Thresholds when 0< γ <1
The following calculations hold when the productivity state remains the same (st = st−1).
Worker reallocation and wages
• Wages in the high-TFP sector exceed wages in the low-TFP sector if bt+γ(1− bt)< Bnc(xt) which is equivalent to
• Wages in the low-TFP sector exceed wages in the high-TFP sector if (1−γ)bt >
Bnc(xt), i.e.
• All workers earn the same wage if
TW L(bt)≤xt ≤TW H(bt).
Borrowing
Using equations (9) and (A.1), one obtains the condition under which high-TFP firms borrow. The conditions under which credit constraints are binding is derived by substitution of (9) into (A.9) and (A.15).
• When wHt > wLt, the high-TFP sector borrows if
The high-TFP firm is credit constrained if
When the low-TFP sector borrows, credit constraints bind if xt > α[γ+ (1−γ)bt] +λ(1−γ)(1−bt)z1−α1
The high-TFP sector is borrowing constrained if xt< α−λ
α
(1−γ)bt z1−α1 + (1−γ)bt
1−z1−α1 ≡THC(bt).
When the low-TFP firm borrows, the credit constraint binds if xt> α(1−γ)bt+λ(1−(1−γ)bt)z1−α1
αh
z1−α1 + (1−γ)bt
1−z1−α1 i ≡TLC(bt).
• If wtL = wHt , always high-TFP firms borrow. To see this, substitute equation (8) into (A.1). This yields
xt≤
α−
αzα1 −λ z1−α1 α−αzα1z1−α1 .
Since the right-hand side is larger than 1, the condition that high-TFP firms borrow is always fulfilled if wLt =wHt . Further, we know from Section 3.1 that credit constraints bind in this case.7
7It was shown in Section 3.1 that credit constraints bind if both firms produce,Rt=M P KtL, andwLt =wHt .
Location of the threshold curves in(bt, xt) space
Proposition 3. WhenwLt > wHt , high-TFP firms borrow and credit constraints bind.
Proof. Simple algebra proves that theTLC curve and theTHB curve are located above the THC curve. It remains to show that the THC curve lies above the TW L curve for all bt ∈ (0,1). Since the TW L curve is convex and the THC curve is concave, and since both curves are increasing and start at bt = 0 and xt = 0, it suffices to show that the THC threshold exceeds the TW L threshold for bt = 1:
α−λ α
1−γ z1−α1 + (1−γ)
1−z1−α1 ≥
αzα1 −λ
(1−γ) αh
1−
1−zα1
(1−γ)i
⇔γ h
α−λ−z(1−α)1
αz1α −λ i
+λ(1−γ)
1−zα1
≥0
The condition is satisfied for 0< γ <1 and λ∈[0, αzα1). QED
Figure 13: Threshold functions for 0< γ <1
It remains to show that TRH(bt) > TB(bt) > TRL(bt) > TW H(bt) > TL(bt) >
TW L(bt) for all bt ∈(0,1). Figure 13 illustrates the threshold functions and includes
their corresponding values atbt= 0 andbt= 1. Since the TRL(bt) curve is increasing and concave, and the TW H(bt) curve is increasing and convex, it suffices to compare both threshold functions atbt= 0 andbt = 1. It can be shown thatTRL(0)> TW H(0) if
γ < α−λ−(αzα1 −λ)z1−α1
α−λ−(αzα1 −λ)z1−α1 −λ(1−zα1).
Since the right-hand side is larger than 1, TRL(0) > TW H(0) is always satisfied for 0< γ < 1. Further, it can be shown thatTRL(1) > TW H(1). The relative location of the other threshold functions is obtained by simple algebra.
Thresholds when γ = 0
When γ = 0, wages are the same in both sectors only if
xt= Equation (B.1) is obtained from (8). The high-TFP firm borrows if (A.1) is satisfied:
xt≤ bt+1
bt+1+ (1−bt+1)z1−α1
≡TB(bt+1).
Credit constraints do not bind if (A.9) or (A.15) is satisfied:
xt ≥ α−λ
C Proofs
Proof of Proposition 1 Step 1. Development of x
st =st−1:
1. If the high-TFP firm borrows, the transitional dynamics of the wealth share of high-TFP firms is described by
xt =X0(xt−1, bt) =
R˜t−1xt−1
R˜t−1xt−1+Rt−1(1−xt−1). The wealth share of the high-TFP firm increases if
X0(xt−1, bt)> xt−1,
which is equivalent to ˜Rt−1 > Rt−1. This condition holds if the high-TFP firm is credit constrained, i.e. if (bt, xt) is located below the TRL curve.
2. If the low-TFP firm borrows, the transitional dynamics of the wealth share of high-TFP firms is described by
xt =X0(xt−1, bt) = Rt−1xt−1
Rt−1xt−1+ ˜Rt−1(1−xt−1). The wealth share of the high-TFP firm decreases if
X0(xt−1, bt)< xt−1,
which is equivalent to ˜Rt−1 > Rt−1. This condition holds if the low-TFP firm is credit constrained, i.e. if (bt, xt) is located above the TRH curve.
3. It follows that the wealth distribution does not change if (bt, xt) is located in the area between the TRL and theTRH curve.
If st 6= st−1, the wealth share of high-TFP firms at the end of period t−1 is given by 1−X0(xt−1, bt).
Step 2. The functions TRL(bt) and TRH(bt) are monotonously increasing in bt and
–, Condition 2 is satisfied if αγ + (2λ−α)(1−γ)z1−α1 ≥0.
IfCondition 1 is satisfied, λ≥ α2. This also ensures that Condition 2 is satisfied.
Hence, λ ≥ α2 is a sufficient condition that (bt, xt) stays in the area between the
TRL and theTRH curve in the long run. QED
Proof of Proposition 2
The inspection of Figures 14 and 15 already suggests that two cases have to be considered. For reasons of clarity leta, e,c, andddenote the line segments between points (1−µ, TRH(1− µ)) and (1 −µ,1), (µ, TRH(µ)) and (µ,1), (1− µ,0) and (1−µ, TRL(1−µ)), as well as (µ,0) and (µ, TRL(µ)), respectively. The length of a line segment is denoted by the symbolk·k. Since theTRH andTRLthreshold functions have the same derivatives, and since they are increasing and concave,kak<kdkand kek<kck.
Figure 14: Case 1: TRL(µ)≤1−TRL(1−µ)
Figure 15: Case 2: TRL(µ)>1−TRL(1−µ)
Case 1. TRL(µ)≤1−TRL(1−µ)
1. Suppose (xt, bt+1) is located ona in Figure 14. As long as the productivity state does not change, the economy converges to TRH(1−µ). When the productivity state changes, the new value of (xt, bt+1) is located on d. Since kak<kdk, the economy eventually converges toTRL(µ) and then fluctuates between two states x1 =TRL(µ) and x2 = 1−TRL(µ).
2. Suppose (xt, bt+1) is located on e. As long as the productivity state does not change, the economy converges toTRH(µ). When the productivity state changes, the new value of (xt, bt+1) is located on c. Since kek < kck, the economy eventually converges to TRL(1−µ) and then fluctuates between two states x1 =TRL(1−µ) and x2 = 1−TRL(1−µ).
Case 2. TRL(µ)>1−TRL(1−µ)
1. Suppose (xt, bt+1) is located on a. As long as the productivity state does not change, the economy converges to TRH(1−µ). When the productivity state changes, the new values of (xt, bt+1) are located ond. Sincekak<kdk, the economy eventually converges to TRL(µ). When the productivity state changes then, the new value of (xt, bt+1) is located oncandxt = 1−TRL(µ).
The economy converges toTRL(1−µ) as long as the productivity state does not change. When the productivity state changes then, the new value of (xt, bt+1) is located on d and xt= 1−TRL(1−µ). The economy converges to a cycle withx∈[1−TRL(µ), TRL(1−µ)]∪[1−TRL(1−µ), TRL(µ)].
2. Suppose (xt, bt+1) is located on e. As long as the productivity state does not change, the economy converges toTRH(µ). When the productivity state changes, the new value of (xt, bt+1) is located on c. Since kek < kck, the