TÍTULO V. Gobierno Escolar
PROHIBICIONES PARA LOS DOCENTES Y DIRECTIVOS
3.3.1 The social planner solution
In order to evaluate and compare the e¢ ciency of the two types of market illustrated before (i.e. fully integrated and labour restricted), I solve the social planner problem and use the solution as a benchmark. More speci…cally, I imagine a benevolent planner that maximizes the utility of a representative household by allocating input factors across all the …rms in the economy directly, subject only to the aggregate resources constraints, i.e.K and L (see Bilbiie, Ghironi and Melitz, 2008; Epifani and Gancia, 2011.) Thus, the social planner optimizes the distribution of input factors over the whole economy. Assuming both capital and labour to be inelastically supplied at the global level, she faces the following optimization problem :
7In the appendix, I will show that in a framework in which both capital and labour are segmented at
the sector level, the two input factors are distributed as labour in (34), that means they both turn out to be functions of the …rm’s relative productivity and its average size.
max ffkn;s;ln;sgNsn=1g S s=1 Y = 2 4 1 S1 S X s=1 1 N1 s s Ns X n n;skn;sl 1 n;s s ! s 3 5 1 :
Thus, capital and labour turn out to be distributed at the …rm level as follows:
kn;s = K s (1 s)(1 ) s n;ss SNs ; (35) ln;s = L s (1 s)(1 ) s n;ss SNs ;
where the global economy average productivity is given by the following function:
= 1 S S X s=1 s:
The model with full integration of input factors matches what found by the social planner only if within sector demand elasticities are equal. Furthermore, the distribution of production factors according to the model in which labour is assumed to be restricted at the sector level also matches (35), if within sector demand elasticities are the same and, moreover, labour supplied at the sector level is equal to
Ls=
L S
s 1 11
:
Finally, the aggregate economy output produced by the Social Planner is a simple CD function, as follows:
Y = K L1 : 3.3.2 E¢ ciency analysis
As it can be easily observed, neither the integrated setting nor the labour segmented one are able to provide a fully e¢ cient allocation of input factors, namely capital and labour, across industries and monopolistic …rms, as long as industry level demand elasticities are heterogeneous. This means that, in both the cases, market solution does not actually match the Social Planner allocation of input factors and, in turn, does not produce as much as the Social Planner does. Here, the aggregate output can be thought of as a measure of welfare for the entire economy. This is because transfers across all the agents 11The right-hand side of that equation comes from aggregating the labour demand given by (35) at
in this economy, aimed at compensating them for eventual losses due to movements in prices, are assumed to be allowed.
Indeed, I have to explain such an ine¢ ciency and how the integrated and the labour restricted settings di¤er in terms of distance from the Social Planner solution. Thus, the ine¢ ciency is simply explained by the fact that in both the competitive market models, in case of heterogeneity of intra-industry demand elasticities, the marginal rate of substi- tution between any two intermediate goods, say 1 and 2, that is the amount of good 2, Y2, that the consumer must be given in order to compensate her for a one-unit marginal
decrease in the consumption of good 1, Y1, is di¤erent from the marginal rate of trans-
formation, that is the rate at which they can be transformed into each other11 Therefore,
suppose that there are only two industrial sectors, 1 and 2 and, particularly, the two sectors di¤er by demand elasticity, that is:
1 6= 2:
As I said above, in order to have an e¢ cient allocation, it must be: M RSY1;Y2 = M RTY1;Y2:
On the one hand, the Marginal Rate of Transformation (MRT) is equal to: M RTY1;Y2 =
2 1
:
On the other hand, the Marginal Rate of Substitution (MRS) takes on di¤erent values according to the economy I set up. Particularly, in case of integrated economy, the MRS is equal to
M RSY1;Y2 =
2 2 1 1
:
So, as evident, there is a wedge between the MRT and the MRS, that is given by the intra- industry demand elasticities ratio, 2= 1.12 Where does that wedge come from? It actually
depends on the asymmetry in monopoly power across sectors. As long as producers have got some monopoly power, the marginal cost of production will be lower than the product price by the markup.13 The larger the markup, the larger the gap between marginal cost
and price of the produced good. If demand elasticities are heterogeneous, then production resources will tend to be overallocated within those sectors where the demand elasticity is lower, that is where the gap between marginal cost and price is relatively lower. However, 11See Adao, Correia and Teles (2003) and Bilbiie, Ghironi and Melitz (2008) as references for ine¢ cient
allocations of input factors.
12Indeed, if
2= 1, then the wedge disappears and we turn to an e¢ cient allocation. 13See also Epifani and Gancia 2011 as a reference for ine¢ cient allocation cases.
as Lerner (1934) argued, when markups are homogeneous, their distorting e¤ect vanishes. Instead, in case of labour segmented economy, the MRS turns out to be di¤erent from the previous case:
M RSY1;Y2 = 2 2L 1 2 1 1L 1 1 ! 1 1 :
Thus, the MRS is not only a function of both the relative productivity and the relative demand elasticity, but also of the exogenous sector-level labour allocation. Intuitively, this might lead to an improvement in terms of e¢ ciency as the distortion implied by the demand elasticity ratio might be, even partially, compensated by the distortion given by the ratio of sector-level labour supplies, if they go to the opposite direction. For example, demand elasticity in sector 1 is larger than demand elasticity in sector 2, then it will result there is overallocation of production resources into sector 1; however, such overallocation can be mitigated if the exogeneous labour allocation in sector 1 is lower than the exogeneous labour allocation in sector 2 (assuming that industry level aggregate productivities are the same).
Particularly, if exogenous labour allocation matches the Social Planner allocation, it results that in the labour segmented context, the marginal rate of substitution between Y1 and Y2 will be:
M RS1;2 = 2 1 2 1 ; where = (1 ) 1 :
As long as the power to which the demand elasticities ratio is raised is lower than one, that is
< 1;
the gap between MRS and MRT is lower than in the integrated economy:
2 1 (1 ) 1 < 2 1 :
So doing, even though I can not turn to the …rst best solution, an exogeneous distribution of labour across industrial sectors can be Pareto improving with respect to the fully integrated market solution. Furthermore, it is also possible to notice that
that is the larger the output elasticity of capital, i.e. , the larger , and then the relatively farer I am to the …rst best solution (i.e. = 0). In conclusion, when 1 is larger than 2 ,
there is overallocation of both capital and labour within sector 1: by labour segmentation, I can exogenously withdraw labour from that sector and move that to sector 2 (where it was originally underallocated), so as to reduce the overallocation of labour. In particular, the lower the output elasticity of capital , i.e. the lower the contribute of capital to production, the more e¤ective labour restriction will be in the direction of a …rst best allocation of input factors.