2.3 Firms’ Heterogeneity in New Economic Geography Models
2.3.1 Firms’ Heterogeneity and Spatial Selection
The model presented by Baldwin and Okubo (2005) can be seen as a marriage between the footloose NEG capital model conceived by Martin and Rogers (1995) extended for allowing heterogeneity in firms’ marginal cost (Melitz, 2003). The basic set up is the same as the footlose capital model already presented. The only extensions concern the already mentioned firms hetero- geneity and the quadratic adjustment costs faced by firms when switching regions.
The Basic Model
The heterogeneity is modeled by assuming firms to have different unit in- put coefficients (different a). Each firms needs a unit of knowledge to start production, hence the source of heterogeneity can be assigned to knowledge capital. It is then assumed that each unit of capital in each region is asso- ciated with a particular level of productive efficiency measured by the unit labor requirements a which are Pareto distributed:
G(a) = a
ρ
aρ0
, 1 ≡ a0 ≥ a ≥ 0, ρ ≥ 1
where a0 < ∞ is the highest possible marginal cost (normalized to unity)
and ρ is the shape parameter.
The second deviation from the footloose capital model stems from the fact that relocation is subject to quadratic adjustment costs. Indeed the cost of switching regions is χ units of labor per firms:
χ = γm
Short Run Equilibrium
Dixit-Stiglitz monopolistic competition maximization yields the following northern and southern operating profits as function of productivity level:
π(a) = a1−σ sE ∆ + φ (1 − sE) ∆∗ Ew Kwσ π∗(a) = a1−σ φsE ∆ + 1 − sE ∆∗ Ew Kwσ where: ∆ = λ (sn+ φ (1 − sn)) , ∆∗ = λ (φsn+ 1 − sn) , λ = ρ 1 − σ + ρ > 0 The deltas are a measure of the degree of competition in the market, while a1−σ can be seen as the competitiveness of a firm with marginal cost a. Combining these two facts we have that a firm’s market share a1−σ/∆Kw
depends upon its relative competitiveness.
Now, according to the Home Market Effect the big market (in the case at hand the north) will a more than proportional share of industry. Let’s see now which firm moves first. The change in operating profits from a single firm moving from south to north, considering an initial situation where no firms have moved (sn = sK) and using the symmetry of region’s relative
factor endowments (sE = sK > 1/2) is:
π(a) − π∗(a) = a1−σ (1 − φ) E w λσKw 2φ s −1 2 ((1 − φ) s + φ) (1 − s + φs)
where s is the north share of E and K. From this equation we can see that southern firms would move to north, that no northern firms would gain from moving to south and that most efficient southern firms would gain more by relocating.
It is pretty obvious that the most efficient firms, gaining most from de- location, are the ones willing to pay the quadratic delocation costs. But of course there is a feedback from migration and the market crowding effect through ∆ :
∆ = λ s + (1 − s) a1−σ+ρR + φ (1 − s) 1 − a1−σ+ρR ∆∗ = λ φs + φ (1 − s) a1−σ+ρR + (1 − s) 1 − a1−σ+ρR
where aR is the threshold level for marginal costs migration. Using this
expression it can be displayed the value of delocation of any southern firm as a function of its own marginal cost and the range of firms already moved:
v(a, aR) = π (a, aR)−π∗(a, aR) = a1−σ
sE ∆(aR) +φ (1 − sE) ∆∗(a R) Ew Kwσ−a 1−σ φsE ∆ (aR) + 1 − sE ∆∗(a R) Ew Kwσ
Given that the southern firms in north is K∗aρR the cost of moving becomes:
χ = γK∗ρaρ−1R ˙aR
Firms will move until the benefits of doing so will be greater or equal to the costs. The marginal cost a will be pinned down by the equality between benefits and costs of migrating, so the value of the marginal firms of migration will be:
v(aR) = γK∗ρaρ−1R ˙aR
This function is declining in aR, hence we have that most efficient firms will
move first.
The Long Run Equilibrium
In the long run equilibrium delocation does no longer take place, hence the marginal adjustment costs and the location condition v(aR) are zero. Solving
for the cut-off level of marginal costs aR:
a1−σ+ρR = 2φ s − 1 2 (1 − φ) (1 − s); sn = s + (1 − s) a ρ R
We can see that if trade gets freer more inefficient firms will delocate. The threshold in trade costs for complete delocation is the same ad in the standard footloose capital model:
φCP = 1 − s s
Summarizing all the stability results we have that heterogeneity leads to a spatial selection effect according to which most efficient firms are the first to move to the bigger market. This implies a Home Market Magnification Effect: the big market attract more than its usual firms’ share.
Selection Bias
For testing the agglomeration the average productivity of a region must be re- lated to the amount of industry in the region itself . The simplest framework is:
ln (lprodr) = c + α ln (snr) + ε
where lprodris the labor productivity in region r and snris the share of indus-
try in region r. In this case α would measure the impact of an increase in the share of industry in one region on the labor productivity in this same region. Performing a similar test for the footloose capital model we have that north’s labor productivity is given by the ratio between the real value of manufac- turing output (i.e. northern manufacturing revenue) np1−σ(E/∆ + φE∗/∆∗) and the total labor input ap−σ(E/∆ + φE∗/∆∗) . Then due to mill pricing the north’s labor productivity will be 1/ (1 − 1/σ). Converting to real terms by dividing for the consumer price index (np1−σ)1/(1−σ) the labor productivity measure becomes: ln (lprodr) = ln s 1 σ−1 n a (1 − 1/σ)
As we can see the labor productivity increases with the share on industry in the north: this is the measure of agglomeration economies. By adding heterogeneity we have that the value of output isR p (i)1−σ
(E/∆ + φE∗/∆∗) and the total labor input R a (i) p (i)−σ(E/∆ + φE∗/∆∗) . Then the labor productivity measure becomes:
ln (lprodr)het= ln λK + λK∗a1−σ+ρR 1 σ−1 a (1 − 1/σ)
Clearly the estimate of agglomeration economies would be overestimated because firms relocated in the north would have systematically higher than average productivity. So there would be a bias in the standard econometric tests for agglomeration.
Regional Policy Implications
Starting form a core-periphery situation assume a regional policy paying firms a subsidy S for moving from the large to the small region. The change
in the operating profits for a firm moving from north to south (not including subsidies) would be:
a1−σ(1 − φ) E w λσ 1 − s φ − s < 0
It is clear that the loss of relocation is decreasing with the firm’s marginal cost a. So the first firms to move to the small region in response to a subsidy would be the less efficient ones. Now let’s see the precise relation between the subsidy and the cut-off marginal cost. If all firms with marginal cost higher than aS would move to the south:
∆ = λ a1−σ+ρS + φ 1 − a1−σ+ρS
∆∗ = λ φa1−σ+ρR + 1 − a1−σ+ρR
Thus the change in the operating profits for a firm moving from north to south including subsidies becomes:
a1−σS Sσ Ew = (1 − φ) s ∆− 1 − s ∆∗
The left hand side of the expression is increasing in aS. By contrast the
right hand side is always decreasing in aS because the competition in north
falls as aS rises. Hence there is only one solution for aS. It is clear that an
increase in subsidies raise the left hand side (the right is not affected) hence there is a decrease in the cut-off level of efficiency. On the contrary a decrease in trade costs would lower the right hand side not affecting the left hand one, so subsidies become more effective ad trade gets freer.
Summarizing it is clear that the subsidy is more effective in promoting relocation the large is the subsidy and the freer is trade, but the relocat- ing firms are the less efficient because are the ones gaining with the lower opportunity cost of leaving the big region.