The comparative static effects of many changes

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3 0001 0035 5465 8 © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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ECONOMICS DEPARTMENT

EUI Working Paper ECO No. 97/33

The Comparative Static Effects

of Many Changes

Lavan Mahadeva

WP 3 3 0

EUR

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without permission of the author.

© Lavan Mahadeva

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The Comparative Static Effects of Many

Changes

Lavan Mahadeva*

21.7.97 JEL Nos: C60, H20

A b strac t

This article explains how changing many structural parameters by dis crete amounts affects many variables of interest when we have no explicit expressions for the variables of interest in terms of the structural para meters. Although the matrix multiplying a vector of discrete changes in parameters into the changes in the vector of variables is not the same ma trix as would multiply a vector of infinitesimal changes, these matrices share some useful properties.

'European University Institute, Florence , Italy

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1. In tro d u c tio n

Even when we cannot write down variables of interest as an explicit function of param eters th a t are changing, we would still like to know something about the effect of many discrete changes in the parameters. For example w hat can we say about the effect of many discrete tax changes on th e supplies of labour of many individuals when we have no explicit expressions for th e supplies of labour? As they are many taxes th a t are changing, there is no mean value theorem which would tell us th a t the change in labour supplies is equal to th e m atrix of deriv atives of th e labour supplies w ith respect to the tax rates (at some set of tax rates) multiplied by th e vector of changes in the taxes. This article shows th a t although th e m atrix describing the transform ation of a vector of discrete changes of arguments into changes in a vector of functions is not th e m atrix describing the transformation of a vector of infinitesimal changes, these matrices share some useful properties. Using this, I show th a t if output is produced by many workers according to a strictly concave production function and th e substitution effect of higher wages always dominates the income effect, then a budget-neutral fiscal policy cannot increase everyone’s labour supply if th e workers are all substitutes in production.

2. T h e E ffect o f M an y C h a n g es in G en eral

Let f = ( /i, .../v ) 2 * * * * 7 be a single-valued and continuously differentiable vector func tion mapping U C R N into V c R N. In an example below, f describes the map of the vector of taxes and subsidies on the earnings of many different individu als into th e vector describing their optimal labour supplies. The purpose of this paper is to show th a t some properties of th e effect of infinitesimal changes on a vector function f may also hold for th e effect of discrete changes. Now the ja-cobian m atrix of th e function f is th e m atrix of derivatives of each of the scalar functions f t w ith respect to each of its arguments. Pre-multiplying by this m atrix transforms a vector of infinitesimal changes in the arguments into their effects on th e functions. I am going to show th a t certain properties of this transform ation of infinitesimal changes hold for discrete changes if th e jacobian m atrix describing th e transform ation is a P-m atrix where:

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D efin itio n 2.1. a P-m atrix is any N x N m a tiix with all its principal minors

positive.

2.1. P relim in aries

The following theorem is a useful property of P-matrices:

T h eorem 2.2. I f A is an N x N P-m atrix then A " 1 is an N x N P-matrix.

Proof. I wish to prove th a t B y , th e Ith principle minor of A ' 1 of size j x j , is positive. B y is in th e top left hand corner of A -1 without loss of generality. Partitioning A such th a t C y is a square subm atrix of the same size as B y gives:

A = C q D y E y F y

B y | - 1*V_{_ |A|}l

(2.1) Prom in A A 1 = I it follows th a t |B y | = (see Johnston (1984) pp.135-8). This is positive as A is a P -m atrix and thus |F y | is positive. As any principle minor of A -1 is positive, A -1 is a P-m atrix. Q E D

Now as before let f = ( /i, . . . /jv)T be a single-valued and continuously dif ferentiable vector function mapping U C R N into V c R N. Assume th a t the inverse of this mapping exists and is single-valued and continuously differentiable everywhere1.

The exist for all h, 0 < h < 1; (3 = (/3„...,/?JV)7’ ; £ = SN)T

€ t / and i , j = 1,..., N . The / , (£ + h/3) are continuous for all h, 0 < h < 1. From a mean value theorem for a function of a vector (see for example Nering (1970) pp.262), we know th a t there exists a vector 0 = {0\ , ..., dN)

th a t

I can write these equations as

, 0 < 9N < 1, such

(2.2)

f ( £ + / 3 ) - f ( £ ) = £ j / ( £ + 0i/3)IW3; (2.3) i— 1

JAs I shall demonstrate in the example below, the single-valuedness and continuously dif ferentiability of the inverse of this function may itself follow from the Jacobean describing the transformation being a P-matrix and may not have to be assumed.

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where each R* is an ./V x N m atrix with the Ith element on th e main diagonal equal to 1 and sill other elements 0 and each J / (£ + 6,(3) is the jacobian m atrix of f with respect to its arguments z, measured at th e values z*= £ + 0,(3.

Define the m atrix V / (£, (3) as :

+ (2.4)

1=1 Hence we can write th e transformation as:

f ( € + j 9 ) - f ( 0 = V / ( t/3 ) /3 . (2.5) L e m m a 2.3. Assum e that X is a rectangular region o f R N . Then for any fixed a = ( a i , ..., aw)T & X , and any other x = ( x i , ..., Xn)t £ X the inequalities

f(x) < < f(a),

x > > a (2.6)

have only the trivial solution x = a in X if the jacobian M atrix J /( x ) o f the mapping f is everywhere a P-m atrix in X 2.

Proof. See Nikaido (1968) pp.366-368. Now consider the following definition:

D e fin itio n 2.4. A N x N m atrix A is said to reverse th e sign o f a vector x = ( x i , x n)t i f Xi [Ax]j < 0 for all i = 1 , . . , N where [z]; denotes the i th element o f a column vector z.

Given th a t definition, we have th e following theorem:

T h e o r e m 2.5. 1) The m atrix V / (£, /3) does not reverse the sign o f any f3 ex

cept the zero vector i f the m atrix J / (p) is a P-m atrix for every p € [£, £ + f3\. 2) The inverse o f V / (£, /3) does not reverse the sign o f a non-zero f (£ + f3) — f (£). Also, 3) each element o f V f (£,/3) has the same sign as the corresponding ele m ent o f J j (p) somewhere in p € [ £ ,£ + /3] and 4) each element o f the inverse o f

V f (£, f3)has the same sign as the corresponding element o f the inverse o f J / (p)

(the m atrix J / (q )” 1) somewhere in q € [f (£), f (£ + (3)\.

2This is also true if the Jacobean Matrix is continuous and everywhere has all its principle minors negative and at least one positive element (Inada 1966).

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Proof. In Lemma 2.3 take x, = at — if /?4 > 0 and a; = £2, x, = £, + f t if /?; < 0. 2.5 shows th a t the m atrix V^ (£, (3) does not reverse the sign of all vectors /3 except the zero vector; property 1 is true.

By Theorem 2.2, the inverse of a P-m atrix is a P-m atrix. The inverse of the mapping f exists, is continuously differentiable and is single-valued. Hence if th e m atrix J j (p) is a P-m atrix for every p € [£,£ + /3], its inverse (the m atrix

J f (q)_1) is a P-m atrix for every q e [f (£), f (£ + (3)]. This proves property 2:

the inverse of V / (£, (3) does not reverse the sign of a non-zero f (£ + /3) — f (£). Properties 3 and 4 then follow from 2.4. QED

Theorem 2.5 applies when th e jacobian m atrix of th e transform ation of infin itesimal changes is a P-m atrix. Along with theorem 2.2, the following theorems show how economies where agents maximise strictly concave objective functions may lead to situations where the jacobian matrices describing th e transform a tion of infinitesimal changes in many structural param eters into changes in the many economic variables are P-matrices. I show first how a negative definite ma trix (such as the m atrix of second derivatives of a strictly concave function) is a P -m atrix and second th a t th e set of P-matrices are closed under certain opera tions. The example in the paper then shows how these theorems can be applied to prove th a t th e jacobian m atrix of the transform ation of infinitesimal changes is a P-m atrix.

I begin w ith th e following definition:

D efin itio n 2.6. A N x N m atrix A is said to be negative (respectively positive) definite i f x TA x < 0 (respectively > 0) for all x = ( x i...x n)t / 0.

Then it is well known th a t

T h eo rem 2.7. i f A is an N x N positive definite matrix, then it is a P-matrix.

The following theorem describes known properties of P-matrices:

T h eo rem 2.8. I f A is any N x N P-m atrix and B is an N x N diagonal m atrix with all its diagonal elements positive, then B A is an N x N P-matrix. I f A is an N x N P-m atrix and B is an N x N diagonal m atrix with all its diagonal elements positive, then A + B is an iV x iV P-matrix.

Proof. See Hofbauer and Sigmund (1988) pp.201.

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3. A n e x a m p le o f th e effect o f ta x e s an d su b sid ie s

I am now going to provide a simple example of how theorems 2.2, 2.7 and 2.8 can be used. In this example, th e economy is composed of N different individuals. The

ith individual’s problem is to choose th e proportion of tim e he spends working, (1 — rii), as opposed to enjoying leisure, n,, so as to maximise his lifetime utility. The

model is static and th e individuals do not derive utility from bequests. His utility is therefore a function only of his enjoyment from leisure in money term s (lei) and his income earned w ith his labour (yt) for his one-period life. His enjoyment from leisure is simply proportional to the tim e he spends in leisure:

let — A in i.

The government can determine the amount of ta x paid by or subsidy paid to a worker of each type for each unit of his labour. Let the policy instrum ent on type

i be called polt where it is positive if the government pays a subsidy and negative

if the government charges a tax on th e hum an capital of worker i. The wage rate received by a representative individual of type i for his unit of tim e spent working would be given as ivt (1 + po/,). The ith individual’s income from work is then:

Vi = Wi( 1 + poli) (1 - Hi) ; (3.1) where w, (n) is the before ta x wage rate he receives for each unit of his tim e which depends on th e labour supplies of all workers (n = ( n j, ..., Un)t ). The utility

function of th e individual is of th e CES form:

( le l-° + y l - " ) / ( l - c r ) . (3.2) where the elasticity of substitution between leisure and income is given by (1 — a) . If th e substitution effect of a higher wage is always greater th a n the income effect then a < 1.

As th e labour market is competitive, th e wage rate per unit tim e of each individual (n) is equal to th e marginal product of th a t unit of tim e in th e pro

duction of final output. The production function of final output is strictly concave w ith respect to all its factors together and there are no production externalities. More specifically, I assume th a t the symmetric m atrix of derivatives of the wage rates, , .d w i(n ). , .

F (n) = t

dn '.

= K>' (n) l'

is positive definite. (3.3) © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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The individuals i and j are said to be substitutes (complements) in production if uiij (n) < 0 ( respectively < 0). For example, according to this definition, if i and j perform an identical role in production they will be substitutes.

The individual’s objective function can be rew ritten as:

which is maximised by his choice of tim e n* subject to th e restriction th a t

and taking th e wage rate w, as given.

From the first order conditions for all individuals, the optim um tim e spent in leisure by all individuals n = (nj, ...,n ^ ) T as a function of the policy variables is given by solving

where cj> = q^T ■ Even for quite simple production functions, it is difficult to derive an explicit solution for the N labour supplies as the function of the N policy variables. I now go on to describe what we can still say about the effect of discrete changes in th e policy variables.

My first step is to show th a t a t least one set of labour supplies exists for each set of policy variables. As th e function A f / ( A f + ((1 + pol,)w, (z))<t,'j is continuous in z = ( z i , . .

.,

z n)t and maps th e closed compact set z = ^ R N s.t 0 < zt < 1 j into itself, it can easily be proved th a t at least one solution for n = (n i,..,n jv )T to equations 3.5 exists.

I now want to show th a t there is only one vector of labour supplies for each vector of policy variables. By the Gale-Nikaido Global Invertibility Theorem (see Nikaido (1968) Theorem 20.4 pp.370 or Woods (1978) pp.228-231 for example), there is only one solution for n to equations 3.5 if th e N x N m atrix of derivatives of the N functions

w ith respect to the N labour supplies is a P-matrix. This m atrix is given by (3.4) 0 < n t < 1 (3.5)

A f /

(A

f

+ ((1 + poli)wt (n))*)

I + Z!F © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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where Z\ is a N x N diagonal m atrix with the Ith main diagonal element <t>n, (1 — n x) jw and F is defined in 3.3. F is negative definite and therefore —F is a P -m atrix by theorem 2.7. By theorem 2.8, (I + Z iF ) is a P-m atrix if </> < 0 and then there is only one set of optim al labour supplies for each set of policy variables. 4> < 0 only if the substitution effect of greater wages always dominates th e income effect and th e labour supply curve was not backward-bending.

The government’s aim is to increase labour supply by altering th e taxes and subsidies which it levies on workers. Now th e jacobian of th e optim um leisure times w ith respect to th e policy variables (p o l = {poll, ■ ■■,PoIn)T ) is the m atrix

P = (I -t- Z ]F )-1 Z2 from

V n = P V p o l; (3.6)

where V q = (Sqi...., <5<7,v ^describes the vector of infinitesimal changes in the components of any vector q = (<ji, ...,qN)T . Differentiating both sides of the first order conditions 3.5 shows th a t P = (I -I- Zi F ) 1 Z2 where Z2 is a N x N diagonal m atrix w ith th e ith main diagonal element (1 — n,) / (1 + poll)) which is positive if (f> < 0. I have already shown th a t (I + Z iF ) is a P-m atrix if <f> < 0. In this case, (I + Z iF ) -1 is a P-m atrix by theorem 2.2. By theorem 2.8, P is a P-matrix.

In the form of 2.4, th e effects of discrete changes in th e tax rates on the demands for human capital are given by:

A n = K (A pol, p ol) A pol. (3.7) By theorem 2.5, if 4> < 0, the m atrix K (A pol, p ol) does not reverse the sign of any non-zero vector of ta x changes as P is a P-m atrix for all ta x rates p o l' = (1 -I- pol\ , . . . , 1 + pol'N)T such th a t p o l < < p o l' < < p o l + A p o l.

Thus when 4> < 0,

a rise in all policy variables (either lowering taxes or raising subsidies on everyone) would encourage at least one worker to work more.

If all the workers were substitutes (respectively complements) then Z iF would have all positive (respectively negative) off-diagonal elements. The inverse of a P -m atrix w ith only positive off-diagonal elements is a P-m atrix with only nega tive off-diagonal elements and vice versa (see Woods (1978) pp.41). So when all the workers are substitutes (respectively complements), th e m atrix P has only negative (respectively positive) off-diagonal elements.

From property 3 of theorem 2.5, when the substitution effect of a higher wage dominates the income effect, then:

7 © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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if all the workers are substitutes, the government can increase everyone’s ’ in centive to work only by either lowering taxes or raising subsidies on everyone and therefore worsening the current budget deficit/surplus.

But if all the workers are complements, the government may be able to increase everyone’s incentive work without worsening the current budget deficit/surplus.

I have shown th a t when the substitution effect of a higher wage dominates the income effect and all workers are substitutes (perform similar roles in production), budget-neutral tax policies to improve everyone’s incentive to work rely on non convexities in production or complementarities. This may not be true if the labour supply curve is backward-bending. But then it may be th a t one set of policy variables can lead to more th a n one combination of labour supplies.

3.1. C onclu sion

I have shown th a t when modelling economies with many different individuals, and where explicit solutions cannot be obtained, we can still stay something about the effect of discrete changes in many structural parameters. In this way, theorem 2.5 tells us about the comparative statics of a vector of changes. Theorem 2.5 can also be used to describe the dynamics of discrete tim e equations for which we have no explicit solution. Here the structural param eters are the last-period values of the sta te variables and we are concerned with the effect of th e vector of discrete changes in th e last-period values of the variables on their current values.

4. R eferen ces

Hofbauer, J. and K. Sigmund, The Theory of Evolution and Dynamical Systems: M athem atical Aspects of Selection, Cambridge University Press, 1988.

Johnston, J., Econometric Methods, McGraw-Hill Publishing Co. ,1984. Nering, E.D., Linear Algebra and M atrix Theory, New York, John Whiley and Sons, 1970.

Nikaido, H, Convex Structures and Economic Theory, Academic Press, 1968. Woods, J.E., Topics in Multi-Sectoral Economies, Longman, 1978.

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The Comparative Static Effects of Many Changes © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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