Chapter 3: the core model

1
Chapter 3 the core model

• Krugman (1991)
• Two countries/regions/cities
• Here North (N) and South (S)
• Two industries agriculture (A) and manufacturing
  (M)
• named farmers and firms
• Location A exogenous, M endogenous
• Labour only production factor
• Firms/workers M mobile farms/workers A immobile
• No commuting workers live and spend their money
  in the region where they work
• M produces varieties/differentiated products
• M internal economies of scale/increasing returns
  one firm produces one variety

2
Innovation

• Key difference from New Trade Theory firms
  decide where to produce
• Geography becomes endogenous

3
Example

• Assumption for world ( NS) size of M is 40
  and size of A is 60, both in terms of production
  and employment
• 2/3 of farmers are located in N and 1/3 in S
  geography of farmers is exogenously given
• Consumers ( M-workers farmers) in N and S
  spend 40 of their income on M-goods and 60 on
  A-goods
• Demand for every M-product is 10 units every
  firm produces 10 units of its product/variety
• Every firm in N and S sells 4 units to M-workers
  and 6 units to farmers worldwide
• Of the 6 units of M-products sold to farmers, 4
  are sold to farmers in N and 2 to farmers in S
• Transport costs 1 euro per unit from N to S

4
Table 3.1 Geography of manufacturing sales of
the individual firm three non-exhaustive
geographies


Sales in North

Sales in South

Total Sales

All firms in North

4 4 8

0 2 2

10

All firms in South

0 4 4

4 2 6

10


25 firms in North




75 firms in South

1 4
5

3 2 5

10

• Note
• workers are where the firm is no commuting
• workers and farmers are the only consumers no
  entrepreneur consumption

5
Table 3.2 Transport costs for individual firm

• assume prices are given firms minimize transport
  costs by choosing their location
• a firm will locate in N if all other firms are
  in N (situation A)
• a firm will locate in S if all other firms are
  in S (situation B)
• situation C a firm is indifferent but one firm
  moving will tip the balance

6
Table 3.2 Transport costs for individual firm

• Cumulative causation once a region has more
  firms others firms will also go there
• Multiple equilibria A and B are both possible
  and stable. In case A and B one firm changing its
  location will not cause others to follow
• C is unstable one firm moving will tip the
  balance and others will follow
• A stable equilibrium can be non-optimal total
  transport costs in A will be 2, in B they are 4
• Interaction of agglomeration and trade full
  agglomeration in A an B leads to pure
  interindustry trade. Intermediate agglomeration C
  leads to also intra-industry trade
• Home market effect products with a large home
  market are also exported

7
Figure 3.1 Structure of the core model of
Geographical Economics

8
The core model in writing

• U F1-d Md 0 lt d lt 1 dfraction of
  income spent on M
• F IM Y price F-good set to 1(numeraire)
• Maximization of U under Y (tn 3.1)
• F ( 1-d) Y IM dY

• (Nc?)1/? N1/? c
  N(1/?)-1 Nc
• (if ci c)
• if ? 1 no love of variety 100 pairs of the
  same shoes same utility as 100 different pairs of
  shoes products are perfect substitutes N100
  and c1 same utility as N1 and c100
• if 0 lt ? lt 1 N100 and c1 more utility than
  N1 and c100

9

• maximize M subject to income budget (tn 3.2)
• Si picidY
• e 1/1-? demand elasticity
• gives cj (pj )-e (Ie-1 dY)
• and M dY/I
• with

10
cj (pj )-e (Ie-1 dY) (pj )-e constant e is
the price elasticity of demand
11
Figure 3.3 Relationship between ? and e


12

• U F1-d Md
• M dY/I increasing N higher M lt-gt lower I
• I is exact price index
• real income y Y I d
• real wage w W I d

13
Figure 3.4 Deviation between assumed demand and
reality
cj (pj )-e (Ie-1 dY) (pj )-e constant but p1
is part of I ???

14
Figure 3.4 continued
15
Supply side

• F (1 ?) L ? fraction labour force in M
• M li a ß xi
• Profits p px W (a ß xi)
• Demand elasticity e leads to simple optimal
  pricing rule known as mark-up pricing (tn 3.3)
• p ßW/? ßW marginal cost and ?lt1
• So
• Every variety i has the same price pi because W
  is the same for every firm/variety the labor
  market is clearing

16
Supply side (cont)

• tn 3.4
• x a(e-1)/ß (when p 0)
• l ae
• N ?L/l ? L/ ae
• So
• Output per firm/variety is fixed
• M only expands or contracts by producing more or
  fewer variaties

17
Economies of scale

• AC/MC gt 1 economies of scale
• AC l/x ae/ a(e-1)/ß ß e/ (e-1)
• MC ß
• AC/MC e/ (e-1) -gt e is also a measure of
  economies of scale

18
Figure 3.6 Division of labor over the regions

Total number of laborers L
g
g
(1-
)
Laborers in the
Laborers in the
g
g
manufacturing sector
L
food sector (1-
)
L
f
f
l
l
1
2
1
2
Laborers in the
Laborers in the
Laborers in the
Laborers in the
food sector in
food sector in
manufacturing sector
manufacturing sector
f
g
f
g
l
g
l
g
region 2
(1-
)
L
region 1
(1-
)
L
in region 1
L
in region 2
L
2
1
1
2
Mobility (section 3.9)
f
f
l
l
Note


1
Note

1

1
2
1
2
19
Iceberg transport costs

• No transport sector
• Need to ship T units from N to S in order for 1
  unit to arrive the rest melts away (send 107
  flowers from Amsterdam to Paris for 100 to arrive
  fresh)
• Distance between N and S set to 1
• Price for N in N pN
• Price for N in S TpN

20
Regional prices (1)

• price charged in region 1 by firm in region 1
  p11 ßW1/?
• price charged in region 2 by firm in region 1
  p12 TßW1/?
• symmetrically
• price charged in region 2 by firm in region 2
  p22 ßW2/?
• price charged in region 1 by firm in region 2
  p21 TßW2/?
• In general
• price charged in region s by firm in region r
  prs Trs ßWr / ?

21
Regional prices (2)

• Remember 3.6
• nr of firms in region 1 and 2 ?1?L / ae
  resp. ?2?L / ae
• the regional price index is a weighted average of
  the prices of local goods and imported goods
• I1 N1 p111-e N2 p211-e 1/(1-e)
• ?1 ?L / ae (ßW1/?) 1-e ?2 ?L / ae
  (TßW2/?) 1-e 1/(1-e)
• (ß/?)(?L/ae)1/(1-e) ?1 W11-e ?2
  T1-eW21-e 1/(1-e)
• In general
• Ir (ß/?)(?L/ae)1/(1-e) Ss ?s Trs1-e
  Ws1-e 1/(1-e)

22
Regional wages (1)

• Only unknown in I is wages
• Solved by demand supply
• From 3.6 cj (pj )-e (Ie-1 dY) and 3.12 p
  ßW/? we get
• demand product 1 in region 1 (dß-e?e )(Y1 W1 -e
  I1 e-1 )
• demand product 1 in region 2 (dß-e?e )(Y2 W1 -e
  T -e I1 e-1 )
• total demand x1 (dß-e?e ) (Y1 W1 -e I1 e-1
  Y2 W1 -e T -e I1 e-1 )
• Supply x a (e -1)/ß (3.13)

23
Regional wages (2)

• Supply x1 demand in region 1 demand in
  region 2
• 
  extra production melted away
• a (e -1)/ß (dß-e?e ) (Y1 W1 -e I1e-1 Y2 W1
  -e T 1-e I1e-1 )
• leads to
• W1e ß1-e?e (d/(e-1)a) (Y1I1e-1 Y2 T1-e
  I2e-1)
• W1 ? ß-? (d/(e-1)a)1/e (Y1I1e-1 Y2 T1-e
  I2e-1)1/e
• In general
• Wr ? ß-? (d/((e-1)a))1/e ( Ss Ys Trs1-e
  Ise-1 )1/e

Remember e1/(1-?) ?(e-1)/e
24
Key equations 3.22 - 3.24 (p117) generalized

• Yr ?r Wr ? L fr ( 1 ?) L
• Ir (ß/?)(?L/ae)1/(1-e) ( Ss ?s Trs1-e
  Ws1-e )1/(1-e)
• Wr ? ß-? (d/((e-1)a))1/e ( Ss Ys Trs1-e
  Ise-1 )1/e

Y Income L labour force W
wage T transport costs ? fraction M-
employment d fraction spending on
M-goods ?r regional share M-employment fr
regional share A-employment ?
substitution elasticity e 1/(1- ?)
demand elasticity a fixed costs ß
variable costs
25
Real wage of farmers and M-workers the same
makes (ß/?)(?L/ae)1/(1-e) 1 Makes ? ß-?
(d/((e-1)a))1/e 1
26
Makes life simple(4.1 - 4.3)

• Yr d ?r Wr fr ( 1 d)
• Ir ( Ss ?s Trs1-e Ws1-e )1/(1-e)
• Wr ( Ss Ys Trs1-e Ise-1 )1/e

1-3 determine the short-term equilibrium (solve
for Y, I and W
27
One more thing

• One parameter T transport costs for 1 distance
  unit. In a 2-region model North/South this
  distance can always be set to 1 so Tns T
• Distance Drs between region r and s
• Transport costs Trs TDrs
• Normally Drs Dsr

28
Long term equilibrium

• N-workers react to regional wage differences
• wr Wr Ir-d
• d?1/?1 ?(w1 ?)
• ? national average real wage
• Long term equilibrium when wr ? in all regions
  -gt no more migration

29
When a worker moves to another region there are
three effects (p124-125)

• Price effect Ir falls with increasing Nr
• Home market effect Nr increases more than
  proportionally with Yr
• Extent of competition effect for an individual
  firm in region r demand falls with falling Ir
• The balance between these forces determine
  whether, once a firm relocates, others will follow