Comparative Statics: Classical and Modern Approaches*

Victor J. Tremblay

Department of Economics303 Ballard Extension Hall

Oregon State UniversityCorvallis, OR 97331-3612

Phone: 541-737-2321Email: V.Tremblay@OregonState.edu

April 2007

If you want literal realism, look at the world around you; if youwant understanding, look at theories. (Dorfman, 1964)

We would, of course, dismiss the rigorous proof as beingsuperfluous: if a theorem is geometrically obvious why prove it? This was exactly the attitude taken in the eighteenth century. Theresult, in the nineteenth century, was chaos and confusion: forintuition, unsupported by logic, habitually assumes that everythingis much nicer behaved than it really is. (Steward, 1975)

I am pleased that the seemingly endless disputes on the role ofmathematics in economics have largely ceased.... Without rigor,the author and the reader simply cannot evaluate whether a resultis right or wrong. (Allen, 2000)

*This is required reading for Econ 513, spring 2007. In spite of being over 15 years old, these modern methods aremissing from undergraduate and graduate textbooks in mathematical economics. This is a rough draft. Commentsare welcome, but please do not quote without the author’s permission. (File Name: Econ-513 – Comparative Statics– 2007).

2

1For a review of the this approach, see Simon and Blume (1994), Hand (2004), Baldani etal. (2005), and Chiang and Wainwright (2005). For a discussion of alternatives, such as therevealed preference approach, see Carter (2001).

2See, for example, Milgrom and Shannon (1994), Shannon (1995), and Eldin andShannon (1998). For a discussion of montone comparative static methods for constrainedoptimization problems, see Quah (2007).

Comparative Statics: Classical and Modern Approaches

1. IntroductionComparative statics or sensitivity analysis investigates how the endogenous variables of a

model are affected by a change in a parameter or exogenous variable. The “comparative” termrefers to a before and after comparison of an optimum or equilibrium value that results from avery small change in an exogenous variable or parameter. The “statics” term refers to the factthat a comparison is made after all adjustments have occurred. That is, the dynamic process ofgoing from one outcome to another is ignored. Such before and after comparisons providetestable predictions and policy implications of economic models. Comparative static analysis is performed on equilibrium and optimization models. Theclassic approach applies the implicit-function theorem to first-order conditions in optimizationmodels and to equilibrium conditions in equilibrium models.1 To apply the implicit-functiontheorem, however, certain regularity conditions must hold. For example, derivatives of relevantfunctions must be continuous, objective functions must be concave, and the equilibrium must bestable. An important weakness of the implicit-function theorem is that it is not applicable fordiscrete changes in parameters or exogenous variables.

Recent work in the area of monotone comparative statics demonstrates, however, thatcomparative static analysis can be done without many of the restrictions required by the implicit-function theorem.2 For example, this approach works for discrete as well as infinitesimally smallchanges in parameters or exogenous variables. The “monotone” term refers to statements aboutorder; that is, an increase in one variable leads to an increase in another variable. Of course,something is given up with this new approach. Unlike the implicit-function theorem, monotonecomparative statics can tell us the direction but not the magnitude of change.

Another concern is that comparative static analysis becomes more difficult to apply ingame theoretic settings. With many players and choice variables, the curse of dimensionality is aproblem. That is, it becomes increasingly tedious (algebra intensive) to calculate the result. Inaddition, multi-equilibria are common in many games, making it difficult to apply the implicit-function theorem. Recent work shows, however, that unambiguous comparative static results canemerge when a game exhibits qualities of super-complementarity. Such a game is said to besupermodular and exists when each player’s own choice variables are complementary and all

3

3For a discussion of strategic complements and supermodular games, see Bulow et al.(1985), Milgrom and Roberts (1990) and Vives (1999, 2005a, and 2005b).

strategic variables across players are strategic complements.3This note is written with three goals in mind. The first is to review comparative statics

using the implicit-function theorem. The second is to show how monotone comparative staticmethods can be used when there are discrete changes. The final goal is to show how comparativestatic analysis can be performed in games that are supermodular.

2. Comparative Statics and Implicit Functions

One of the most important professional activities of economists isto carry out exercises in comparative statics: to estimate theconsequences and merits of change in economic policy and oureconomic environment. (Scarf, 1994, 111)

In economics, most comparative static problems involve answering the followingquestion: How does a change in a parameter (or exogenous variable) affect the equilibrium oroptimal value of an endogenous variable in a model? Although tedious, this is relatively easywhen the structural equations of a model take on a specific functional form and can be solvedexplicitly for the optimal value of an endogenous variable.

To illustrate, consider a simple demand and supply model with the following structuralequations:

Q a bP cY

Q eP

D

S

= − +

=where QD is quantity demanded, QS is quantity supplied, P is price, and Y is consumer income.Price and quantity are endogenous variables, and income is exogenous. The parameters a, b, c,and e of the are assumed to be positive. That is, demand has a negative slope, the commodity inthis model is a normal good, and supply has a positive slope. To determine the effect of incomeon the equilibrium price (P*) , we can solve explicitly for P* and differentiate the function withrespect to income. In equilibrium, QD = QS and the equilibrium price is

P a cYb e

* .=++

In this case, the effect of income on the equilibrium price isdPdY

cb e

*

,=+

which is positive, given the assumptions of the model. That is, the model predicts that anincrease in income will lead to an increase in the equilibrium price of a normal good.

In more complex models, this approach may be impractical or impossible to use. Forexample, it may be difficult to solve explicitly for the equilibrium price, which can occur whendemand and supply are non-linear. More importantly, how do we perform comparative static

4

4In three dimensions, this neighborhood literally includes an open ball. With two choicevariables, it is an open disk. With one choice variable, it is an open interval. In the case of asingle choice variable, for example, the neighborhood around point x* = 2 might be: 1 < x < 3,written as x , (1, 3).

analysis when the specific functional forms of demand and supply are unknown? For example,let demand and supply take the following general forms:

Q Q P Y

Q Q P

D D

S S

=

=

( , )

( )Assuming a unique interior solution, excess demand (ED) equals zero when the market is inequilibrium [i.e., ED(P, Y) = QD(P, Y) - QS(P) = 0]. Such a function is referred as in implicitfunction. In general cases such as these, it is impossible to explicitly solve for the equilibriumprice and perform simple comparative static analysis. To address these general types ofproblems, we must know when there exists an explicit solution to an implicit function such asED(P, Y) = 0.

The implicit-function theorem identifies conditions that assure that such an explicitfunction exists and provides a technique that produces comparative static results.

Implicit-Function Theorem: Let F(x, y) be a function with partial derivatives that exist and arecontinuous in a neighborhood (called an open ball B) around the point (x1

*, ..., xn*, y*),4

such that (x1*, ..., xn

*, y*) satisfies:F x x y c

F x x yy

n

n

( ,... , , ) ;

( ,... , , ).

* * *

* * *

1

1

0

0

− ≡

≠∂

∂Then F(x, y) - c uniquely implies a function y = f(x1, ..., xn) that has derivatives that arecontinuous within the open ball B around point (x1

*, ..., xn*) such that:

A. F(x1, ..., xn, y(x1, ..., xn)) = c for all (x1, ..., xn) within B,B. y* = f(x1

*, ..., xn*), and

C. for each i = 1, ..., n,

∂∂

∂∂

∂∂

y x xx

F x xx

F x xy

n

i

n

i

n

( ,... , )( ,..., )

( ,..., )

* (

* *

* *1

1

1

= −

The result in part C is sometimes called the implicit-function rule.

5

5For a proof of the implicit-function theorem, see Lang (1983).

6The assumption of an open ball is required to rule out corner solutions. With two choicevariables, this is an open disk. With one choice variable, this is an open interval.

Proof of the Implicit-Function Rule:5 Take the total differential of F(x1, ..., xn, y(x1, ...,xn)) = c around the point (x1

*, ..., xn*, y*),

∂∂

∂∂

∂∂

Fx

dx Fx

dx Fy

dyn

n

1

1 0+ + + =...

Note that the asterisks are suppressed for convenience. Assuming we are interested inMy/Mx1, set dx2 = dx3 = ... = dxn = 0.6 Solving for dy/dx1,

dydx

Fx

Fyx x xn1

1

2 3, ,...

.=− ∂ ∂∂

∂

Note by definition that this equals My/Mx1 because we have set dx2 = dx3 = ... = dxn = 0. Q.E.D.

Part C of the implicit-function theorem can determine the magnitude of change whenspecific functions are used to describe the structural model. With general functional forms, weare only able to determine the direction of change. Several examples illustrate this approach.

2.1 Comparative Statics in Equilibrium or Optimization ModelsThe most common economic applications of the implict-function theorem are to

equilibrium models and models that involve optimization techniques. Several examples areillustrated below.

Example 1 (Equilibrium Problem): Consider the demand and supply problem described abovewith the following excess demand function: ED(P, Y) = QD(P, Y) - QS(P). Let the partialderivatives of the excess demand function be continuous, the demand function have a negativeslope (i.e., MQD/MP < 0), the commodity be a normal good (i.e., MQD/MY > 0), and supplyfunction have a positive slope (i.e., MQS/MP > 0). As before, our goal is to determine how anincrease in Y will affect the equilibrium price. Assuming an interior equilibrium exists, theexcess demand function will be identically equal to zero at equilibrium values of price andquantity. Under these conditions, the implicit function implies the following:

dPdY

EDY

EDP

QY

QP

QP

D

D S

*

.= − =−

−>

∂∂

∂∂

∂∂

∂∂

∂∂

0

6

Note that stability of the demand and supply model requires that MED/MP < 0. When thiscondition holds, price rises when there is excess demand (ED > 0), causing excess demand tofall; price falls when there is excess supply (ED < 0), causing excess supply to fall. Stabilityconditions are commonly used to derive comparative static results in equilibrium problems.

Example 2 (Optimization Problem): Consider a monopoly firm that faces a per-unit tax (t), andour goal is to determine the effect of the tax on the firm’s optimal (profit-maximizing) outputlevel. The firm’s profit is: B(q, t) = TR(q) - TC(q) - tq, where B is profit, q is output, TR is totalrevenue, and TC is total production cost. Assuming a differentiable and strictly concave profitequation, the respective first- and second order-conditions of profit maximization are:

∂π∂

∂∂

∂∂

∂ π∂

∂∂

∂∂

qTRq

TCq

t

qTRq

TCq

= − − =

= − <

0

02

2

2

2

2

2

;

.

Although it cannot be derived explicitly, embedded in the first-order condition is the profitmaximizing level of output. At this optimal value, the first-order condition is identically equal tozero and the implicit-function theorem implies that

dqdt

q t

qTR

qTC

q

*

.= − = −−

−

∂ π∂ ∂

∂ π∂

∂∂

∂∂

2

2

2

2

2

2

2

1

Because the denominator is negative from the second order condition, this derivative is negative. That is, an increase in an excise tax will reduce a monopolist’s profit maximizing output level. This example shows how the second-order condition is used to derive comparative static resultsin optimization problems.

Example 3 (Optimization Problem): Consider a problem where we want to determine how anincrease in the price of an input will affect a firm’s profit maximizing quantity of that input (i.e.,the slope of the firm’s profit-maximizing input demand). Assume the firm is a price taker (i.e.,both input and output markets are perfectly competitive) and uses just two inputs: labor (L) andcapital (K). The firm’s profit is: B = pq(L, K) - wL - rK, where w is the price of labor and r is therental rate of capital. Assume the problem is short tun, where capital is fixed and labor isvariable (the long-run problem will be considered next). Assuming a differentiable and strictlyconcave profit equation, the respective first and second order conditions of profit maximizationare:

∂π∂

∂∂

∂ π∂

∂∂

Lp q

Lw

Lp q

L

= − =

= <

0

02

2

2

2

;

.

Embedded in the first-order condition is the optimal value of labor, and from the implicit-function theorem:

7

∂∂

∂ π∂ ∂

∂ π∂

∂ π∂

Lw

L w

L p L

*

.= − =

2

2

2

2

2

1

Again, this example illustrates how the second-order condition plays a key role in thecomparative static analysis in optimization problems. When the second-order condition is met,the marginal product of labor will have a negative slope and the firm’s profit maximizingdemand for labor will have a negative slope.

Most optimization problems in economics have more than one choice variable. Forexample, General Motors uses varying degrees of skilled to unskilled labor, different types ofraw materials (e.g., steel, plastic, leather), and different types of physical capital to produce cars,trucks, and refrigerators. In cases such as these, first-order conditions become a system of linearequations that must be solved simultaneously to find the optimal values of the choice variables. When these functions are implicit and include many variables, it becomes convenient todifferentiate the first-order conditions and solve the resulting system using matrix algebra andCramer’s rule (Simon and Blume, 1994, p. 194). To illustrate, consider the previous problem,but now more than one input is variable.

Example 4 (Optimization Problem with a System of Equations): Assume that a perfectlycompetitive firm wants to maximize its long-run profit with respect to all of its inputs. For ninputs, the profit equation is

π = −=∑pq x w xii

n

i( ) ,1

where xi is the quantity of input i, wi is the corresponding input price, and x is a vector of ninputs. For simplicity, let n = 2. The system of first-order conditions is

π

π

1 1 1

2 2 2

0

0

= − =

= − =

pq w

pq w

,

.For notational convenience, let Bi equal the first derivative of the profit with respect to input iand qi equal the first derivative of the production function with respect to input i. Because p andw are exogenous, M2B/MxiMxj = M2q/MxiMxj for all i and j equal to 1 and 2. Again, for convenience,we write this as qij. Thus, the matrix of second derivatives of the profit equation, the Hessianmatrix (H), can be written as

Hq q

q q=⎛

⎝⎜⎜

⎞

⎠⎟⎟

11 12

21 22

For a unique maximum, the profit (production) equation must be strictly concave. For this tooccur, the Hessian matrix is negative definite (second-order conditions): qii must be negative andthe determinant of the Hessian matrix must be positive. In this problem, our goal is to determinehow a change in wi will affect xi

* and xj*. By substituting the optimal input values into the first-

order conditions, making them identically equal to zero, we can differentiate each first-ordercondition with respect to wi. This yields the system of equations:

8

pqxw

pqxw

pqxw

pqxw

iii

i

ij

j

i

jii

i

jj

j

i

∂∂

∂

∂

∂∂

∂

∂

* *

* *

,

.

+ ≡

+ ≡

1

0

This is a system of two linear equation in two unknowns (Mxi*/Mwi and Mxj

*/Mwi), which can bewritten in matrix form as

pq pq

pq pq

xw

xw

ii ij

ji jj

i

i

j

i

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

≡⎛

⎝⎜⎜⎞

⎠⎟⎟

∂∂

∂

∂

*

*

1

0

Applying Cramer’s rule to this system produces the comparative static results of interest

∂∂

∂

∂

xw

pq

pq

Hpq

p q q q

xw

pq

pq

Hpq

p q q q

i

i

ij

jj jj

ii jj ij

j

i

ii

ji ji

ii jj ij

*

*

( ),

( ).

= =−

= =−

−

1

0

1

0

2

2

As in previous optimization problems, the denominators in both results are strictly positive bythe second-order conditions. Because second-order conditions also require qii to be negative, theown price effect will always be negative. That is, the long-run demand for all inputs will have anegative slope. The cross price effect is indeterminate, however. If the two inputs arecomplements in production (qij > 0), then Mxj

*/Mwj < 0; if the two inputs are substitutes inproduction (qij < 0), then Mxj

*/Mwj > 0.

Comparative static analysis of the objective function itself is also important in economicsand involves the use of the envelope theorem. This says that if we plug in the optimal values ofthe choice variables into the objective function, the derivative of this function (frequently calledthe value function or indirect objective function) with respect to a parameter is equal the directeffect of the parameter on the original objective function. When applied to a firm’s profitfunction, this is called Hotelling’s lemma, which is illustrated in the following example.

Example 5 (Optimization Problem and the Envelope Theorem): Let us continue with theexample above. Optimal input values, xi

* and xj*, derive from the system of first-order

conditions. Plugging them into profit equation produces the profit function (or indirect objectivefunction) is

9

π * * * * *( , , ) ( , ) .w w p pq x x w x w xi j i j i i j j= − −

To determine how maximum profit changes with a change in wi, we can differentiate the profitfunction,

( ) ( )

∂π∂

∂∂

∂

∂∂∂

∂

∂

∂∂

∂

∂

* * * **

*

* **

,

.

wp q x

wq

xw

w xw

x wxw

pq w xw

pq wxw

x

i

ii

i

jj

i

ii

i

i jj

i

i ii

i

j jj

i

i

= +⎛

⎝⎜

⎞

⎠⎟ − − −

= − + − −

Because the terms in parentheses are the first-order conditions and equal zero at the optimum,

∂π∂

** ,

wx

i

i= −

which equals MB/Mwi when evaluated at xi*. This means that the marginal effect on profit of a

change in wi has only a direct effect. That is, there is no indirect effect through the optimalvalues of inputs. It also implies that the negative of the demand function for input i equals thederivative of the profit function with respect to input i. This provides a proof of the envelopetheorem for the special case of a profit function, Hotelling’s lemma.

2.2 Comparative Statics in Problems with Both Equilibrium and OptimizationThe preceding section discussed comparative static analysis for either an equilibrium or

an optimization problem. Solutions to game theoretic problems, however, typically involvesolutions to both maximization and equilibrium problems. The most widely used equilibriumconcept in non-cooperative game theory is the Nash equilibrium, and it will be our focus here. The first applications of the Nash equilibrium concept to firm behavior are the Cournot (1838)and Bertrand (1883) models of duopoly. In the Cournot model, the choice variable is output; inBertrand, it is price. Because price is more common than output competition, we focus on theBertrand model.

Example 6 (Problem with Both Equilibrium and Optimization): In a differentiated Bertrandmodel, two firms (1 and 2) produce differentiated products in a single market and compete bysimultaneously choosing price. The demand function for firm i (1 or 2) is qi(pi, pj), where qi isthe firm’s output, pi is the firm’s price, and pj is the price of the firm’s rival. Let demand have anegative slope (Mqi/Mpi < 0) and, because they are substitute products, Mqi/Mpj > 0. Firminterdependence is revealed in the demand functions; an increase in the price of one firm raisesthe other firm’s demand, ceteris paribus. The total revenue of firm i is TRi(pi, pj) = pi qi(pi, pj). Let total cost for firm i [TC(qi, a)] increases with its own output and parameter a, where a is acost parameter such that MTC/Ma > 0 and M(TC/M qi)/Ma > 0. For example, a could be the price ofan important input or a per-unit tax. Our goal is to determine how a change in a will affect Nashequilibrium prices (pi

* and pj*).

Given these definitions, profit for firm i is TRi(pi, pj) - TCi(qi(pi, pj), a). Note that anincrease in a will raise a firm’s marginal returns of raising price for a negatively sloped demand

10

function. That is, M(MB /M pi)/Ma / Bia > 0. Respective first- and second-order conditions of profitmaximization for firm i are:

∂π∂

∂∂

∂∂

∂ π∂

∂∂

∂∂

pTRp

TCp

pTRp

TCp

i i i

i i i

= − =

= − <

0

02

2

2

2

2

2

;

.

Note that subscripts are suppressed in profit, total revenue, and total cost. The best reply function for firm i is determined by solving the firm’s first-order condition

for pi: piBR(pj) = pi

*. This identifies the optimal pi for all values of pj. The optimal value of pi isembedded in firm i’s first-order condition, which is identically equal to zero at pi

*. Thus, we canapply the implicit-function theorem to firm i’s first-order condition to determine the slope of itsbest-reply function. This is

dpdp

i

j

ij

ii

*

.=− π

π

As in previous examples, Bij is defined as the second derivative of firm i’s profit with respect topi and pj, and Bii is the second derivative of firm i’s profit function with respect to pi . The slopeof the best reply will be positive, because prices are strategic complements (i.e., Bij > 0) asdefined by Bulow et al. (1985) and because Bii < 0 from the second-order condition of profitmazimization. Thus, best reply functions in a Bertrand game will have a positive slope.

We determine the effect of an increase in a on Nash prices as follows. By substitutingthe optimal prices into the first-order conditions of each firm, we can differentiate them withrespect to a. This yields the system of equations:

π∂∂

π∂∂

π ∂∂

π∂∂

π∂∂

π ∂∂

111

122

1

212

222

2

0

0

pa

pa

aa

pa

pa

aa

a

a

+ + ≡

+ + ≡

,

.

This linear system can be written in matrix form as

π π

π π

∂∂

∂∂

π

π

11 12

21 22

1

2

1

2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟≡

−

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

pa

pa

a

a

*

*

Applying Cramer’s rule,

∂∂

π π

π πpa

a

a1

1 12

2 22*

.=

−

−

Π

where A is the 2 x 2 matrix of second derivatives of profits from the matrix form of the previouslinear system above. To have a stable Nash equilibrium, the determinant of A must be positive(i.e., B11 B22 - B12 B21 > 0). This is proven in Appendix A. In addition, second order conditions

11

7For a review of lattice theory, see Milgrom and Shannon (1994) and Topkis (1998).

require that Bii < 0. Because prices are strategic substitutes, Bij > 0. Given these and the fact thatBia > 0, an increase in a will cause p1

* to increase. Because the problem is symmetric, p2* will

also increase with a. This example shows how both stability conditions and second-orderconditions are important to comparative static analysis in problems that involve bothoptimization and equilibrium concepts.

3. Monotone Comparative StaticsTo apply classic comparative static methods, certain assumptions are necessary. These

typically include concavity of the objective function, convexity of constraint sets, andsmoothness of objective functions and constraints. Recent work in monotone comparative staticsdemonstrates that many comparative static conclusions can be obtained with weaker assumptions(Milgrom and Shannon, 1994, Shannon, 1994, and Edlin, Shannon, 1998, and Quah, 2007).

For example, an important weakness of the implicit-function theorem is that it requiresdifferentiability. This is particularly problematic in policy analysis, where a particular policy hasa discrete character. For example, an occupational safety regulation is either in effect or not.Likewise, a pollution abatement policy may completely ban the use of certain inputs. Both typesof regulations may cause a discrete jump in a firm’s objective function, making it impossible touse the implicit-function theorem. The tools needed to analyze such problems are monotonemethods, which show that differentiability and convexity are not always required to performcomparative static analysis. The application of this approach is simple when there is just onechoice variable, but higher order problems require a knowledge of lattice theory.7 For thisreason, the illustrations below focus on choice problems in just two dimensions.

To compare classic and new methods, assume that an economic agent wants to maximizean objective function, f(x, a) with respect to x, where a is a continuous policy variable. If aninterior solution exists and the second derivative of f is continuous, then, from the implicit-function theorem,

dxda

fx a

fx

*

.=− ∂ ∂ ∂∂

∂

2

2

2

From the second order condition, M2f /Mx2 < 0. Thus, the effect of a on x* will be positive(negative) when M2f /MxMa is positive (negative). This method is invalid, however, for a discretechange in the policy variable (e.g., when the discrete variable takes on a value of 0 or 1). To docomparative static analysis in this case, we must use Edlin and Shannon’s (1998) strictmonotonicity theorem. To distinguish the continuous from the discrete policy variable, let aequal " in the discrete case.

Strict Monotonicity Theorem: Let f: ú6ú, S d ú, x* = argmax x 0S f(x, "*), and xN = argmax x 0Sf(x, "N). Suppose that x* is a unique interior solution and that df /dx is continuous and hasstrictly increasing marginal returns with respect to the parameter ". Then x* > xN if "*

12

8Similarly, x* > xN if "* > "N.

9Note that in the continuous case, strictly increasing marginal returns means that theparameter and the choice variable are complements. That is, M2f/MxM" > 0.

> "N.8

Proof: The proof of the strict monotonicity theorem hinges on the assumption of strictlyincreasing marginal returns, which means that Mf /Mx (x*, ") is increasing in ".9 Giventhis definition, the basic argument is as follows. Because x* is the unique argmax at "*,f(x, "*) must increase as we move from "N to "*. It must be true that x* > xN, because ifx* < xN, then strictly increasing marginal returns implies that increasing x from x* to xNmust lead to an increase in f(x, "*). But this contradicts the definition that x* is theoptimal choice at "*. Q.E.D.

To illustrate the intuition behind the theorem, consider a specific functional form. Suppose the objective function is f(x, ") = g(")x - x2. The parameter " can take on two discretevalues, such that g("N ) = 2 and g("*) = 3. These two objective functions are illustrated in Figure1a and are labeled fN and f*. The function exhibits strictly increasing marginal returns in "because the slope of the tangent to the objective function increases as " increases from "N to "*. In other words, M f*/Mx > M fN /Mx as shown in Figure 1b. Thus, by the strict monotonicitytheorem, the argmax of f(x, ") increases from xN to x* as " increases from "N to "*. Thisexamples highlights the role of first-order conditions and illustrates how to apply the theorem –essentially all that needs to be checked is whether or not the function exhibits strictly increasingmarginal returns with respect to the parameter in question.

Example 7 (Optimization Problem with Discrete Policy Change): Consider a monopolyproblem where government policy (") causes a discrete decrease in the cost of doing business. This would include a discrete reduction in an excise tax or a policy that causes a discrete cut inbureaucratic red tape that is imposed on the firm. Our goal is to determine how this policy willaffect the firm’s optimal (profit-maximizing) level of output. The firm’s profit is: B(q, ") =TR(q) - TC(q, "). Profit is assumed have a derivative in q that is continuous, and an increase in" from "N to "* causes a discrete reduction in total and marginal cost. Assume further that andqN is the unique argmax of B(q, "N) and that q* is the unique argmax of B(q, "*). Under theseconditions, the profit equation exhibits increasing marginal returns as " increases from "N to "*. This is demonstrated below.

13

∂π α∂

∂π α∂

∂∂

∂ α∂

∂∂

∂ α∂

∂ α∂

∂ α∂

( ) ( ' ) ;

( ) ( ' ) ;

( ) ( ' ) ,

*

*

*

q q

TRq

TCq

TRq

TCq

TCq

TCq

− >

−⎛⎝⎜

⎞⎠⎟ − −

⎛⎝⎜

⎞⎠⎟ >

− + >

0

0

0

which holds because the marginal cost under regime "N is greater than the marginal cost underregime "* by definition. Thus, by the strict monotonicity theorem, the profit-maximizing levelof output increase with a government deregulation that reduces marginal cost.

It is also easy to see from examples 2 and 8 that the implicit-function theorem is a specialcase of the strict monotonicity theorem. When the policy parameter (" = a) is continuous, as in aper-unit subsidy (or excise tax reduction), for example, there will be strictly increasing marginalreturns to a. In the continuous case, this implies that M2B/M q/Ma > 0. From the second-ordercondition, M2B/M q2 < 0. Thus, the strict monotonicity theorem implies that a marginal increase inthis parameter will cause an increase in the firm’s profit maximizing output level. By theimplicit-function theorem, these conditions imply the same result.

∂∂α

∂ ∂ ∂∂α

∂ π∂

qTC q

q

*( / )

.= −−

>2

2

0

A weaker version of the theorem applies when the objective function is neither smoothnor concave (Milgrom and Shannon, 1994). One example is provided in Figures 2a and 2b. Inthis case, f(x, ") exhibits strictly increasing differences in ", which is a discrete version ofstrictly increasing marginal returns. That is, f(x, ") has strictly increasing differences in x and " when for all x’ > x, f(x’, ") - f(x, ") is increasing in ". Another example, this time where theobjective function is not concave, can be seen in Figure 3. Under the weaker conditions of thesetwo examples, however, the Milgrom and Shannon (1994) theorem states that the optimal valueof x will be non-decreasing in ". This can be seen in the example in Figure 4 where " has noeffect on the optimal value of x even though the objective function exhibits increasingdifferences in ". When f(x, ") is not differentiable in x, this weaker monotonicity theoremimplies that Mx*/M" $ 0. This example illustrates why differentiability is required for the strictmonotonicity theorem to hold.

The problem is more complicated when there are multiple choice variables. In the singlechoice problem, checking for increasing marginal returns or increasing differences is essentiallyall that is needed to do monotone comparative static analysis. With multiple choice variables,the problem is more complicated due to possible interact effects. In this case, monotonicitytheorems also requires that all choice variables be complementary. In the continuous case wherethe objective function is f(x1, x2, "), complementarity of choice variables means that M2f/Mx1Mx2 $

14

10This implies that best-reply functions are differentiable, but one could assume moregenerally that best-replies are complete lattices instead of smooth functions without affecting themain conclusions. For further discussion, see Milgrom and Roberts (1990) and Vives (1999).

0. When this conditions holds for all choice variables, the objective function is said to besupermodular. Thus, the application of monotone comparative static analysis when there aremultiple choice variables requires that all parameters and choice variables be complementary. That is, one must check for both supermodularity and increasing marginal returns (or increasingdifferences). The concept of supermodularity will be especially important in the next sectioninvolving comparative static analysis in game theoretic settings.

4. Monotone Methods and Game Theory

This class [of supermodular games] turns out to encompass manyof the most important economic applications of noncooperativegame theory (Milgrom and Roberts, 1990, 1255)

In a game theoretic setting, both optimization and equilibrium concepts are required tofind the solution. This issue was illustrated in Example 6 above, where two firms completed bysimultaneously choosing price (i.e., Bertrand duopoly). Each firm was a profit maximizer andthe equilibrium concept was Nash. With two firms, there was a first-order condition for eachfirm, and comparative static analysis required us to differentiate each first-order condition withrespect to the parameter in question and to use Cramer’s rule to solve the resulting system ofequations. This approach suffers from the so called curse of dimensionality, as finding thesolution to such games becomes excessively tedious as the number of firms and the number ofchoice variables (e.g., price, advertising, product quality) increase. In addition, this techniquecannot be used when the relevant functions are not differentiable.

Fortunately, comparative static analysis can still be derived using monotone methods. What is required is for the game to be supermodular. Because the main ideas are the same in thedifferentiable and non-differentiable cases, we will focus on problems where the curse ofdimensionality is at issue, not differentiability.10 These are called smooth supermodulargames. To illustrate, assume an industry has n firms and each firm has two strategic variables:price (p) and marketing expenditures (M). Each firm’s profit depends on its own and rival pricesand marketing. Firms compete in a smooth supermodular game, defined below.

Definition: Firms play a smooth supermodular game if the following conditions hold for eachfirm i and each rival j (Milgrom and Roberts, 1990, p. 1264).

(A1) Bounds on Strategies: pi and Ai each lie within a closed interval where {pi | 0 < piL #pi # piH} and {Mi | 0 < MiL # Mi # MiH}.

(A2) Differentiability: The profit equation is twice continuously differentiable withrespect to pi and Mi.

(A3) Complementary Strategies: M2Bi /Mpi MMi $ 0.

15

11A strict inequality will hold if the best-reply functions in prices have a positive slopeand if an increase in advertising shifts the best-reply functions away from the origin. Best-replyfunctions will generally have a positive slope in a differentiated Bertrand-type game where firmscompete in prices (Bulow et al., 1985, and Tirole, 1988, chapter 5). As will be seen shortly,advertising can shift best-reply functions toward or away from the origin, causing prices todecrease or increase.

(A4) Strategic Complements: M2Bi /Mpi Mpj $ 0, M2Bi /Mpi MMj $ 0, M2Bi /MMi Mpj $ 0, and M2Bi /MMi MMj $ 0.

Assume further that a policy parameter, a, has the following qualities: M2Bi /Mpi Ma $ 0and M2Bi /MMi Ma $ 0. That is, each firm’s profit equation exhibits increasing marginal returns inprice and marketing for an increase in a. This assumption is identified as (A5). Such a policycould include an excise tax that raises the marginal returns to raising price or a governmentsubsidy to marketing that raises the marginal returns in marketing. Under these conditions andassuming a unique Nash equilibrium, the following comparative static results hold for all firms(Milgrom and Roberts, 1990, Theorem 6):

∂∂

∂∂

pa

Ma

i i* *

; .≥ ≥0 0

That is, an excise tax that increases prices will have a non-negative effect on the Nashequilibrium level of marketing, and a subsidy that increases marketing will have a non-negativeeffect on Nash equilibirum prices in this supermodular setting.11

The proof of this result requires the use of lattices, so it will not be presented here. Themain idea is intuitive, however. The driving force behind the proof in the case of a strictinequality is the super-complementarity assumption. That is, any policy that increases pi

* (Mi *),

causes Mi * (pi

*) to rise because own choice variables are complements (assumption A3), and italso causes pj

* and Mj * to rise for all j because rival choice variables are strategic complements

(assumption A4). This causes a chain of events that reinforces these increases. That is, theresulting increases in pj

* and Mj * cause further increases in pi

* and Mi * etc. In terms of best-

reply functions, this means that the policy change causes one or both of the best reply functionsfor each choice variable to shift away from the origin (e.g., from equilibrium A to B in Figure 5for choice variable x). Thus, the Nash equilibrium, where the best reply functions intersect, willsupport higher levels of the strategic variables. The following example illustrates three comparative static approaches to an extension ofExample 6 above for linear demand and cost functions.

Example 8 (Example 6 with Specific Functional Forms): In a differentiated Bertrand model,two firms (1 and 2) face linear demand and costs. Firm i’s respective demand and costs are: qi =a - b pi + d qj and TCi = (c - t) qi, where a > c > 0 and b > 2d > 0. Policy Parameter t represents aper-unit tax on the firm. Thus, the firm i’s profit equals: B = ( pi - c - t)(a - b pi + d pj ). In thiscase, the respective first and second derivatives of the profit equation are:

16

π

π π

π π

π

i i j

ii jj

ij ji

it

a bp dp b c t

b

d

b

= − + + +

= = −

= =

=

2

2

( ),

,

,

.With this notation, Bi / MB /Mpi, Bii / M2B /Mpi

2, Bij / M2B /Mpi Mpj, and Bit / M2B /Mpi Mt for firm i. To compare approaches, each of the three techniques discussed in this paper is used to determinethe effect of the excise tax on Nash equilibrium prices.

(1) Brute Force Method: Because specific functional forms are used, we can solve thesystem of first-order conditions for p1 and p2 and differentiate. Because the problem issymmetric, the Nash equilibrium prices will be the same fore each firm and equal

p a b c td bi

* ( ) .=+ +

−2As a result, Mpi

* /Mt = b/(2d - b) > 0, because 2d - b is greater than 0 by definition.

(2) Implicit-Function Rule and Cramer’s Rule: From Example 6, we have seen that

∂∂α

π π

π πpi

it ij

jt jj*

.=

−

−

Π

The determinant of A = 4b2 - d2 and must be positive for stability. This condition holds,because b > 2d by assumption. The numerator in the above equation equals -Bit Bjj + BjtBij = -(b)(-2d) + (b)(d) > 0. Thus, Mpi

* /Mt > 0.

(3) Supermodularity Theorem: Alternatively, because the game is supermodular, Mpi

*/Mt > 0 by the supermodularity theorem. The supermodularity and complementarityparameter assumptions (A1-A5) are required for the theorem to hold and are verifiedbelow.

A1. This condition is met as long as pi 0 [0, 4).A2. The second derivatives of B are continuous.A3. This assumption is not relevant, because each firm has only one choice

variable.A4.Bij = d > 0, verifying that prices are strategic complements.A5.Bit = b > 0, the excise tax rate is a complementary exogenous parameter.

A strict inequality holds in this case, because an excise tax causes the best reply functionsof both firms to shift away from the origin and shift the Nash equilibrium from point A topoint B in Figure 6.

This example demonstrate how easy it is to use the supermodularity theorem.

17

5. ConclusionIn this note, I have reviewed the implicit-function theorem and its limitations. I also

present recent work using monotone methods. The strict monotonicty theorem provides a moregeneral method of doing comparative static analysis. When the assumptions required of theimplicit-function theorem are met, I have also shown that the implicit-function theorem is aspecial case of the strict monotonicity theorem. Finally, I have shown how tedious and difficultit can be to derive comparative static results from games with many players and strategicchoices. Fortunately, when the game is supermodular, the supermodularity theoremdemonstrates that comparative static analysis can be done with ease. I hope that his convincesyou to learn these modern methods and use them in future research.

18

Appendix A: Stability of the Bertrand-Nash Equilibrium

Consistent with the duopoly model in section 2.1, assume smooth best reply functionsand an unique interior Nash equilibrium. The graph of the best reply functions assumes that p1 is on the vertical axis and p2 on the horizontal axis. Stability requires that for anydisequilibrium set of prices in the neighborhood of the Nash equilibrium, the dynamic process ofeach firm responding to its rival’s disequilibrium price converges to the Nash prices. This willoccur when firm 2's best reply function is steeper than firm 1's best reply function. You shouldverify this fact by starting from a disequilibrium point.

As demonstrated above, best reply functions for firm’s 1 (p1* / b1) and 2 (p2

* / b2) willhave a positive slope. Recall that they are

b b112

11

221

22

=−

=−π

πππ

; .

Because we are interested in solving each best reply function for p1 (i.e., p1 is on the verticalaxis), the slope firm 2's best reply when p1 on the vertical axis is 1/b2. Thus, the Bertrand-Nashequilibrium will be stable iff 1/b2 > b1. Thus:

−>−

<

ππ

ππ

ππ

ππ

22

21

12

11

22

21

12

11

.

Because B21 > 0 and B11 < 0, this becomesπ π π π

π π π π

11 22 12 21

11 22 12 21 0

>

− > . Q.E.D.

19

References

Allen, Beth, “The Future of Microeconomic Theory,” Journal of Economic Perspectives, 14 (1),Winter 2000, 143-150.

Baldani, Jeffrey, James Bradfield, and Robert W. Turner, Mathematical Economics, ThomsonSouth-Western, 2005.

Bulow, Jeremy I., John D. Geanakoplos, and Paul Klemperer, “Multimarket Oligopoly: StrategicSubstitutes and Complements,” Journal of Political Economy, 93 (3), June 1985, 488-511.

Carter, Michael, Foundations of Mathematical Economics, MIT Press, 2001.

Chiang, Alpha C., and Kevin Wainwright, Fundamental Methods of Mathematical Economics,McGraw Hill Irwin, 2005.

Edlin, Aaron S., and Chris Shannon, “Strict Monotonicity in Comparative Statics,” Journal ofEconomic Theory, 81 (1), July 1998, 201-219.

Hands, Wade E., Introductory Mathematical Economics, New York: Oxford University Press,2004.

Iwasaki, Iwasaki, Yasushi Kudo, and Victor J. Tremblay, “Supermodularity and the Advertising-Price Relationship: Theory and Evidence,” working paper, Department of Economics, OregonState University, 2007.

Lang, Serge, Undergraduate Analysis, New York: Springer-Verlag, 1983.

McAfee, R. Preston, Competitive Solutions: The Strategist’s Toolkit, Princeton: PrincetonUniversity Press, 2002.

Milgrom, Paul, and John Roberts, “Rationalizability, Learning, and Equilibrium in Games ofStrategic Complementarities,” Econometrica, 58 (6), November 1990, 1255-1277.

Milgrom, Paul, and Chris Shannon, “Monotone Comparative Statics,” Econometrica, 62 (1),January 1994, 157-180.

Ok, Efe A., Real Analysis with Economic Applications, Princeton: Princeton University Press,2007.

Quah, John K.-H., “The Comparative Statics of Constrained Optimization Problems,”Econometrica, 75 (2), March 2007, 401-431.

20

Scarf, Herbert E., “The Allocation of Resources in the Presence of Indivisibilities,” Journal ofEconomic Perspectives, 8 (4), Fall 1994. 111-128.

Shannon, Chris, “Weak and Strong Monotone Comparative Statics, Economic Theory, 5, 1995,209-227.

Silberberg, Eugene, The Structure of Economics: A Mathematical Analysis, New York: McGrawhill Publishing Company, 1990.

Simon, Carl P., and Lawrence Blume, Mathematics for Economists, W.W. Norton and Company,1994.

Sundaram, Rangarajan, A First Course in Optimization Theory, New York, NY: CambridgeUniversity Press, 1996.

Topkis, Donald M., Supermodularity and Complementarity, Princeton University Press, 1998.

Van Zandt, Timothy, “An Introduction to Monotone Comparative Statics,” INSTEAD,November 14, 2002.

Vives, Xavier, “Nash Equilibrium with Strategic Complementarities,” Journal of MathematicalEconomics, 19 (3), 1990, 305-321.

__________, Oligopoly Pricing: Old Ideas and New Tools, Cambridge, MA: The MIT Press,1999.

__________, “Complementarities and Games: New Developments,” Journal of EconomicLiterature, 43 (2), June 2005, 437-479.

__________, “Games with Strategic Complementarities: New Applications to IndustrialOrganization,” International Journal of Industrial Organization, 23 (7), September 2005, 625-637.

21

22

f x

x

slope

= 2

slope = 1

-1

-2

f '

f *

x ' x *

Figure 2a

( ) xxf ∂∂

x

Figure 2b

'x ∗x

2

1

-1

-2 ( ) xxf ∂∂ '

( ) xxf ∂∂ ∗

23

'x ∗x 'f

∗f

x

( )xf

Figure 3

( )xf

x 'f

∗f

∗= xx'

Figure 4

24

1x

2x

A B

BRx2

BRx1

Figure 5

1p

2p

A

B

BRp2

BRp1

Figure 6