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#### Nonparametric Comparative Statics and Stability

**By Douglas Hale, George Lady, John Maybee, James Quirk**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 1999 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-00690-1

All rights reserved.

ISBN: 978-0-691-00690-1

CHAPTER 1

Nonparametric Analysis

**1.1 INTRODUCTION**

In the early 1950s, the federal government decided to stockpile strategic materials to protect against the possibility of hostilities with the USSR. Planners constructed a massive 400 × 400 input-output model of the U.S. economy, entailing the estimation of a very large number of input-output coefficients, mainly based on data from the 1947 Census of Manufactures, to assess the effects of alternative inventory holdings. After several years of data gathering and analysis, they tested the resulting model, using the final bill of goods for the economy for 1951. Among other findings from that test was that the calculated domestic steel requirement for producing the 1951 final bill of goods was 40% more than the capacity of the U.S. steel industry in 1951, a physical impossibility. What had gone wrong was that relative prices of different grades of steel (alloy, stainless, carbon) had changed between 1947 and 1951, leading to a change in the output mix of the steel industry. Price changes in the steel industry had also led to changes in steel-using technology, and substitution of other metals for steel, by the customers of the steel industry. In the space of four years, the adaptiveness of the American economy had clearly revealed the volatility of the input-output coefficients underlying the Defense Department's economic planning models. Problems with the volatility of input-output and other such coefficients continue to be commonplace in economic research.

Since the quantitative particulars of the interrelationships that constitute economic phenomena are so often volatile and transitory, it is natural to inquire about aspects of the interrelationships that might be more stable and robust. This book explores what can be said about the collective outcome of interdependent quantitative phenomena when the precise nature and magnitudes of their separate influences are not known. Although the particular case of economic phenomena motivated the original work in this area, the problem of inference with limited quantitative information is endemic to scientific inquiry. Scientific explanations of observable phenomena are based on structural relations:

F (*y; α, e*) = (*f*1, ..., *f*n),

where the *fl* are functions linking the phenomena to be explained (the endogenous variables), *y* = (*y*1 ... *yn*), to conditioning numbers (parameters and exogenous variables, collectively, the data), α = (α1, ..., α*m*), determined outside the theory, and unobserved random disturbances, *e.* The random disturbance, *e*, is suppressed in most of this book. Scientific predictions are derived under the assumption that observed values of the phenomena, *y,* are equilibrium values, *y** = *y** (α, *e*), defined by

F (y*; α, e) = (f1, ... fn) = On (1.1)

Much of the daily work of scientists involves making observations and conducting experiments to estimate the form of the equation systems such as (1.1), the values of conditioning numbers, and the distribution of the unobserved disturbances. The result of a successful research program is a complete and internally consistent explanation of the phenomena. Comparison of its predictions to data can then test the validity of the theory.

**1.2 QUANTITATIVE ANALYSIS**

Quantitative analysis in economics has traditionally focused on comparative statics: the problem of computing changes in the equilibrium values of endogenous variables induced by changes in the data. Analysis of the local direction of change in economic magnitudes in response to changes in technology, resource endowments, people's preferences, and public policy naturally results in locally linear systems of equations under appropriate differentiability assumptions (see Samuelson 1947). In this book, most of the systems we analyze are local comparative statics models. In recent years, the comparative statics problem has been reformulated by Milgrom and Shannon (1994), Milgrom (1994), and Milgrom and Roberts (1990, 1994). This approach is directed at establishing conditions necessary and sufficient for global qualitative comparative statics results. Because of its generality, the approach is less useful as a device for identifying the specific comparative statics results that economic models can generate. The precise links between this approach and that taken in this book have not been completely established. However, in at least one case, that involving the maximization hypothesis, the correspondence between the approach taken here and the Milgrom-Shannon monotonicity theorem is easily established. See the discussion in Chapter 5.

**Comparative Statics**

The analysis is initiated by noting that in the neighborhood of an equilibrium, *y**, the changes induced in *y** by changes in α can be written in differential form as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

where *i, j* = 1, ..., *n* and *k* = 1, ..., *m*. If only one exogenous variable changes, the most common case in economic modeling, the differential system becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)

Determining the change in the equilibrium values of phenomena with respect to a change in an exogenous variable, *dy***j/d*α, is the subject matter of comparative statics ("statics" because time does not explicitly enter (1.1)-(1.3). Define the square *n* x *n* matrix, *A = [alj]* = [[partial derivative]*fl*/[partial derivative]*yj*], the *n* x 1 vector, *x* = (*dy**l/*d*α), and the *n* x 1 vector, b = [bj] = [-[partial derivative]*fl*/[partial derivative]α]. The local comparative statics problem then can be written as

*Ax = b,* (1.4)

where *x* is to be determined. The matrix *A* is called the *Jacobian matrix,* corresponding to a solution to the system (1.3).

A theory is locally scientific in the sense of Popper ([1934] 1959) if for a given, potentially observable *b*-vector, a particular *x*-vector could never arise as a solution to (1.4). The theory would be "refuted" if the particular ^-vector were in fact observed. From the standpoint of refutable hypotheses, the content of a theory is represented by the characteristics of its Jacobian matrix.

**Dynamics**

The equilibrium *y** defined by (1.1) is often interpreted as the stationary state associated with a dynamic adjustment process operating on the phenomena, *y,* over time, *t*. Formally, the adjustment process is

[??] = dy/dt = g(y; α, e) = (g1, ... gnn), (1.5)

where *dy/dt* is the time derivative of *y.* When the rate of change of each *yt* is increasing with its "distance" from equilibrium as measured by *fl*, the adjustment mechanism can be written as

[??] [equivalent to] dy/dt = g(f) = (g, (f)) = (gl(fl)). (1.6)

A linear approximation of (1.6) in the neighborhood of *y** is obtained by a first-order Taylor series, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], evaluated at the equilibrium *y*.* Writing this expression in matrix form yields

[??] [equivalent to] dy/dt = DA{y -y*), (1.7)

where *D* is a diagonal matrix with *dll = dgl/dfl* > 0 and *A* is as defined in (1.4). Global stability analysis is concerned with determining conditions that ensure that (1.5) or (1.6) devolves to zero in the limit, for arbitrary initial conditions. Linear approximation stability analysis seeks conditions on *D* and *A* ensuring that (1.7) devolves to zero in the limit, in a neighborhood of *y**.

**1.3 NONPARAMETRIC ANALYSIS**

The quantitative approach to scientific explanation breaks down when the theory's underlying equations, conditioning numbers, and unobserved disturbances are only vaguely known. In the natural sciences, where the relationships are presumed immutable in time and space, nascent theories may reveal the import of novel data sets that may not be available for decades. In the social sciences, both the underlying relationships and the magnitudes of the conditioning numbers may change with place and time. People change, institutions change, and the technology changes: careful observation and estimation may still yield only provisional approximations of a transient reality. The demands of quantitative analysis are often simply not feasible. The validity of plausible inferences from theories, mental models, and computer programs can all be rejected on the basis of quantitative information, but the problem of inference with limited quantitative information remains. Social scientists in particular have little hope of ever achieving precise knowledge of people and their organizations.

Quantitative results in economic applications consequently have limited predictive power. Little is known about the actual form of the underlying relationships, and controlled field experiments to resolve magnitudes are seldom feasible. Even in the linearized structure of (1.4) or (1.7), precise quantitative information is typically absent. Formally, the magnitudes of the entries of *A* and *b* in (1.4) and *D* in (1.7) are not completely known. Given this, the immediate issue becomes, What *can* be safely assumed to be known? For the purposes of this book, any state of knowledge about the nature of (1.4) of (1.7) that is less than fully quantified will be termed *nonparametric.*

A basic difficulty in performing scientific work outside of a fully quantitative environment is that the set of nonparametric information available to researchers can come in a seemingless endless variety of forms. Depending on his or her progress, a researcher may know only which variables appear in individual relationships (i.e., which entries of *A* in (1.4) are zero and which are not), or the direction of influence of parameters on variables (i.e., the signs [+, -, 0] of the entries of *A* in (1.4)), or the relative magnitudes of some of the entries of *A* (i.e., a ranking of the entries of *A* in (1.4)). There is no single nonparametric environment. Researchers have pursued two related approaches to this curse of riches.

One approach focuses on the types of information that are likely to be available to researchers concerning the entries of the Jacobian matrix. The classic example, developed in Chapter 2, is to assume researchers know only whether entries are greater than, less than, or equal to zero. This emphasis on sign information arose historically in economics because economists are most secure in their beliefs about which variables appear in relationships and the nature of their direct influence, i.e., whether [partial derivative]*fl*/[partial derivative]*yj* and [partial derivative]*[fl*/[partial derivative]α are zero, positive, or negative. An analysis based upon sign pattern information alone is termed a *qualitative analysis.* Thus, a qualitative analysis deals with the matrices in equations (1.4) and (1.7) under the assumption that sign pattern information is available concerning the Jacobian matrix *A* and the vector *b.* Chapter 2 presents the results available for qualitative analyses. Sometimes a researcher may assume to know additional information about the matrix's entries, such as their relative sizes or bounds upon their magnitudes. Chapter 3 organizes sign pattern information analysis with these additional categories of information into a hierarchy analogous to that of measurement scales. Results are derived that show how the different categories of information about the entries of the Jacobian matrix can lead to definitive conclusions about the entries of the inverse Jacobian matrix. The other approach, developed in Chapters 5–8 is to hypothesize underlying principles, such as maximization or stability, governing the Jacobian matrix of the systems described by (1.4) and (1.7). These principles, combined with qualitative information, can sometimes yield definite results.

In all of these the fundamental mathematical questions are the same. First, under what conditions, given the information assumed to be available about the entries of the Jacobian matrix and the vector *b* (in (1.4)), can we solve (partially or completely) for the sign pattern of the vector *x*? And second, when can the same information enable us to determine the stability of the differential equation system (1.7)?

**1.4 AN EXAMPLE**

It might be helpful to consider an example of a qualitative analysis. In a simple full-employment economy, current output, *X*, is fixed at a level *Xf*. Total real output is divided among investment, consumption, and government expenditure, all expressed in real (inflation-adjusted) dollars. Equivalently, output can be viewed as the sum of real savings, consumption, and taxes. Investment, *I*, is assumed to decrease with the interest rate, *i*. Consumption, *C*, is assumed to increase with disposable income, *Xd*, which is defined as output less taxes, i.e., *Xd = X - T*. For the purposes of this example, taxes and government expenditures are assumed to be exogenous variables set by government policy, i.e., they can be chosen independently of other economic variables.

In the money market, it is assumed that the money supply, *M* is set by the central bank. The real money supply is *M/P*, where *P* is the price level. Demand for real money balances consists of transactions demand, *kX*, where *k* is a constant and *L(i)*, a decreasing function of the interest rate. In equilibrium, the demand for real money balances equals the supply of real balances. The equilibrium equations governing this simply economy, based upon market clearing in the goods and money markets, are

G + I(i) + C(Xd) = X

and

M/P = kX + L(i),

where *X = Xf* and *Xd = X - T*.

Differentiating the two equilibrium equations totally with respect to all endogenous and exogenous variables yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The differentials *dG, dT,* and *dM* are policy changes selected by the government or the central bank and result in changes in the interest rate and the price level, *di* and *dP,* respectively. The signs (+, 0, -) of the derivatives are established by the assumptions made earlier. The qualitative system corresponding to (1.4) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Inverting the coefficient matrix and solving yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consider what happens if government expenditures are increased (*dG* > 0), while taxes and the money supply are held constant (dT = dM = 0):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, this system is fully sign solvable—the signs of both *di* and *dP* are determined by the sign pattern information given. An increase in real government expenditures in this economic model gives rise to an increase in the interest rate and an increase in the price level, assuming taxes and the money supply are held fixed. Further, it is easy to verify that changing taxes or changing the money supply, holding the other exogenous variables fixed, also leads to a fully sign solvable system. Thus, within this model, increasing government expenditure increases the interest rate and the price level (both *di/dG* and *dP/dG* are positive). Increasing taxes reduces both the interest rate and the price level (both *di/dT* and *dP/dT* are negative). Increasing the money supply increases prices *(dP/dM* > 0), but does not affect the interest rate (*di/dM* = 0). To say that this simple economy is sign solvable means that, assuming that equilibrium is reestablished, it is possible to deduce the direction of change in all the economic variables as government policy changes, *independent of the magnitudes of the influences expressed by the Jacobian matrix so long as the directions of the influences (i.e., signs of the entries) are those assumed.*

*(Continues...)*

Excerpted fromNonparametric Comparative Statics and StabilitybyDouglas Hale, George Lady, John Maybee, James Quirk. Copyright © 1999 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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