and for the cost function to be convex this matrix must be positive definite, i.e. we
must have:
H1 > 0 H2 > 0
With reference to the first condition we have:
H1 > 0 ⇒
∂2C
∂n2
> 0 ⇒ ξ (ξ - 1) nξ 2 (1 - Pt)θ Pη
>0⇒ξ>1
i.e. it is satisfied if ξ>1, while with reference to the second condition we have:
that is:
H2 > 0 ⇒ det H>0 ⇒
∂ 2C
∂n2
∂2C
dρ2
∂C^]>
∂ρt∂nt
ξ(ξ - 1) ntξ-2 (1 - ρt)θ ρtη
• n (1 - Pt)θ-2 ρη-2 [η (η - 1)(1 - Pt) - Ptn (θ - 1) +
dt
-Ptη(η+θ)(1- Pt) -Pt2 (η+θ)(θ - 1) -
that leads to:
ξnt2ξ-2 (1 - Pt)2θ-2 Pt2η-2
dt2
ξ-1
ξ (1 - Pt)θ-1 pη-1 [n - (n + θ) Pt]
dt
[(ξ- 1)η(η- 1) (1 -Pt) -Ptη(ξ- 1) (θ - 1)+
>0
-Ptη (ξ - 1) (η+θ)(1- Pt) - Pt2 (ξ- 1) (η+θ)(θ - 1) +
-ξn2 + 2ξnPt (n + θ) + ξ (n2 + 2nθ + θ^ Pt2] > 0
The fraction outside the square bracket is positive, while considering the expression
inside the square bracket and letting n tend to 0 (both in the case of n positive and
inthecaseofn negative) we get:
-Pt2 (ξ - 1) θ (θ - 1) + ξθ2Pt2 > 0,
which holds when ξ>1 and θ<1, and hence the cost function is convex under the
conditions of the proposition. ■
At this point the profit function writes:
∏t = At • £(1 - ρt) • ht • Ttd • L]1-a n? - nξ ^ (1 -dpt) ^ pη - wt • ht • Td • Lt
where dt > 0 and where Ttd denotes now the working time demanded by the firm.
In the decentralized economy the firm’s optimization program is then given by:
max At £(1 - ρt) htTtdLt] 1-α n? - П (1 -Pt) P - WthTdLt
nt ,Ttd,ρt dt
12
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