The Economics of Space 433: Session 1 An …...The Economics of Space 433: Session 1 An Introduction...

The Economics of Space 433: Session 1

An Introduction to Dixit-Stiglitz Preferences

Costas Arkolakis and Eduardo Fraga1

1Yale University

25 September 2018

The Economics of Space, ECON 433 Yale c© Costas Arkolakis and Eduardo Fraga 1

Dixit-Stiglitz Preferences

I In ‘‘classic’’ Intermediate Micro, we were used to dealing with Cobb-Douglas(C-D) utilityI Consumer’s utility function is: U(x1, x2) = xα

1 x1−α2

I x1 and x2 are the quantities consumed of each good (e.g. x1 = 2 oranges, x2 = 4apples)

I The consumer then proceeds to maximize this utility function subject to budgetconstraint: p1x1 + p2x2 = y

I It this class, we will focus on another type of utility: Dixit-Stigliz (D-S)

I Consumer’s utility function is: U(x1, x2) = [xσ−1σ

1 + xσ−1σ

2 ]σσ−1

I As in the C-D case, x1 and x2 are the quantities consumed of each goodI Also as in the C-D case, consumer maximizes utility subject to budget constraint:

p1x1 + p2x2 = y


Dixit-Stiglitz Preferences

I In ‘‘classic’’ Intermediate Micro, we were used to dealing with Cobb-Douglas(C-D) utilityI Consumer’s utility function is: U(x1, x2) = xα

1 x1−α2

I x1 and x2 are the quantities consumed of each good (e.g. x1 = 2 oranges, x2 = 4apples)

I The consumer then proceeds to maximize this utility function subject to budgetconstraint: p1x1 + p2x2 = y

I It this class, we will focus on another type of utility: Dixit-Stigliz (D-S)

I Consumer’s utility function is: U(x1, x2) = [xσ−1σ

1 + xσ−1σ

2 ]σσ−1

I As in the C-D case, x1 and x2 are the quantities consumed of each goodI Also as in the C-D case, consumer maximizes utility subject to budget constraint:

p1x1 + p2x2 = y


Setup

I In this Appendix, we derive facts and properties regarding consumer choice underD-S preferences

I We also want to show that these facts and properties can be derived in the sameway as in the C-D case

I Here’s our roadmap:I Derive facts and properties of C-D consumer choiceI Use exactly analogous procedure to derive facts and properties of D-S consumer

choice


Setup




choice


Setup



I Here’s our roadmap:

I Derive facts and properties of C-D consumer choiceI Use exactly analogous procedure to derive facts and properties of D-S consumer

choice


Setup



I Here’s our roadmap:I Derive facts and properties of C-D consumer choice

I Use exactly analogous procedure to derive facts and properties of D-S consumerchoice


Setup




choice


Before we Start

I A quick note on nomenclature:

I In this class we will often denote quantities consumed by c instead of the habitualxI E.g. c1 = 3, c2 = 6 means I consume 3 oranges and 6 apples

I Another change is that we will denote consumer income by w instead of y

I Hence, consumer maximization problem is:

maxc1,c2

U(c1,c2)

subject to p1c1 + p2c2 = w


Before we Start


I In this class we will often denote quantities consumed by c instead of the habitualx

I E.g. c1 = 3, c2 = 6 means I consume 3 oranges and 6 apples



maxc1,c2

U(c1,c2)



Before we Start





maxc1,c2

U(c1,c2)



Before we Start





maxc1,c2

U(c1,c2)



Before we Start





maxc1,c2

U(c1,c2)



C-D: the Set-up

I The consumer’s choice problem with C-D preferences can be written as:

maxc1,c2

cα1 c1−α2


I Note that from the consumer’s perspective, (p1, p2,w , α) are all exogenousvariables/parametersI That is, they are just ‘‘given’’ to the consumer. She has no direct influence on

their valueI Consumer takes them as inputs when making her consumption choice

I On the other hand, from the consumer’s perspective, c1, c2 are endogenousvariablesI The consumer chooses their value within her maximization program in order to

maximize utilityI We denote the specific ‘‘maximizing’’ values of c1, c2 that the consumer chooses as

(c∗1 , c∗2 ) and the corresponding ‘‘optimized’’ level of utility as U∗


C-D: Lagrangian and FOCs

I To solve the maximization probem, we start by writing the Lagrangean function

L = cα1 c1−α2 + µ (w − p1c1 − p2c2)

I The variable µ is the Lagrange multiplier

I Then take the first-order condition (FOC) of the Lagrangian with respect to c1and c2

∂L∂c1

= αcα−11 c1−α2 − µp1 = 0 (1)

∂L∂c2

= (1− α)cα1 c−α2 − µp2 = 0 (2)

I And don’t forget the budget constraint

p1c1 + p2c2 = w (3)


C-D: Lagrangian and FOCs

I To solve the maximization probem, we start by writing the Lagrangean function

L = cα1 c1−α2 + µ (w − p1c1 − p2c2)

I The variable µ is the Lagrange multiplier

I Then take the first-order condition (FOC) of the Lagrangian with respect to c1and c2

∂L∂c1

= αcα−11 c1−α2 − µp1 = 0 (1)

∂L∂c2

= (1− α)cα1 c−α2 − µp2 = 0 (2)

I And don’t forget the budget constraint

p1c1 + p2c2 = w (3)


C-D: Algebra 1

I Rewrite equations (1) and (2) to isolate the Lagrange multiplier µ in theright-hand side

αcα−11 c1−α2 = µp1

(1− α)cα1 c−α2 = µp2

I Divide the first of these two equations by the second

αcα−11 c1−α2

(1− α)cα1 c−α2

=µp1µp2

=⇒

αc2(1− α)c1

=p1p2

=⇒

c2 =(1− α)p1c1

αp2(4)


C-D: Algebra 2

I Now substitute c2 from equation (4) in the budget constraint (3)

p1c1 + p2[(1− α)p1c1

αp2] = w =⇒

p1c1[1 +(1− α)α

] = w =⇒

c∗1 =αw

p1(5)

I We have expressed c1 as a function of exogenous parameters only! Thus, we’vefound the consumer’s utility-maximizing choice, c∗1

I What about c∗2 ? To find it, substitute the c1 in equation (4) using the optimal c∗1from equation (5)

c∗2 =(1− α)p1αp2

[αw

p1]

c∗2 =(1− α)w

p2(6)


C-D: Algebra 2


p1c1 + p2[(1− α)p1c1

αp2] = w =⇒

p1c1[1 +(1− α)α

] = w =⇒

c∗1 =αw

p1(5)



c∗2 =(1− α)p1αp2

[αw

p1]

c∗2 =(1− α)w

p2(6)


C-D: Algebra 2


p1c1 + p2[(1− α)p1c1

αp2] = w =⇒

p1c1[1 +(1− α)α

] = w =⇒

c∗1 =αw

p1(5)



c∗2 =(1− α)p1αp2

[αw

p1]

c∗2 =(1− α)w

p2(6)


C-D: Solution, Utility

I We have found the utility-maximizing consumption bundle! It is the following

(c∗1 , c∗2 ) = (

αw

p1,(1− α)w

p2)

I To find the ‘‘optimized’’ utility level achieved by this optimal consumption choice,plug c∗1 , c

∗2 into the utility function:

U∗ = U(c∗1 , c∗2 ) = (c∗1 )

α(c∗2 )1−α =⇒

U∗ = (αw

p1)α(

(1− α)wp2

)1−α =⇒

U∗ = w(αα(1− α)1−α

pα1 p1−α2

) =⇒

U∗ =w

pα1 p1−α2

αα(1−α)1−α

(7)


C-D: Price Index

I From last slide, the ‘‘optimized’’ level of utility achieved by the consumer is

U∗ =w

pα1 p1−α2

αα(1−α)1−α

I If we defined a new variable P as P =pα1 p1−α

2

αα(1−α)1−α , we can then rewrite the utility

level simply as

U∗ =w

PI This saves a lot of notation!

I P is known as the price indexI Note that P is increasing in p1 and p2I Also note that the formula for P resembles that of C-D utility

I Thus, the precise form that the index assumes is influenced by the underlying utilityfunction

I If we had used a different utility function, the index may have been different!


C-D: Price Index


U∗ =w

pα1 p1−α2

αα(1−α)1−α


2


level simply as

U∗ =w






C-D: Price Index


U∗ =w

pα1 p1−α2

αα(1−α)1−α


2


level simply as

U∗ =w






C-D: Expenditure Shares

I Finally, let’s compute the expenditure shares of each goodI That is, the fraction of the consumer’s income that is spent on each goodI E.g. expenditure share of good 1 is: λ1 =

money spent on good 1total income

I Total money spent on good i is pic∗i , for i ∈ {1, 2}

I So expenditure share of good i is λi =pic∗i

w

I Substituting the values for from equations (5) and (6), we get

λ1 = α

λ2 = 1− αI Note that expenditure share is independent of prices or income

I E.g. if p1 goes up, consumer buys fewer units of c1 but spends more money oneach unit

I These two effects cancel out so that expenditure share on good 1 stay constant atα







w


λ1 = α










w


λ1 = α










w


λ1 = α

λ2 = 1− α

I Note that expenditure share is independent of prices or incomeI E.g. if p1 goes up, consumer buys fewer units of c1 but spends more money on

each unitI These two effects cancel out so that expenditure share on good 1 stay constant atα







w


λ1 = α





Moving on to D-S

I Having shown the facts and properties of consumer choice under Cobb-Douglaspreferences, we now move on to Dixit-Stiglitz preferences

I We will use the exact same sequence of steps as the ones we used to deriveresults for Cobb-Douglas preferences

I To make it even more explicit, we will present derivations for C-D and D-Sside-by-side


C-D vs. D-S: Set-Up

I The consumer’s utlity-maximization problem can be written in each case (C-Dand D-S) as follows

Cobb-Douglas case

maxc1,c2

cα1 c1−α2

subject to:

p1c1 + p2c2 = w

(8)

Dixit-Stiglitz case

maxc1,c2

[cσ−1σ

1 + cσ−1σ

2 ]σσ−1

subject to:

p1c1 + p2c2 = w

(9)


C-D vs. D-S: Lagrangean and FOCs

I Now, let us write the Lagrangean and take first-order conditions:

Cobb-Douglas case

L =cα1 c1−α2

+ µ(w − p1c1 − p2c2)(10)

Dixit-Stiglitz case

L =[cσ−1σ

1 + cσ−1σ

2 ]σσ−1

+ µ(w − p1c1 − p2c2)(11)

FOC(c1) :∂L

∂c1

= αcα−11

c1−α2

− µp1 = 0 (12)

FOC(c2) :∂L

∂c2

= (1 − α)cα1 c−α2− µp2 = 0 (13)

∂L

∂c1

=(σ

σ − 1)[c

σ−1σ

1+ c

σ−1σ

2]σσ−1−1

(σ − 1

σ)c

σ−1σ−1

1

− µp1 = 0

(14)

∂L

∂c2

=(σ

σ − 1)[c

σ−1σ

1+ c

σ−1σ

2]σσ−1−1

(σ − 1

σ)c

σ−1σ−1

2

− µp2 = 0

(15)


C-D vs. D-S: Algebra 1

I Algebraically manipulating equations (12)-(15) a bit, we get the following:

Cobb-Douglas case

αcα−11

c1−α2

= µp1

(1 − α)cα1 c−α2

= µp2

Dixit-Stiglitz case

[c

σ−1σ

1+ c

σ−1σ

2]σσ−1−1

c

σ−1σ−1

1= µp1

[c

σ−1σ

1+ c

σ−1σ

2]σσ−1−1

c

σ−1σ−1

2= µp2

I Dividing the top equation by the bottom equation, we get:

αc2(1− α)c1

=p1p2⇒

c2 =(1− α)p1αp2

c1

(16)

cσ−1σ −1

1

cσ−1σ −1

2

=p1p2⇒

c2 = (p2p1

)−σc1

(17)


C-D vs. D-S: Algebra 2

I Substituting equations (16)-(17) into budget constraints, we get:

Cobb-Douglas case

p1c1 + p2

((1− α)p1αp2

c1

)= w ⇒

c∗1 =αw

p1(18)

Dixit-Stiglitz case

p1c1 + p2

((p2p1

)−σc1

)= w ⇒

c∗1 =p−σ1 w

p1−σ1 + p1−σ2

(19)

I We just obtained c∗1 , the optimal consumption choice for good 1!I Now plug it back into equations (16)-(17) to obtain the optimal consumption

choice for good 2:

c∗2 =(1− α)w

p2(20) c∗2 =

p−σ2 w

p1−σ1 + p1−σ2

(21)The Economics of Space, ECON 433 Yale c© Costas Arkolakis and Eduardo Fraga 34

C-D vs. D-S: Solution, Utility

I Thus, the utility-maximizing choice bundle in each case is:

Cobb-Douglas case

(c∗1 , c∗2 ) =

(αw

p1

,(1 − α)w

p2

)Dixit-Stiglitz case

(c∗1 , c∗2 ) =

(p−σ1

w

p1−σ1

+ p1−σ2

,p−σ2

w

p1−σ1

+ p1−σ2

)

I We can then find the achieved level of utility U∗ = U(c∗1 , c∗2 ) by plugging c∗1 and

c∗2 into the utility function:

U∗ = (c∗1 )α(c∗2 )1−α

=w(

pα1

p1−α2

αα(1−α)1−α

) (22)

U∗ = [(c∗1 )σ−1σ + (c∗2 )

σ−1σ ]

σσ−1

=

[(

p−σ1

w

p1−σ1

+ p1−σ2

)σ−1σ + (

p−σ2

w

p1−σ1

+ p1−σ2

)σ−1σ

] σσ−1

=w

[p1−σ1

+ p1−σ2

]1

1−σ

(23)


C-D vs. D-S: Price Index

I For each case, let us conveniently define the price index, P, so that we canexpress utility as U∗ = w

P :

Cobb-Douglas case

P ≡ pα1 p1−α2

αα(1− α)1−α(24)

Dixit-Stiglitz case

P ≡ [p1−σ1 + p1−σ2 ]1

1−σ (25)

I Note that, in both cases, increases in either p1 or p2 cause the price index toincrease

I Also note that, in both cases, the formula of the price index bear some similaritiesto the utility functionI Once more, the form of the index is influenced by the underlying utility function


C-D vs. D-S: Expenditure Shares

I As we mentioned above, the expenditure share of a good i is λi =pic∗i

w , fori ∈ {1, 2}

I So, substituting the values of c∗i from equations (18)-(21), we get in each case:

Cobb-Douglas case

λ1 = α

λ2 = 1− α(26)

Dixit-Stiglitz case

λ1 =p1−σ1

p1−σ1 + p1−σ2

λ2 =p1−σ2

p1−σ1 + p1−σ2

(27)

I Expenditure shares are affected by prices in the D-S case, but not in the C-D caseI In the D-S case, a higher p1 makes the consumer spend less money on good 1I That is, the ‘‘fewer units’’ effect dominates the ‘‘higher price per unit’’ effect

I In both the C-D and D-S case, λ1 + λ2 = 1I This makes sense: if I add up the fractions of my income that I spend on each

good, I must get 100%


D-S: Expenditure Shares

I Note the structure of the formula for D-S expenditure shares:

λ1 =p1−σ1

p1−σ1 + p1−σ2

λ2 =p1−σ2

p1−σ1 + p1−σ2

I For each good, the formula has the structureK”that good”

K”summed across all goods”

I It turns out that this structure generalizes to the D-S case with not only 2 but Ngoods (which we didn’t tackle in this Appendix)

I In that case, the expenditure share of each good i would be: λi =p1−σi∑N

s=1 p1−σsfor

i ∈ {1, 2, ...,N}


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The Economics of Space 433: Session 1 An …...The Economics of Space 433: Session 1 An Introduction...

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