Public debt in an OLG model with imperfect competition: long-run effects of austerity programs and changes in the growth rate

Appendix A: Existence and stability of steady states

Existence To see the existence of the solution to (24), let us define

(A1)f(R)≡s[(εε−1)11−γα−11−γσ−γ1−γR11−γ−R]−1−n (A1)

where R=r+δ. We then have two cases:

1. 0≤γ≤1: f(0)<0, lim_R→∞f(R)→∞ and f(R) is continuous. Therefore there exists R∈(0, ∞) such that f(R)=0. The convexity of f(R) ensures the uniqueness.

2. γ<0: f(0)<0, lim_R→∞f(R)→–∞ and f(R) is continuous. f(R) is concave and initially increasing but decreasing eventually. Therefore if σ is sufficiently large (for given values of ε and α), there exist two distinct roots, namely, R₁ and R₂ with R₁<R₂. Note that the concavity of f(R) implies

   (A2)f′(R1)=s[11−γ(1+1+nsR1)−1]>0 (A2)

   (A3)f′(R2)=s[11−γ(1+1+nsR2)−1]<0 (A3)

Stability The left-hand and the right-hand side of (23) are strictly decreasing in r_t+1 and r_t, respectively. Thus, (23) implies that r_t+1 is strictly increasing in r_t, i.e., dr_t+1/dr_t>0 for all r_t. It follows that a fixed point of (23) is locally stable if and only if dr_t+1/dr_t<1 at the point.

r_t+1 is a decreasing function of K_t+1/L_t+1. To show that dr_t+1/dr_t<1 at a stationary point is therefore equivalent to showing that d(K_t+1/K_t)/dr_t is positive. Intuitively, at a steady state the capital stock grows at the same rate as the labor force; the steady state is stable if a value of r above the steady growth solution (corresponding to a K/L below the equilibrium) generates a growth rate of the capital that exceeds the growth rate of the labor force. We have

(A4)Kt+1Kt=s(wt+πt)LtKt=s[(εε−1)11−γα−11−γσ−γ1−γ(rt+δ)11−γ−(rt+δ)]=f(rt+δ)+1+n (A4)

The stability results now follow from the properties of the f–function (see above): (i) if 0≤γ≤1, the unique stationary solution is stable; (ii) if γ<0, the low solution for r is locally stable and the high solution is unstable.

Appendix B: Effects of a rise in the markup

Re-stating equation (24) we have

1+n=s[(εε−1)11−γα−11−γσ−γ1−γ(r+δ)11−γ−(r+δ)]

By the implicit function theorem,

d(r+δ)d(εε−1)=−11−γ(εε−1)11−γ − 1(r+δ)11−γ11−γ(εε−1)11−γ(r+δ)11−γ − 1−α11−γσγ1−γ<0

Hence,

d[εε−1(r+δ)]d(εε−1)=r+δ+εε−1d(r+δ)d(εε−1)=−(r+δ)α11−γσγ1−γ(r+δ)11−γ(εε−1)11−γ(r+δ)11−γ−α11−γσγ1−γ(r+δ)<0

Appendix C: The Leontief case

If γ→–∞, the production function converges to the Leontief form,

(C1)yjt=min{σkjt, λljt} (C1)

In order for full-employment growth to be technically feasible, the aggregate capital stock must grow at least as fast as the labor force when all output is being invested. Algebraically,

σKt≥Yt≥(n+δ)Kt

or

(C2)σ≥n+δ (C2)

This technical feasibility condition is necessary but not sufficient. With a logarithmic utility function and a saving rate of 1/(2+ρ), the parameters need to satisfy the more restrictive condition

sYt=Yt2+ρ≥Kt+1=(1+n)Kt≥1+nσYt

or

(C3)σ≥(1+n)(2+ρ) (C3)

The economy has two steady growth paths. There is a full-utilization path with σk_jt=λl_jt=y_jt, K_t=(λ/σ)L_t and ct=wtλ+rt+δσ. Along this path the pricing equation (11) takes the form 1=pt=pjt=εε−1(wtλ+rt+δσ); the real wage and the amount of profits are given by wt=λ(1−1ε)−λrt+δσ and πt=λε, respectively. Using these expressions, equation (22) can be written as

(C4)Kt+1Lt=Kt+1Lt+1(1+n)=λσ(1+n)=s(wt+πt)=s[λ−λrt+δσ] (C4)

Solving (C4) for the rate of interest and substituting the result back into the factor-price frontier, we have:^[13]

(C5)r+δ=σ−(2+ρ)(1+n)>0 (C5)

(C6)w=λ(2+ρ)(1+n)σ−λε (C6)

The steady growth path described by equations (C5)-(C6) is dynamically efficient: the net marginal product associated with a reduction in the capital-labor ratio exceeds the growth in the labor force (σ–δ>n). This high-interest path is unstable, however. The saving rate, s, is constant (given the logarithmic specification of the utility function); starting from the efficient path, a positive shock to w therefore raises the amount of saving, and the capital intensity increases in the next period to give K_t+1/L_t+1>λ/σ. The young workers’ income (the wage rate plus profits) then rises to λ in subsequent periods,^[14] and the economy will be following a steady growth path with excess capacity:

(C7)KL=λ(2+ρ)(1+n)>λσ (C7)

This steady-growth path is dynamically inefficient; the net marginal product of capital is equal to –δ<n; consumption could be increased by reducing investment (eliminating the excess capacity) and having each young generation use some of its saving to finance consumption for the old generation.

Appendix D: Effects of debt elimination on the marginal product

With a Cobb-Douglas production function, Y=K^αL^1–α, we have

∂Y∂K=αKα−1L1−α

and

Δlog∂Y∂K=(α−1)Δlogk

where k=K/L. The effect on K of eliminating debt follows from the saving equation. If x=(K+B)/K, equation (31) implies that in a steady state

(D1)xk=s1+n(w+π−τ)+(1−s)11+rϕ=s1+n(1−α+αε)kα−s1+nτ+(1−s)11+rϕ (D1)

Assume that the values of g and φ are kept unchanged. If the optimal steady state (associated with the initial debt-capital ratio of 1/3) has r*=n, it follows from (32) that the value of τ in the zero-debt steady state will also be unchanged, compared to its value in the optimal steady state. Using (D1) and assuming (plausibly) that φ≥0 and s1+nτ≥(1−s)11+rϕ, we now have

dlog[s1+n(1−α+αε)kα−s1+nτ+(1−s)11+rϕ]dlogk>α

Hence,

−(1−α)Δlogk<Δlogx

and

Δlog∂Y∂K=−(1−α)Δlogk<Δlogx=log1−log43

It follows that

log∂Y∂Kno  debt<log(34∂Y∂Kwith  debt)