Rearranging the first-order conditions yields,
Rk
Rt
ψ' ( ωit )
ψ (ωit ) ξ' (ωit ; μ ) - ψ' (ωtt, ) ξ (ωtt, ; μ) ;
(4)
Bit-1
Nit-1
ψ'(ωit )ξ (ωit ; μ)
ψ ( ωtt )ξ '( ωt ; μ ) ■
(5)
Given the aggregate rate of return on capital Rtk and the risk-free rate Rt , equa-
tion 4 determines the default productivity threshold ωt that characterizes the debt
contract. Equation 5, in turn, determines the firm’s leverage ratio N— ■ Together,
the optimal debt contract generates a schedule relating the firm’s leverage ratio to
the external finance premium, RRt. Because the external finance premium is unobserv-
able, we can use the equation 1—which defines the contractual rate Ribt—to obtain a
relationship linking the observable credit spread for firm i, RRit ,to the firm’s leverage
Rt
Nit-1, the default productivity threshold ωt, and the external finance premium RRt :
R=ωtt (1+
Bit-1 ʌ Rk
Nit-1) Rt ■
(6)
It is worth noting that even in the frictionless case, in which μ = 0 and Rt/ Rt = 1,
equation 6 implies a positive spread between the contractual rate Ribt and the risk-free
rate Rt . This spread exists to compensate the lender for the fact that a certain fraction
of firms will inevitably default on their debt obligations, and the equilibrium stipulates
that the lender earns the risk-free rate of return. In the presence of bankruptcy
costs, credit spreads also include an external finance premium component. However,
movements in the external finance premium do not lead to one-to-one movements
in the credit spread, because when μ > 0, the first-order conditions 4 and 5 imply
endogenous changes in the default productivity threshold and leverage.
2.2 Comparative Statics
To gain a better insight into how various model-implied relationships vary with
changes in the key structural parameters—namely the degree of financial frictions
μ and the variance of the idiosyncratic productivity shock σ2—we turn to compar-
ative statics. Following BGG, we assume that the productivity disturbance ωit is
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