Closed Economy Equilibrium

Firms’ choices of the level of technology in production and the related quantities of in-termediate inputs are embedded in an autarkic Dixit-Stiglitz economy. A representative household has a taste for variety implied by the utility function

uc=

Z M_c,d 0

Y_c,d,i^β di

!_β¹

, 0< β <1,

and supplies low- and high-skilled labor (L^s, H^s) inelastically. c denotes variables in closed economy. There exists a continuum of final goodsY_c,d,i, withi∈[0, M_c,d], that are supplied by a (symmetric) mass M_c,d of domestic firms of technology type κ_d. _1−β¹ > 1 is the elasticity of substitution between final goods. The above preferences imply the demand function

Y_c,d,i =

p_c,d,i P_I

−_1−β¹ A_c

P_c,I (1.17)

where pc,d,i is the price of good i, Ac is the aggregate spending level, and Pc,I ≡

RMc,d

0 p⁻

β 1−β

c,d,i di

!−^1−β

β

is the price index of final goods. Defining P_c,I as the numeraire (P_c,I ≡1), the implied demand function for each firm, A_cp⁻

1 1−β

c,d,i , in Section 1.2.2becomes identical to (1.17). Whenever there is no loss of clarification I abstract from the firm index i. With optimal firm choices in Section 1.3.1 and market clearing, equilibrium is defined as:

Definition 1.1 Equilibrium in a closed economy with symmetric firms is given by a set of prices {p_c,d, w_c,H, w_c,L}, quantities {Y_c,d, H_c,d, L_c,d}, and a level of technology N_c,d such that with free entry of firms consumers choose consumption of each final good optimally, firms choose output, level of technology and labor inputs optimally, and labor and product markets clear.

Note that intermediate inputs do not show up directly. They are produced within each firm with high- and skilled labor and are aggregated to firm-specific high- and low-skilled labor production demands.

1.4.1 Wages in Closed Economy

There is free entry, but firms have to incur f_d units of low-skilled labor to set up pro-duction. Adding this to the production low-skilled labor demand results in a firm’s total low-skilled labor demand (L_c,d+f_d). The following free entry condition

p_c,dY_c,d−C(Y_c,d)−w_c,Lf_d= 0 ⇐⇒ (1−β)p_c,dY_c,d=w_c,Lf_d (1.18) fixes the wage level given a firm’s revenue. The latter is derived by multiplying the optimal price (1.8) by the optimal output (1.7). Using subsequently the optimal technology choice (1.9) as well as minimal unit costs, kj =wLw¯^µj, results in the following revenue function

p_c,dY_c,d=β^1−ββ A_cN

κdβ 1−β

c,d w

β β−1

c,L w¯

Nc,dβ µ(β−1)

c (1−κ_dσ)^σ(β−1)β . (1.19) Plugging this expression into the free entry condition (1.18) and using total labor income (w_c,LL^s+w_c,HH^s=A_c) shows that the low-skilled wage

w_c,L=βN_c,d^κdw¯⁻

Nc,d

c µ (1−κ_dσ)^−σ1 (1−β)(L^s+ ¯w_cH^s) fd

!^1−β_β

(1.20) is a function of labor endowments, parameters, and the skill premium. The wage gap is computed from setting relative labor supply equal to total relative labor demand, ^H_Ls^s =

Mc,dHc,d

Mc,d(Lc,d+fd). Using (1.11) and (1.12), this implies w¯c

H^s

L^s = κ_d

ln ¯wc

β −κ_d. (1.21)

The above equation implicitly and uniquely determines the skill premium. It depends on the relative scarcity of high-skilled labor^H_Ls^s

, the elasticity between final goods (β), and the firms’ scope for technology in production (κ_d).

Lemma 1.4 There exists a unique skill premium. Furthermore, w¯_c> e^κdβ.

The proof is given in Appendix 1.7.7. The proof builds on the properties of the left and right hand side of (1.21). The existence of an unique skill premium implies that there

exists an unique choice of the level of technology in equilibrium, since the wage gap is the only endogenous variable in the equation of the optimal technology choice (1.9).

Proposition 1.4 The wage gap is lower for higher relative skill endowments (^H_L^SS), larger for higher technology types (κd), and increases in the elasticity of market demand (β).

The proof is given in Appendix 1.7.7. A country’s high- and low-skill labor endowments are supplied to firms that use technology-complementary production processes. The skill premium represents a measure of the relative scarcity of skills. Holdingκandβ constant, the wage gap increases when the relative supply of high- to low-skilled labor decreases.

This is illustrated inFigure 1.2 where the left and right hand side of (1.21) are simulated Figure 1.2: Skill Premia in Closed Economy

I

s s

L w H ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

II

s s

L w H ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

m m

w κ β

κ )− ln(

d d

w κ β

κ )− ln(

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ^I

s s d

c L

w κ , H _⎟^⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ^I

s s m

c L

w κ , H _⎟^⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ^II

s s m

c L

w κ , H

w

Skill premia in closed economy for different skill endowments and technology types. The graphs in red depict the left hand side of (1.21) while the graphs in blue depict the right hand side of (1.21).

for an array of skill premia. In particular, a fall in the share of skilled workers from 25%

to 15%, i.e. a relative skill supply decrease from ^H_Ls^s

I

to ^H_Ls^s

II

, implies a rise in the skill premium from ¯w_c

κ_m,^H_Ls^s

I

to ¯w_c

κ_m,^H_Ls^s

II

22. Firms with a technologically

22Note that this section analyzes autarky. κmandκd are used to depict a closed economy with a high and low technology type, respectively. In open economy analysis, I refer to these cases.

more efficient production function (i.e. a higher κ) choose a higher level of technology and employ more skill intensive intermediate inputs. This is best illustrated by comparing two economies with equal skill endowments, but with firms of different technology types.

One is populated by domestic firms such that all firms are endowed withκ_d= 0.16 while the other consists of multinationals withκm = 0.24. As a consequence, the skill premium in an economy populated exclusively by domestic firms, ¯w_c

κ_d,^H_Ls^s

I

, is well below the wage gap of an economy populated exclusively by multinational firms, ¯w_c

κ_m,^H_Ls^s

I

(see Figure 1.2). As technology is skill-complementary, multinationals demand relatively more high-skilled labor what is settled in equilibrium by a higher wage gap.

Furthermore, higher technology types imply a greater productivity, changing the free entry condition (1.18). For a detailed analysis of how parameters effect free entry in general and equilibrium firm numbers in particular, (1.18) is plugged into the equality of household income and total expenditures (w_c,LL^s+w_c,HH^s = M_c,dp_c,dY_c,d). This results in the number of firms

M_c,d= 1−β fd

(L^s+ ¯w_cH^s), (1.22)

which clearly increases in the skill premium. Since a larger technology type rises the wage gap, a greater scope for technology in production leads to an increase in the number of firms. Intuitively, an increase in the efficiency of technology in production implies higher expected profits. More firms enter the market while simultaneously the relative renumeration of high-skilled workers increases.

Whereas a greater market elasticity has no effect on relative production labor demands, it increases total relative labor demands and, consequently, the skill premium. The latter, indirect effect, rises the number of firms while the direct effect of β in (1.22) decreases the equilibrium number of firms. However, the total impact on the number of firms is negative²³ as a higher market elasticity increases competition among firms, reduces mark-ups and decrease firms’ profits.

23∂Mc,d/∂β= 1/fd[−L^s−w¯cH^s(1−β(1−β)_w¯ ^{ln ¯wc}

c−κdβ+1)]<0 asβ(1−β) ln ¯wc+κdβ <w¯c+ 1.

1.4.2 Technology Levels in Closed Economy

The relative endowment of high-skilled labor determines (given κ_d, β) the skill premium that, in turn, is crucial for the decision on the optimal technology level. Very low skill premia that arise from an abundant supply of high-skilled labor may induce firms to choose levels of technology that are not defined (i.e. N_c,d > µ). The explicit approximation of the implicitly given level of technology (1.10) leads to the determination of a threshold level of relative skill supply which ensures optimal technology choices withinN_c,d ∈(1, µ].

Proposition 1.5 There exists an approximated threshold H^s

L^s ≤exp

−2κ_d 1−κ_dσ

| {z }

<1

1−κ_dσ

2

β −1 +κ_dσ

| {z }

<1

. (1.23)

such that firms optimally choose N_c,d∈[0, µ]. Furthermore, N_c,d>1 and N_c,d∈(1, µ]. The proof is given in Appendix 1.7.7. The optimal level of technology in production, given the skill premium, can be approximated from (1.9) by (1.10). Then, given the wage gap from equation (1.21), an approximated threshold to the relative supply of skills is derived that restrains firms’ choices to the admitted interval. As a consequence, a relative skill endowment smaller than or equal to the threshold ensures an optimal choice of N_c,d within [0, µ]. Clearly, the scarcer high-skilled labor is the larger may be the right hand side of (1.23) such that the resulting skill premium still delivers a well defined level of technology.

Im Dokument Firm-level technology choices and endogenous production structures in the global economy (Seite 32-36)