Keywords: Linearization, linear-quadratic method, second-order approximation, welfare.
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,
Two endowment levels, denoted as and , are independent and lognormally distributed,
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:
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:
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:
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:
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
This example shows how to apply the revised LQ approach to a simple deterministic growth model:
To convert the linear terms of the utility function into a purely quadratic form, we need the following relationship among the steady-state values,
Using this relationship, we transform the linear terms in the criterion function as follows:
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
Rewriting this constraint forward, we have
The optimality conditions corresponding to the transformed criterion function are
Suppose that the criterion function of the social planner is
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.