MARCO TEÓRICO
2.4. SISTEMA SOLAR FOTOVOLTAICO
2.4.2. SUBSISTEMA DE ALMACENAMIENTO
For illustration, I assume that utility and production in each sector obey a Cobb-Douglas functional form. Doing so allows me to establish sufficiency of the KKT conditions, existence of an interior solution and uniqueness of the market outcome. Furthermore, functional form assumptions are required for the estimation of agglomeration elasticities.
Local labor demand
Production of tradables I assume production of the tradable good as
FX(.) =AjX
KXj αLjXβNXj1−α−β. When firms take regional population as given, first
order conditions imply the log local labor demand curve lnwj = 1 1−αlnA j X− α 1−αlni j+ β 1−αlnL j X− β 1−αlnN j X+ α 1−αlnα+ ln (1−α−β) (A.7) where the first order condition for capital has been substituted.
Equation (1.4) follows directly from the definition of σj and the fact that the wage in a region is proportional to average output per worker:
dFXj dNX = FXj NX 1−α−β+σj=wj + w j 1−α−βσ j (A.8)
Assumption on the agglomeration function If the agglomeration function is log-linear (as suggested by the results inKline and Moretti (2014a)), i.e.
g(.) = σlnNX
R +c (A.9)
where σ is the constant agglomeration elasticity, we have ∂lnwj
∂lnNXj = σ−β
1−α. If, in addition,
σ < β (also suggested by the results in Kline and Moretti (2014a)), the labor demand curve
economies such that the crowding of the fixed factor land outweighs the productivity gain of adding an additional worker. Notice that when estimating agglomeration elasticities, I do not restrict the functional form or the elasticities. The empirical evidence reported in section 1.6 supports the assumption of log-linearity andσ < β.
Overall local labor demand Cobb-Douglas production of housing and the publicly provided good is analogous to the tradable good except that productivity is exogenous. This implies downward sloping labor demand as in equation (A.7) in both sectors. Summing labor demand across sectors yields overall downward-sloping labor demand.
Local labor supply
ForU(x, y, gj, Qj) =Qj(gj)˜γ(y)α˜(x)β˜ with ˜α+ ˜β+ ˜γ = 1, agents solve (in a given location)
maxx,y lnQ+ ˜γlng+ ˜αlny+ ˜βlnx s.t. pYy+x=w(1−τwF) +F =: ˜w,
where ˜w denotes income after taxes and transfers. This yields demand proportional to
income: x= α˜+ ˜β˜βw˜ and y= ( ˜αα+ ˜˜β)wr˜. Regional log housing demand then reads
lnHd= lnN + ln ˜w−lnpY +c1
for some constantc1.
Production of housing is analogous to production of the tradable good (with land being a fixed factor) but with exogenous productivity: FY(.) =AjY
KYjα y LjYβ y NYj1−α y−βy . The implied log housing supply is a linear function of the log wage and price:
lnHs = 1−β
y
βy lnpY +
αy +βy −1
βy lnw+c2
Solving for effective local labor supply amounts to solving for local housing market equilibria, i.e. imposing lnN + ln ˜w−lnpY +c1 = 1−βy βy lnpY + αy+βy−1 βy lnw+c2 which yields lnpY = (1−αy−βy) lnw+βyln ˜w+βylnN +c3
Equilibrium across locations requires ln ¯U = lnUj ∀j such that, usingg = NGω, local labor supply Nj is determined by
Figure A.1: Local labor market equilibrium
ln N
ln
w
log labor supply log labor demand
ln ¯U = lnQj+ ˜γlnGj−(ω˜γ+ ˜αβy) lnNj + (˜α+ ˜β−αβ˜ y) ln(wj −Tj +Fj)
−(˜α(1−αy −βy)) lnwj+c4
Upward-sloping local labor supply is obtained if (˜α+ ˜β−αβ˜ y)wj >α˜(1−αy−βy)(wj−Tj+Fj). Intuitively, this requires that an increase in local wages makes local agents better off, i.e. the increase in the cost of housing due to higher local wages is relatively less important than the increase in consumption. Notice that this condition is satisfied in particular if Tj ≥Fj.2
The descriptive statistics reveal that in Germany, wage income tax payments due to differing wage premia are considerably larger than per-capita equalization payments. Finally, the condition P
jNj = 1 determines ¯U. A higher level of ¯U causes an inward shift of local labor supply in all regions.
Uniqueness of the market solution
Downward-sloping local labor demand and upward-sloping local labor supply imply unique- ness of the local labor market equilibrium, as illustrated in figure A.1.
Existence of an interior solution
The first order condition necessary as written in equation (A.1) is valid only at an interior solution. It is not clear that Inada conditions can be assumed to hold for the production
2 Also note that incorporating housing in the model is not essential for the main result of this paper. Without housing, upward-sloping labor supply obtains due to the congestion of the publicly provided good if ω >0.
function in the presence of agglomeration economies in general. Given that I assume log- linearity of the agglomeration function as in equation (A.9) andσ < β however, an interior
solution obtains.
First consider what happens when NXj approaches infinity. Then, the first two terms on
the left hand side of equation (A.1), i.e. the entire marginal product of labor decrease. To see why this is the case, note that ∂FXj
∂NX =w
j and that ∂lnwj ∂lnNXj =
σ−β
1−α <0. The second term on the left hand side of equation (A.1) is a multiple of the first term according to equation (1.4). The resource cost of housing is increasing inNj. I assume that it is not optimal for regions to be empty, i.e. that the last term of equation (A.1) relating to the congestion of the publicly provided good does not dominate asNj approaches zero (Nj can approach zero if and only if NXj, NYj and NGj all approach zero).
Sufficiency
With a strictly quasi-concave utility function such as the one chosen above, the KKT conditions are sufficient for a unique global maximum if the constraint functions satisfy quasi-concavity (Sundaram, 1996). Cobb-Douglas production of the housing good and the publicly provided good exhibiting decreasing returns (due to the fixed factor) directly yields concavity since productivity is exogenous. Concavity of the constraint function relating to production of tradables,
P jFX KXj , LjX, NXj, AX NXj−P jNjxj , is less straightforward due to endogenous productivity. However, with Cobb-Douglas production, provided that the agglomeration function fulfills equation (A.9) and σ < β, concavity is
immediate since the exponents of the variable factors then add up to less than one.
Production in the dynamic model
In the dynamic version of the model, production of tradable goods in decade t occurs
according to FX,t(.) = AjX,t
KX,tj αLjX,tβNX,tj 1−α−β. Agglomeration is assumed to
operate with a decadal lag, i.e.
lnAjX,t=g NX,tj −1 Rj +νj+φt+ j t (A.10)
where φt is a period fixed-effect (equal across locations) and νj represents time-constant region-specific productivity. The errorjt is a shock to productivity in region j and decade t,