Real factor prices and factor-augmenting technical change

Proof of Proposition 1

For ease of notation, let ψ=(σ–1)/σ. Then σ→0 means ψ→–∞, and we want to prove that

limψ→−∞{Γ[bKψ+a(AL)ψ]μ/ψ}=Γmin [Kμ,(AL)μ].

Let us suppose that K≤AL such that ΓK^μ=Γmin ⌊K^μ, (AL)^μ⌋. Without loss of generality, let Γ=1. We show that

Kμ=limψ→−∞{[bKψ+a(AL)ψ]μ/ψ}.

First, we note that bK^ψ≤bK^ψ+a(AL)^ψ so that

(6.1)[b1/ψK]μ≥[bKψ+a(AL)ψ]μ/ψ (6.1)

since μ>0>ψ. Next, observe that K≤AL implies K^ψ≥(AL)^ψ since ψ<0. Therefore, we have bK^ψ+a(AL)^ψ≤aK^ψ+bK^ψ=(a+b)K^ψ so that

(6.2)[bKψ+a(AL)ψ]μ/ψ≥[(a+b)1/ψK]μ. (6.2)

Combining inequalities (6.1) and (6.2) yields

[b1/ψK]μ≥[bKψ+a(AL)ψ]μ/ψ≥[(a+b)1/ψK]μ.

Since limψ→−∞b1/ψ=limψ→−∞(a+b)1/ψ=1, the CES production function converges to K^μ which is min[K^μ, (AL)^μ]. Mutatis mutandis, if K≥AL the same steps prove the convergence of the CES to (AL)^μ=min[K^μ, (AL)^μ].     ■

Proof of Proposition 2

The competitive equilibrium is defined as in Section 2. To build intuition, let me start with Claim 3. Suppose R>0 and w>0. Then, according to (2.1), there must be full employment of both factors of production, i.e., K^d=K and L^d=L. Moreover, firm i’s profit-maximizing factor demands satisfy K(i)=AL(i). Using the latter and factor market clearing delivers immediately

(6.3)Kd=∫01K(i)di=A∫01L(i)di=ALd=AL=K. (6.3)

This proves the first part of Claim 3. As to the second part observe the following. Since the production plan (0,0,0) yields zero profits, any profit-maximizing production plan with strictly positive output must generate non-negative profits, i.e.,

Γmin [K(i)μ,(AL(i))μ]−wL(i)−RK(i)=Γ(AL(i))μ−wL(i)−RK(i)=AL(i)Γ(AL(i))μ−1−wA−R≥0.

For μ=1, this immediately gives

(6.4)1≥1Γ(wA+R). (6.4)

For μ<1, it becomes

AL(i)≤[1Γ(wA+R)]1μ−1

or, accounting for full employment of labor,

(6.5)1≥[1Γ(wA+R)](AL)1−μ. (6.5)

Conditions (6.4) and (6.5) assure that unit costs cannot exceed the output price. Hence, any set of production plans {Y(i), K(i), L(i)}_i∈[0,1] and any pair of strictly positive factor prices (w, R) qualify as an equilibrium if they satisfy K(i)=AL(i) for all i∈[0,1], (6.3), and (6.5) for 0<μ≤1. Accordingly, the level of equilibrium factor prices remains undetermined. This completes the proof of Claim 3.

Next, consider Claim 1, where K>AL. From Claim 3 it is clear that full employment of both factors of production at strictly positive factor prices cannot be an equilibrium. Hence, if an equilibrium exists, equilibrium factor prices satisfy either R=0 and w>0, or R>0 and w=0, or R=w=0. I shall show that only the first of the three alternatives is compatible with a competitive equilibrium and that equilibrium factor prices satisfy indeed (3.3).

1. Suppose R=0 and w>0. Then, firm i chooses K(i) and L(i) to maximize

   (6.6)Γmin [K(i)μ,(AL(i))μ]−wL(i). (6.6)

   For μ=1 this gives factor demands

   (6.7)L(i)={0if w>ΓA, and K(i)≥0,[0,K(i)A]if w=ΓA, and K(i)≥0,K(i)Aif w<ΓA, and K(i)=∞.  (6.7)

   Clearly, w>ΓA leads to an excess supply of labor and the factor market clearing condition for labor cannot be fulfilled for w>0. If w<ΓA, then the demand for capital is unbounded which cannot arise in equilibrium either. Hence, in equilibrium we have w=ΓA and L^d=L. Since K>AL=AL^d any K^d∈[AL, K] is an equilibrium capital employment level.

   If μ<1, then factor demands satisfy K(i)≥0 and

   (6.8)L(i)=min[K(i)A,  (ΓμAμw)11−μ]. (6.8)

   Intuitively, as long as the firm’s demand for capital is large enough, its (unconstrained) labor demand results from the first-order condition

   (6.9)ΓμAμL(i)μ−1=w. (6.9)

   Accordingly, aggregate labor demand satisfies

   (6.10)Ld=∫01min[K(i)A,  (ΓμAμw)11−μ]di=min[KdA,  (ΓμAμw)11−μ]. (6.10)

   Now, two cases must be considered.

   • The first case has

     min[KdA,  (ΓμAμw)11−μ]=(ΓμAμw)11−μ

     and arises as long as

     w≥ΓμA(Kd)1−μ.

     Since, by assumption, w>0, the labor market clearing condition requires

     (6.11)Ld=(ΓμAμw)11−μ=L, (6.11)

     and the market-clearing wage is

     (6.12)w=ΓμAμL1−μ, (6.12)

     which coincides with (3.3). At this wage, it holds that K^d/A≥L^d=L which is feasible since K>AL and K^d∈[0, K] imply K≥K^d≥AL^d=AL. Hence, the wage given by (6.12), L^d=L, and K^d∈[AL, K] is a competitive equilibrium.

   • The second case has

     min[KdA,  (ΓμAμw)11−μ]=KdA

     and arises as long as

     w≤ΓμA(Kd)1−μ.

     By assumption, w>0, so that the labor market clearing condition requires

     (6.13)Ld=KdA=L ⇔ Kd=AL (6.13)

     which implies a real wage

     w≤ΓμAμL1−μ.

     However, any real wage that satisfies this inequality strictly cannot constitute an equilibrium wage. To see this, consider a configuration involving (6.13) and a wage level w_∈(0,ΓμAμ/L1−μ). Since K>K^d=AL, there is idle capital of which individual firms may avail themselves for free. Moreover, at w_ the (unconstrained) labor demand defined by (6.9) delivers L(i)>K^d/A=L. Hence, firms would want to hire more workers provided they can increase their capital demand proportionately. Since there is idle capital, this is feasible. To put it formally, (6.13) requires A=K^d/L^d. Suppose the firms hire (1+ε)K^d<K units of capital where ε>0 but small. At the same time, they increase their demand for labor to (1+ε)L^d. This strategy increases profits and leads to an an excess demand for labor. Hence, any constellation involving (6.13) and w_ does not qualify as an equilibrium. The only real wage level to which the preceding argument does not apply is w_=ΓμAμ/L1−μ which coincides with (6.12).

     Hence, the only strictly positive equilibrium wage level is given by (6.12) which restates (3.3) and includes the case w=ΓA for μ=1. Clearly, the equilibrium real wage increases in A. Moreover, with R=0, s=0, too.

2. Next, suppose that R>0 and w=0. I show that this cannot arise in equilibrium. Firm i chooses K(i) and L(i) to maximize

   (6.10)Γmin [K(i)μ,(AL(i))μ]−RK(i). (6.14)

   For μ=1 this gives factor demands

   (6.15)K(i)={0if R>Γ, and L(i)≥0, [0,   AL(i)]if R=Γ, and L(i)≥0,AL(i)if R<Γ, and L(i)= ∞. (6.15)

   Clearly, R>Γ leads to an excess supply of capital contradicting R>0, whereas R<Γ implies an unbounded demand for labor. Hence, in equilibrium it must be that R=Γ and K^d=K. However, this leads to a contradiction since K^d≤AL^d≤AL<K. Hence, R>0 and w=0 are not equilibrium factor prices for μ=1.

   If μ<1, then factor demands satisfy L(i)≥0 and

   (6.16)K(i)=min[(ΓμR)11−μ,   AL(i)], (6.16)

   i.e., if capital is not binding the (unconstrained) demand for capital results from the first-order condition

   (6.17)ΓμK(i)μ−1=R. (6.17)

   Full employment of capital requires

   (6.18)Kd=∫01min[(ΓμR)11−μ,   AL(i)]di=min[(ΓμR)11−μ,   ALd]=K. (6.18)

   However, full employment of capital implies an excess demand for labor. Indeed, if K^d=AL^d then K^d=AL^d=K⇒L^d>K/A>L. If Kd=(Γμ/R)11−μ, then K^d≤AL^d=K⇒L^d>K/A>L. Hence, a constellation with factor prices satisfying R>0 and w=0 cannot arise in equilibrium for 0<μ<1 either.

3. Finally, consider the case where R=w=0. Then, both factor demands become unbounded which cannot arise in equilibrium. This completes the proof of Claim 1.

Mutatis mutandis, the proof of Claim 2 involves the same steps as the one of Claim 1. It leads to the results of (3.4) and is left out for brevity.     ■

Proof of Proposition 3

From (3.1) it is straightforward to show that

(6.19)R=∂F∂K=μs1+sYK  and  w=∂F∂L=μ1+sYL. (6.19)

It follows that

(6.20)∂R∂A=∂2F∂K∂A={RA11+s(μ−σ−1σ)μaRA>0    if σ ≠ 1,if σ=1 (6.20)

and

(6.21)∂w∂A=∂2F∂L∂A={wA11+s(σ−1σs+μ)μawA>0     if σ ≠ 1,if σ=1.  (6.21)

Hence, the inequalities indicated in (3.8) follow since R, w and s are strictly positive for σ>0.     ■

Proof of Proposition 4

Let me first derive an expression for lim_σ→0∂w/∂A that will be used later. From (6.19), (6.21), and Proposition 1 I deduce that

(6.22)limσ→0∂w∂A=limσ→0{wA11+s(σ−1σs+μ)}=μALlimσ→0{F(1+s)2(σ−1σs+μ)}=μΓmin [Kμ,(AL)μ]ALlimσ→0{1(1+s)2(σ−1σs+μ)}. (6.22)

Now, I turn to the two claims of the proposition.

1. Consider the case K>AL,

   • Then, k>1 and the relative share of capital given in (3.6) satisfies lim_σ→0s=0. Hence, the limit in (6.22) becomes

     (6.23)limσ→0∂w∂A=μΓ(AL)μ−1limσ→0{σ−1σs+μ}=Γμ2(AL)1−μ>0, (6.23)

     since, lim_σ→0s(σ–1)/σ=0. Observe that the limit in (6.23) coincides with the expression for ∂w/∂A of (3.3) derived for the equilibrium with σ=0. Hence, by continuity there is σ_>0 such that for all σ∈(0,   σ_) we have ∂w/∂A>0. To prove the existence of σ¯>0 with the properties mentioned in the proposition recall ∂w/∂A of (6.21). Then,

     limσ→1∂w∂A=limσ→1wA11+s(σ−1σs+μ)=aμlimσ→1wA=aμlimσ→1YL>0,

     since lim_σ→1Y is the Cobb-Douglas function. Hence, by continuity there is σ¯<1 such that ∂w/∂A>0 for all σ¯<σ<1. This completes the proof of Claim 1.(a).

   • Next, I derive the critical value (b/a)¯ and show that ∂w/∂A<0 arises if and only if b/a>(b/a)¯. To see this consider (6.21). With (3.6) I have

     (6.24)∂w∂A<0 ⇔ σ−1σ(ba)kσ−1σ+μ<0. (6.24)

     Since σ<1, a larger ratio b/a reduces the first summand. Accordingly, the critical value (b/a)¯ solves the latter inequality as an equality and is given by

     (ba)¯≡ρ(σ, μ, k)=μσ1−σk1−σσ.

     It is straightforward to show that ρ(σ, μ, k) is ∪-shaped with limσ→0ρ(σ, μ, k)=limσ→1ρ(σ, μ, k)=∞.

     Moreover, ρ(σ, μ, k) attains a minimum on (0,1) at σ*=In k/(1+In k) where ρ(σ*, μ, k)=e·μ·In k. Partial differentiation delivers (3.10). This completes the proof of Claim 1.

2. Consider the case K<AL⇔k<1. The relative share of capital given in (3.6) satisfies lim_σ→0s=∞. Hence, the limit in (6.22) becomes

   limσ→0∂w∂A=μΓKμALlimσ→0{(σ−1)sσ(1+s)2}=0,

   since the term in curly brackets converges to zero. Accordingly, in the limit A has no effect on w.

   To prove the existence of σ^ with the properties mentioned in the proposition recall ∂w/∂A of (6.21). Since w/A(1+s)>0 except if σ=0, critical values that determine the sign of ∂w/∂A must satisfy (σ–1)s/σ+μ=0. To find out whether such values exist, define the function g : (0, 1)→ ℝ, where

   (6.25)g(σ)=σ−1σs+μ. (6.25)

   Clearly, g is continuously differentiable on its domain with dg/dσ>0. To see that there is a single critical value, σ^, that satisfies g(σ)=0 observe that

   limσ→0g(σ)=limσ→0σ−1σs+μ=−∞  and  limσ→1g(σ)=limσ→1σ−1σs+μ=μ>0.

   Hence, there is one value σ^∈(0,   1) that satisfies g(σ^)=0 or

   (6.26)σ^−1σ^(ba)kσ^−1σ^+μ=0. (6.26)

   For 0<σ<σ^ we have g(σ)<0 such that ∂w/∂A<0, for σ^<σ<1 we have g(σ)>0 and ∂w/∂A>0. Implicit differentiation of (6.26) reveals the existence of the function ς(μ, b/a, k) with the derivatives signed as in (3.12).     ■