco International Macro Finance 1/39 Flashback Mundell-Fleming MF on the whiteboard Francesco
Uncertainty aversion and heterogeneous beliefs in linear models
Uncertainty aversion and heterogeneous beliefs in linear models Cosmin Ilut Duke & NBER Pavel Krivenko Stanford March 2016 Martin Schneider Stanford & NBER Abstract This paper proposes a simple perturbation
1. Money in the utility function (start)
Monetary Economics: Macro Aspects, 1/3 2012 Henrik Jensen Department of Economics University of Copenhagen 1. Money in the utility function (start) a. The basic money-in-the-utility function model b. Optimal
Part A: Answer question A1 (required), plus either question A2 or A3.
Ph.D. Core Exam -- Macroeconomics 5 January 2015 -- 8:00 am to 3:00 pm Part A: Answer question A1 (required), plus either question A2 or A3. A1 (required): Ending Quantitative Easing Now that the U.S.
Econ 5110 Solutions to the Practice Questions for the Midterm Exam
Econ 50 Solutions to the Practice Questions for the Midterm Exam Spring 202 Real Business Cycle Theory. Consider a simple neoclassical growth model (notation similar to class) where all agents are identical
Solving Nonlinear Rational Expectations Models by Approximating the Stochastic Equilibrium System
Solving Nonlinear Rational Expectations Models by Approximating the Stochastic Equilibrium System Michael P. Evers September 20, 2012 Revised version coming soon! Abstract Dynamic stochastic rational expectations
Finite State Markov-chain Approximations to Highly. Persistent Processes
Finite State Markov-chain Approximations to Highly Persistent Processes Karen A. Kopecky y Richard M. H. Suen z This Version: November 2009 Abstract The Rouwenhorst method of approximating stationary AR(1)
Country Portfolios in Open Economy Macro Models 1
Country Portfolios in Open Economy Macro Models 1 Michael B Devereux 2 and Alan Sutherland 3 April 2009 1 We are grateful to Philip Lane, Klaus Adam, Pierpaolo Benigno, Gianluca Benigno, Berthold Herrendorf,