Dixit-Stiglitz (1977) type model (version #1)(Lecture Notes, Thomas Steger, University of Leipzig, summer term 13)

� The setup

Consider a static economy with two sectors. Production technologies are as follows

(1)Final output : X = âi=1

N

xiΑ

1

Α

(2)Intermediate goods : xi = Li - F " i Î 81, ..., N<Production takes place according to the following sequence: (i) x-production; (ii) X-production. On the household side there is a continuum of

mass one of identical households. Every household is endowed with L units of time which are supplied inelastically to the labor market. Total

labor supply hence is L. Total labor demand is Úi=1N Li (in symmetric equilibrium this is simply N Li).

We also assume that there is an infinite number of possible varieties xi. Setting up a new firm is associated with fixed costs F > 0.

Observation: The reduced form technology (under symmetry and noting that N xi = N Li - N F = L - N F implies x =LN

- F) reads as follows

(3)X = âi=1

N

xiΑ

1

Α

= N1

Α x = N1

Α

L

N- F = N

1-Α

Α L - N1

Α F

That is, labor productivity is positively associated to the number of varieties N.

Definition: A (general) equilibrium in this economy consists of quantities 8Li, xi< and prices {P, pi, w} such that

(1) X-producers maximize profits.

(2) x-producers maximize profits.

(3) The labor market clears, i.e. LD = LS (where LD = Úi=1N Li and LS = L).

(4) The intermediate goods market clears, i.e. Úi=1N xi

S= Úi=1

N xiD).

(5) The market for the final output good clears, i.e. XS = XD.

Assignments(i) Assume that F increases by 10% (this may reflect a drop in the productivity of the bureaucracy as a result of political change). By how

much does output decline? Provide a concise economic reasoning.(ii) Assume that the labor force L increases by 10% (this may reflect immigration). By how much does output per capita change? Provide a

concise economic reasoning.

(iii) Determine the price index P for intermediate goods prices in equilibrium, which allows us to write P X = Úi=1N xi pi.

� Solution

We proceed in the following steps:(1) Determine the demand, as articulated by X-producers, for xi.

(2) Determine the optimal supply price of the typical xi-producer.

(3) Determine the demand for labor, as articulated by the typical xi-producer.

(4) Determine the equilibrium wage rate.(5) From the free-entry condition (Πi = pi xi - w Hxi + FL = 0) determine N.

Demand for x-goods. The typical X-producer solves

(4)maxxi

:P âi=1

N

xiΑ

1

Α

- âi=1

N

pi xi>The first-order conditions read as follows

(5)P 1

Α â

i=1

N

xiΑ

1

Α-1

Α xiΑ-1

- pi = 0 for all i Î 81, ..., N<

Using X1-Α = IÚi=1N xi

ΑM 1

Α-1

one gets

Using X1-Α = IÚi=1N xi

ΑM 1

Α-1

one gets

(6)P X1-Α xiΑ-1

- pi = 0

(7)X1-Α xiΑ-1

=pi

P

(8)Þ xi = K pi

PO

1

Α-1

X Hdemand scheduleL

(9)Þ pi = K xi

XOΑ-1

P Hinverse demand scheduleLOptimal supply price of x-producers. Profit of the typical xi-producer is given by

(10)Πi = pi xi - w Li = P X1-Α xi

Α

pi xi

- w Hxi + FL

The first-order condition for a profit maximum may be stated as

(11)¶ Πi

¶ xi

= Α P X1-Α xiΑ-1

- w = 0 Þ xi =w

Α P X1-Α

1

Α-1

The associated price is easily found by plugging the optimal amount xi into the inverse demand schedule

(12)pi =

J w

Α P X1-ΑN 1

Α-1

X

Α-1

P =

w

Α P X1-Α

XΑ-1 P =

w

Α

Demand for labor. Noting xi = I pi

PM 1

Α-1 X, pi =wΑ

, and xi = Li - F, the demand for labor is given by

(13)Li =w

Α P

1

Α-1

X + F

Aggregate labor demand under symmetry reads

(14)N Li =w

Α P

1

Α-1

X N + F N

The equilibrium wage rate. Using N Li = L (equilibrium in the labor market) the wage rate may be expressed as

(15)w = Α PX N

L - F N

1-Α

Noting the reduced form technology X = N1

Α I LN

- FM one gets

(16)w = Α PN

1

Α I LN

- FM N

L - F N

1-Α

= Α PN

1

Α H L - F NLL - F N

1-Α

= Α P N1-Α

Α

The number of varieties. The number of varieties in equilibrium results from the free entry condition Πi = 0. To evaluate this condition one

needs to determine equilibrium profits.

(17)Πi = pi xi - w Hxi + FL = piHxi - Α xi - Α FL = 0

Noting that w = Α pi and xi = Li - F one gets

(18)xi =Α F

H1 - ΑL Li =F

H1 - ΑLSince N xi = N Li - N F = L - N F, the number of varieties in equilibrium is determined by

(19)N* =H1 - ΑL L

F

� Economic interpretation of N*

2 Dixit_Stiglitz_SS13.nb

�

Economic interpretation of N*

The higher the fixed costs F associated with setting up a firm, the lower is the number varieties in equilibrium N*. The lower N*, the lower is

labor productivity and hence output. Now the economically interesting aspect is the following. F can be given several distinct interpretations:

It may be interpreted as an (excessive) fee paid to the government (® red tape, unproductive bureaucracy), it may be interpreted as anoutcome of corruption (® consequences of corruption/ bad institutions);

It may also be interpreted as the wage bill for skilled labor (® low education and / or emigration of skilled labor).

Þ The model therefore provides a model-based reasoning for why an economic environment with either unproductive bureaucracy, corrup-tion, or a shortage of skilled labor may induce a low labor productivity and hence a low level of per capita income.

Source: Djankov et al. (QJE, 2002)

� Assignments

(i) Output elasticity I. Notice X = N1-Α

Α L - N1

Α F from (3) and substitute N* to get

(20)X =H1 - ΑL L

F

1-Α

Α

L -H1 - ΑL L

F

1

Α

F = BH1 - ΑL 1-Α

Α - H1 - ΑL 1

Α F L1

Α FΑ-1

Α

The elasticity of output X with respect to F is given by

(21)¶ X

¶F F

X=

Α - 1

Α< 0

If F increases by 1 percent, output declines by Α-1Α

percent. Higher fixed costs F lead to a lower number of varieties N* and therefore to lower

labor productivity and output in equilirbium.

(ii) Output elasticity II. Output per capita is given by

(22)X

L= X

~

= BH1 - ΑL 1-Α

Α - H1 - ΑL 1

Α F L1-Α

Α FΑ-1

Α

The elasticity of output per capita X~

with respect to L is given by

(23)¶ X

~

¶L L

X~

=1 - Α

Α> 0

If L increases by 1 percent, output per capita increases by 1-Α

Α percent. An increase of the labor force L increases the number of varieties N*

and therefore increases labor productivity and output per capita in equilibrium.

(iii) The CES-price index. Noting the demand schedule xi = I pi

PM 1

Α-1 X we have

(24)âi=1

N

pi xi = âi=1

N

pi pi

1

Α-1 P1

1-Α X = P1

1-Α X âi=1

N

pi

Α

Α-1 note : 1 +1

Α - 1=

Α - 1 + 1

Α - 1=

Α

Α - 1

The X-sector is perfectly competitive and there is free entry. Hence, profits must vanish in equilibrium, i.e. P X = Úi=1N pi xi. This gives us

(25)P X = P1

1-Α X âi=1

N

pi

Α

Α-1

(26)P1-

1

1-Α = âi=1

N

pi

Α

Α-1

(27)P-Α

1-Α = âi=1

N

pi-

Α

1-Α

(28)P = âi=1

N

pi-

Α

1-Α

-1-Α

Α

Notice that this ideal price index allows us to write P X = Úi=1N xi pi. The price index times a quantity index (P X) equals total expenditures for

intermediates goods (Úi=1N pi xi).

Without loss of generality, we can set P = Úi=1N pi

Α

Α-1

Α-1

Α

= 1. This implies that pi becomes a relative price (initially pi

P has denoted the relative

price of xi).

4 Dixit_Stiglitz_SS13.nb