Finance and Economics Discussion Series: 2007-64 Screen Reader version ^{♣}

Keywords: Linearization, linear-quadratic method, second-order approximation, welfare.

Abstract:

This paper illustrates two types of pitfalls in using linearization methods. First, if constraints are linearized before deriving optimality conditions, the derived conditions are not correct up to first order. Second, even when the behavior of the economy is correct to the first order, applying
this behavior to welfare implications may lead to incorrrect results.

JEL Classification: C6.

The economics profession has long been using linear models due to their simplicity, and linear modeling has been applied to forward-looking models as well as backward-looking ones. A related phenomenon is a wide usage of a linear quadratic (LQ) framework in analyzing optimal policy problems, since the combination of a quadratic criterion function with a linear constraint is well known to yield a linear behavior for the optimal solution. However, this attractiveness of linear models has sometimes misled the profession to adopt certain kinds of linearization methods that can deliver spurious results.

In this paper, we present two types of pitfalls that arise when linearization methods are improperly applied in the dynamic macroeconomics literature. Our paper is not the first to point out such pitfalls; rather, this paper is intended to illustrate these two pitfalls in a succinct way and to
distinguish between the two types of pitfalls.^{1} In solving an optimal policy problem whose criterion function and constraints are nonlinear, some researchers
have linearized the constraint--as well as quadratically approximating the criterion function--*before* deriving the optimality conditions.^{2} The resulting optimality conditions from this procedure are not a correct linear representation of the original nonlinear optimality conditions. A simple two-agent model will show that this pitfall leads to an implication that risk sharing can reduce the welfare level of the economy.

While the first pitfall originates from the improperly derived optimality conditions, the second pitfall happens even after the optimality conditions are properly derived. In this case, since the nonlinear optimality conditions are a correct representation of the original model, a linear
approximation of derived optimality conditions would yield a correct linear representation of the original model. However, when applied to calculate welfare levels, this linear approximation can generate incorrect welfare implications. Using the same two agent model as in the first pitfall, we
demonstrate that linearization can yield a spurious result that the welfare level of the complete markets economy can be *lower* than that of the autarky. This pitfall can occur in welfare evaluation of any equilibrium models, regardless of whether the optimality
conditions are derived from optimization problems or given by ad hoc assumptions.

The remainder of this paper proceeds as follows. Section 2 presents a model economy, which is used as an example for the discussion of two types of pitfalls. Section 3 illustrates the first pitfall, and Section 4, the second pitfall. Some additional examples are included in the Appendix.

For both types of pitfalls, we use a simple two-agent endowment economy model with complete asset markets.^{3} Assuming symmetry, the competitive equilibrium of
this economy is equivalent to a social planner problem maximizing the average of two agents' utilities,

subject to the aggregate resource constraint

We assume that utility is a power function of consumption where the degree of risk aversion is ,

Note that the second order approximation of this power utility function with respect to around is

where the constant term, , is suppressed for brevity since we are not interested in the absolute level of welfare.

Two endowment levels, denoted as and , are independent and lognormally distributed,

for

The optimality condition of this maximization problem is
and the solution for consumption in this complete-markets economy is
where the superscript `exa' represents an exact solution for the complete markets economy. The (log-)linear approximation of this solution is

where the superscript `lin' represents the linear approximation of the exact solution and the bar variables represent steady state values.

Throughout the paper, we use the second order approximation of the utility function (4)--instead of the original utility function (3)--for welfare evaluation due to its simplicity and convenience for welfare comparison purposes. Substituting (4) into the social planner's objective function (1), we have the following quadratic welfare criterion for the social planner:

We first show how the first pitfall arises and then discuss ways to avoid this pitfall.^{5} Suppose we start by approximating the constraint (2) to the first order in logs before deriving optimality conditions:

where . The first pitfall can occur when we maximize the quadratic criterion function (7) subject to this linearized budget constraint (8).

Due to the symmetry, the necessary condition for optimality is
. In the case of the objective function is concave with
respect to log consumption so this condition is also sufficient. Therefore, the solution of this maximization problem would be the same as (6), a correct loglinear approximation of the exact solution. However, when the sufficient condition for concavity is not satisfied, and
does not correspond to an optimum.^{6} In
fact, optimality comes under a corner solution. That is, risk sharing reduces the level of welfare.^{7} The solution from this LQ problem--maximizing the
quadratic objective function subject to a linearized budget constraint--can be different from a correct linear representation of the exact solution. This improper LQ setup is referred to as a "naive" LQ problem by Benigno and Woodford (2006).

An evident way to avoid the first pitfall is to derive the optimality conditions based on the original nonlinear constraints and then to linearize the optimality conditions as well as the constraints. This method is widely used in dynamic macroeconomics. Recently, Benigno and Woodford (2004, 2005) provided another way to avoid this pitfall while keeping the linear quadratic framework in solving a general class of nonlinear optimization problems.

Their critical step is to approximate the constraint (2) quadratically rather than linearly:

This quadratic constraint (9) can now be used to substitute out the linear terms in the objective function (7), which is then transformed into the following quadratic form (without any linear terms for endogenous variables):

Now we maximize this transformed quadratic objective function (10) subject to the linear constraint (8). The solution of this problem is (6) implying full risk sharing, and it is the correct loglinear approximation of
the exact solution: avoiding the first pitfall. This "revised" LQ approach can be applied to dynamic models, including models with backward-looking constraints such as stochastic growth models and ones with forward-looking constraints such as optimal monetary policy.^{8}

From now on, we assume that one is equipped with a correct linear representation of the optimality conditions. The second pitfall takes place when this set of linear equations (or its solution that expresses endogenous variables in terms of exogenous and predetermined variables) is used for welfare evaluations. Our illustration of how the second pitfall arises is based on the same model as was used for the case of the first pitfall.

Linear representation of the correctly derived optimality conditions produces the loglinear solution (6). By plugging this equation into the quadratic welfare criterion function (7), we can calculate the welfare level of the complete markets economy based on the loglinear approximation of the optimality conditions:

In contrast, the level of welfare under autarky is

Comparison of these two welfare levels reveals that, when , the welfare level of autarky is higher than that of the complete markets economy. This result is clearly spurious, given that it violates the first welfare theorem. In general, this second pitfall yields incorrect welfare implications.

The revised LQ approach in the previous section, proposed by Benigno and Woodford, avoids the second pitfall as well. Suppose that we plug the loglinear solution (6) into the purely quadratic objective function (10). Then, this criterion yields the following level of welfare under the complete markets economy:

This value is higher than the level of welfare under autarky evaluated with the purely quadratic objective function (10),

and we have avoided the second pitfall. This approach can also be applied to evaluate the welfare implications of any sub-optimal economies.

Another way to avoid the second pitfall, employed in Kim and Kim (2003), is the second order approximation of the economy--including the optimality conditions as well as the constraints.^{9} That is, the solution need be of second order with respect to the exogenous and predetermined variables--as well as the standard deviation of shocks in a stochastic case. In the complete markets economy, the solution based on the
second order approximation becomes

where the superscript `quad' denotes a quadratic approximation.

The welfare level is always greater than that under autarky as in (12) for any positive values for the risk aversion parameter (): avoiding the second pitfall.

"Optimal Taxation in an RBC Model: A Linear-Quadratic Approach," Journal of Economic Dynamics and Control, 30, 1445-1489.

"Inflation Stabilization and Welfare: The Case of a Distorted Steady State," Journal of European Economic Association, 3:6, 1185-1236.

"Linear-Quadratic Approximation of Optimal Policy Problems," Manuscript, New York University.

Numerical Methods in Economics, MIT Press, Cambridge, MA.

" Inflation Targeting and Nominal-income-growth Targeting: When and Why are They Suboptimal?" Journal of Monetary Economics, 52, 1463-1495.

"Spurious Welfare Reversals in International Business Cycle Models," Journal of International Economics, 60, 471-500.
# Appendices

# A. Growth Model

where is the Lagrangian multiplier.

Multiplying the former equation by , we have and the latter by where By adding these two equations, we have the following relationship: where

The parenthesized term in the last expression (i.e. ) involves the first-order terms of the second-order approximation of the capital accumulation equation. Therefore, except for that is predetermined and independent of policy, the last expression can be expressed as purely quadratic terms.
# B. Optimal Monetary Policy

where is the Lagrangian multiplier. Under the naive LQ approach, the system would not contain 's and the resulting dynamics do not match the first order behavior of the exact solution.

That is, The criterion function can be written as follows:

The first term of the last expression reflects the inflation gain in the first period, which would disappear under the timeless perspective.

where is the Lagrangian multiplier. Note that the steady state of this multiplier is zero in this transformed system. Even though two 's appear in the first optimality condition, this parameter does not influence the linear solution of this system. However, the that showed up in the transformed criterion function would still affect the level of welfare.
# C. Third-Order Properties of Utility and Constraint

That is, four third-order parameters do in fact affect the quadratic solution without affecting the level of welfare based on its quadratic approximation--a criterion of the revised LQ approach as well the quadratic perturbation approach.

#### Footnotes

^{♣} This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to Text

This example shows how to apply the revised LQ approach to a simple deterministic growth model:

subject to This model can be generalized to a stochastic one without loss of generality. For the notational convenience, the approximations will be implemented in terms of the logarithms rather than the levels. That is, the optimality conditions arewhere is the Lagrangian multiplier.

To convert the linear terms of the utility function into a purely quadratic form, we need the following relationship among the steady-state values,

Multiplying the former equation by , we have and the latter by where By adding these two equations, we have the following relationship: where

Using this relationship, we transform the linear terms in the criterion function as follows:

The parenthesized term in the last expression (i.e. ) involves the first-order terms of the second-order approximation of the capital accumulation equation. Therefore, except for that is predetermined and independent of policy, the last expression can be expressed as purely quadratic terms.

This example illustrates an application of the revised LQ approach to an optimal policy problem with a nonlinear forward looking constraint.^{11} The policy
problem is to

where is the Lagrangian multiplier. Under the naive LQ approach, the system would not contain 's and the resulting dynamics do not match the first order behavior of the exact solution.

Rewriting this constraint forward, we have

That is, The criterion function can be written as follows:

The first term of the last expression reflects the inflation gain in the first period, which would disappear under the timeless perspective.

The optimality conditions corresponding to the transformed criterion function are

where is the Lagrangian multiplier. Note that the steady state of this multiplier is zero in this transformed system. Even though two 's appear in the first optimality condition, this parameter does not influence the linear solution of this system. However, the that showed up in the transformed criterion function would still affect the level of welfare.

Suppose that the criterion function of the social planner is

and that the constraint is In this case, the quadratic solution isThat is, four third-order parameters do in fact affect the quadratic solution without affecting the level of welfare based on its quadratic approximation--a criterion of the revised LQ approach as well the quadratic perturbation approach.

Intuition for this property comes from the reason why the revised LQ approach works in the first place. The reason why this approach works is that, though (10) is not any kind of approximation of (7) in general, the two expressions are equivalent up to the second order for all outcomes that are consistent with the constraint. In the same vein, modification of the criterion function and the constraint would not affect (10)--hence no influence on the level of welfare up to the second order.

* A version without appendices was published in the Journal of Money, Credit and Banking, Vol. 39, No. 4. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of
the Federal Reserve System or any other person associated with the Federal Reserve System. We appreciate discussion with Pierpaolo Benigno, Andrew Levin and Mike Woodford, and editorial comments by Jason Steinman. Return to Text

* Corresponding Author, Division of Monetary Affairs, Mailstop 71, Federal Reserve Board, Washington, DC 20551. Tel: (202) 452-2981, E-mail: [email protected]. Return to Text

* Department of Economics, Tufts University, Medford, MA 02155. Tel: (617) 627-3662, E-mail: [email protected]. Return to Text

1. See Judd (1998) and Benigno and Woodford (2006) for discussion on the first pitfall. Kim and Kim (2003) documented the second pitfall. Return to Text

2. In this paper, we use the term `optimality conditions' rather than `first order conditions', since the latter could be confusing due to the usage of `first order' Taylor approximations throughout the paper. Return to
Text

3. This model is a static example. See Appendices A and B for dynamic examples. Return to Text

4. For the same reason, we do not consider mean preserving spreads for the endowment distribution. Return to Text

5. Part of this section draws on Michael Woodford's plenary lecture at the 2004 Society of Computational Economics Conference in Amsterdam, to be included in Benigno and Woodford (2004). Return to Text

6. In general, the first pitfall leads to an optimality condition different from a linear representation of the correct optimality condition. For example, Benigno and Woodford (2006) used the stochastic growth model as an example and illustrated how the first
pitfall alters the first-order behavior of the optimal taxation policies. Return to Text

7. In the borderline case of log utility
, the social planner is indifferent to whether risks are shared between the two agents or not. Return to Text

8. See Kim and Henderson (2005) for an application to a simple forwarding-looking case. Return to Text

9. A tradeoff exists between these two ways of avoiding the second pitfall. The latter method is computationally more burdensome. However, it allows us to handle cases when we are interested in first moments other than welfare (e.g. average level of
consumption) or do not accept the welfare criterion according to the timeless perspective. Return to Text

10. In the previous section on the first pitfall, the modification of the criterion function into a purely quadratic form involves the second-order properties of the constraint. Another element that affects the quadratic solution is the third-order properties
of the constraint as well as the criterion function. It is interesting to note that this element does not affect the level of welfare. See Appendix C for the details. Return to Text

11. This problem modified that of Clarida, Gali and Gertler's "The Science of Monetary Policy" minimally to illustrate the revised LQ approach. Return to Text