Bounded rationality and the ineffectiveness of big push policies

Wei Xiao; Junyi Xu

doi:10.1515/bejm-2017-0182

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Bounded rationality and the ineffectiveness of big push policies

Wei Xiao and Junyi Xu

Published/Copyright: September 26, 2018

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From the journal The B.E. Journal of Macroeconomics Volume 19 Issue 2

Abstract

If a poverty trap exists, can a big-push policy lift the economy out of it? This paper applies Sargent’s [Sargent, Thomas J. 1993. Bounded Rationality in Macroeconomics. New York: Oxford University Press] bounded rationality approach to study the post-policy transition of an economy from a low-income equilibrium to a high-income equilibrium. The effectiveness of the policy diminishes if individuals are adaptive learners who cannot make optimal decisions instantaneously. This paper contributes to the renewed discussion on the effectiveness of massive aid policies for developing countries from a theoretical perspective.

Keywords: big push policy; bounded rationality; learning; massive aid policies; poverty trap

JEL Classification: D83; O11; E1

Appendices

A Baseline model

A.1 Proof of Lemma 1

Combine (10) and (11). Saving can be expressed as:

(22)st=st−1−γtU′(st−1)(1−γt)Mt−1+γtU′′(st−1).

Next, incorporate the first and second order derivatives of the utility function U′(s)=ϕ[(1−τ)w−s]−s[(1−τ)w−s]s and U′′(s)=−s2+ϕ[(1−τ)w−s]2[(1−τ)w−s]2s2 into (22). Now saving becomes

(23)st=st−1−γtϕt−1[(1−τ)wt−1−st−1]−st−1[(1−τ)wt−1−st−1]st−1(1−γt)Mt−1−γtst−12+ϕt−1[(1−τ)wt−1−st−1]2[(1−τ)wt−1−st−1]2st−12,

where longevity ϕ_t−1 and wage income w_t−1 are both functions of k_t−1 and are both increasing in k_t−1.

Let c_t−1 be the consumption of the old generation. It is an increasing function in k_t−1 because ct−1=(1−τ)wt−1−st−1=(1−τ)A(1−α)kt−1α−st−1. We can re-write (23) to express saving as a function of c_t−1 and ϕ_t−1, conditional on s_t−1 and M_t−1:

(24)st=st−1+(ϕt−1ct−1−st−1)ct−1st−1−1−γtγtMt−1ct−12st−12+st−12+ϕt−1ct−12

(25)=st−1+ϕt−1st−1ct−12−st−12ct−1[γt−1γtMt−1st−12+ϕt−1]ct−12+st−12.

M_t–1 < 0 because the second derivative of the utility function is negative. γt−1γt is also negative as the learning gain γ_t ∈ (0, 1).

Take the derivative of s_t with respect to c_t−1 to obtain:

(26)dstdct−1=dϕt−1dct−1Γt−1st−13ct−14+dϕt−1dct−1s3ct−12+2ϕt−1st−13ct−1−st−14+dϕt−1dct−1st−12ct−13+Γt−1st−14ct−12+ϕt−1st−12ct−12[(Γt−1st−12+ϕt−1)ct−12+st−12]2.

Γt=γt−1γtMt−1>0.

Note that longevity ϕ_t−1 can be written as

ϕt−1=βΛ(ct−1+st−1)1+Λ(ct−1+st−1)

where Λ=τ1−τ>0.

Take the derivative of longevity ϕ_t−1 with respect to c_t−1, we will get

dϕt−1dct−1=βΛ[1+Λ(ct−1+st−1)]2.

Thus, (26) becomes

(27)dstdct−1=st−12[Act−14+Bct−13+Cct−12+Dct−1+E][1+Λct−1+Λst−1]2[(Γt−1st−12+ϕ)ct−12+st−12]2.

The coefficients are

A=Λ2Γst−12+βΛΓst−1+βΛ2,

B=2[Λ2Γst−13+ΛΓst−12+2βΛ2st−1+2βΛ],

C=Λ2Γst−14+2ΛΓst−13+[5βΛ2Γ2−Λ2Γ2+Γ]st−12+4βΛst−1,

D=2(1+Λst−1)(β−1)Λst−12

and

E=−[1+Λst−1]2st−12.

The numerator of the derivative (27) is a polynomial in c_t−1 of order 4. Since negative saving is not allowed, it is straight-forward to see that A > 0, B > 0, D ≤ 0 and E ≤ 0. Although the sign of coefficient C is undetermined, the Descartes’ Rule of Signs indicates that the numerator of (27), and thus dstdct−1, has at most one positive root. Therefore, s_t, as a function of c_t−1, has at most one turning point in the domain ct−1∈(0,+∞). This, along with the fact that dstdct−1>0 as c_t−1 approaches +∞, suggests that as consumption c_t−1 increases, saving s_t either increases or first decreases and then increases. Since c_t−1 is an increasing function in k_t−1, according to the property of composite functions, saving s_t is either a monotonically increasing function, or a U-shaped function in k_t−1. The lemma thus follows. □

A.2 Proof of Lemma 2

According to Lemma 1, in the limit, the only way to increase saving is to increase capital. From (25), we have

(28)limkt−1→∞st=st−1+limkt−1→∞ϕt−1st−1[(1−τ)wt−1−st−1]2−st−12[(1−τ)wt−1−st−1][γt−1γtMt−1st−12+ϕt−1][(1−τ)wt−1−st−1]2+st−12

(29)=st−1+limkt−1→∞st−1[(1−τ)wt−1−st−1]2−st−12[(1−τ)wt−1−st−1]ϕt−1[γt−1ϕt−1γtMt−1st−12+1][(1−τ)wt−1−st−1]2+st−12

(30)=st−1+st−1γt−1βγtMt−1st−12+1

(31)<2st−1.

The equality (30) holds because limkt−1→∞wt−1=+∞ and limkt−1→∞ϕt−1=β. The lemma then follows. □

B Learning to optimize and forecast

B.1 Properties of equilibria

With “learning to optimize” and “learning to forecast”, it is still possible to generate three steady-state equilibria in the model economy. The middle steady state is unstable, and the other two are stable. As Figure 7 shows, when disturbed, the economy deviates from the medium-income steady state and converges to either the high-income or the low-income steady state.

Figure 7:

Stability of steady states under learning to optimize and learning to forecast. Upper panel: the economy converges to the low-income steady if initial capital is lower than the threshold. Lower panel: the economy converges to the high-income steady if initial capital is higher than the threshold.

B.2 Proof of Proposition 2

To obtain a theoretical result, we take several steps. We begin with the following lemma.

Lemma 3

If the learning algorithm is as described by (17),(18) and (19), we can write the saving decision s_t as a function of capital k_t, conditional on lagged saving s_t−1, expected capital return in the previous period rte and lagged perceived second derivative of utility M_t−1. Let k¯=inf {kt:kt>0 and U′(st−1|kt)>0}. Then s_t increases in k_t in the domain (k¯,+∞).

Proof.

According to the saving decision (17), when kt<k¯, st≤st−1. This implies that financial aid should at least raise capital level above k¯ in order to increase saving. Now we prove that s_t monotonically increases as k_t rises above k¯. Combine (17), (18) and (19) to express the saving decision as

(32)st=st−1+−[(1−τ)wt−st−1]−θ+ϕtθst−1−θ[rte+1t(rt−rte)]1−θγt−1γtMt−1+θ{[(1−τ)wt−st−1]−θ−1+ϕtθst−1−θ−1[rte+1t(rt−rte)]1−θ}

(33)=st−1+−ct−θ+ϕtθst−1−θ[rte+1t(rt−rte)]1−θΓt+θ[ct−θ−1+ϕtθst−1−θ−1[rte+1t(rt−rte)]1−θ].

Γt=γt−1γtMt−1>0.ct=(1−τ)wt−st−1 is consumption. Since c_t is an increasing function of k_t, we now turn to prove that s_t increases in c_t at the domain (c¯,+∞), where c¯ is the corresponding consumption level when kt=k¯. Differentiate both sides of (33) with respective to c_t and we have

dstdct=θct−θ−1+θϕtθ−1ϕt′st−1−θ(rt+1e)1−θ+1t(1−θ)ϕtθst−1−θ(rt+1e)−θrt′Γt+θct−θ−1+θϕtθst−1−θ−1(rt+1e)1−θ−{−ct−θ+ϕtθst−1−θ(rt+1e)1−θ}{−θ(θ+1)ct−θ−2+θ2ϕtθ−1ϕt′st−1−θ−1(rt+1e)1−θ+1tθ(1−θ)ϕtθst−1t+−θ−1(rt+1e)−θrt′}(Γt+θct−θ−1+θϕtθst−1−θ−1(rt+1e)1−θ)2.

Here longevity function ϕ_t and capital return r_t are written as functions of c_t:

(34)ϕt=βτ1−τ(ct+st−1)1+τ1−τ(ct+st−1),

(35)rt=Aαktα−1=A1α(1−τ)1−αα(1−α)1−ααα(ct+st−1)α−1α=Ω(ct+st−1)α−1α.

The derivative of s_t with respect to c_t can thus be simplified as

(36)dstdct=g1(ct)+g2(ct)+g3(ct)(Γt+θct−θ−1+θϕtθst−1−θ−1(rt+1e)1−θ)2,

where

g1(ct)=ϕtθ−1st−1−θ−1(rt+1e)−θ[Γtst−1+θ+st−1ct−θ−1+θct−θ][θdϕtdctrt+1e+1t(1−θ)ϕtdrtdct],

g2(ct)=θct−θ−1Γt+θ2ct−2θ−2+θ2ct−θ−1ϕtθst−1−θ−1(rt+1e)1−θ,

and

g3(ct)=θ(θ+1)ct−θ−2[−ct−θ+ϕtθst−1−θ(rt+1e)1−θ]=θ(θ+1)ct−θ−2U′(st−1|kt).

Since Γ_t > 0 and U′(st−1|kt)>0 when ct>c¯, we can conclude that g2(ct)>0 and g3(ct)>0. Now we prove that g1(ct) is positive.

We re-write g1(ct):

g1(ct)=ϕtθ−1st−1−θ−1(rt+1e)−θ[Γtst−1+θst−1ct−θ−1+θct−θ][θdϕtdctrt+1e+1t(1−θ)ϕtdrtdct]=ϕtθ−1st−1−θ−1(rt+1e)−θ[Γtst−1+θst−1ct−θ−1+θct−θ]h1(ct).

Take derivative of h1(ct) with respect to c_t and we have

dh1(ct)dct=−θβΛ2[1+Λ(ct+st−1]3rt+1e+θβΛ[1+Λ(ct+st−1]21tdrtdct−1t(1−θ)Ωα−1α(ct+st−1)−1/α−1βΛ1+Λ(ct+st−1)[ct+st−11+Λ(ct+st−1)−ct+st−1α]<0when α<1.

The inequality holds because drtdct=α−1αΩ(ct+st−1)−1α<0. Hence, h1(ct) is a decreasing function. We can also show limct→∞h1(ct)=0. This guarantees that h1(ct)>0 when ct∈(0,+∞), and g1(ct) is thus positive.

Altogether, according to (36), we have dstdct>0 when ct>c¯. Thus, saving is increasing in k_t in the domain (k¯,+∞). The lemma follows. ⊡

Next, we prove another result.

Lemma 4

Under forward looking behavior, a financial aid can at most increase saving by a factor of 1 + 1θ.

Proof. Combine the learning processes (17), (18) and (19), and the saving decision follows

(37)st=st−1+−[(1−τ)wt−st−1]−θ+ϕtθst−1−θ[rte+1t(rt−rte)]1−θγt−1γtMt−1+θ[(1−τ)wt−st−1]−θ−1+ϕtθst−1−θ−1[rte+1t(rt−rte)]1−θ.

Notice that wage income w_t, longevity ϕ_t and capital return r_t are functions of capital k_t. Suppose a huge financial aid is landed. Take limit on both sides of (37) as k_t → ∞ and we have

(38)limkt→∞st=st−1+−limkt→∞[(1−τ)wt−st−1]−θ+limkt→∞ϕtθst−1−θ[rte+1t(rt−rte)]1−θγt−1γtMt−1+limkt→∞θ[(1−τ)wt−st−1]−θ−1+limkt→∞ϕtθst−1−θ−1[rte+1t(rt−rte)]1−θ

(39)=st−1+βθst−1−θ(rte)1−θ(1−1t)1−θγt−1γtMt−1+θβθst−1−1−θ(rte)1−θ(1−1t)1−θ

(40)=st−1+1θst−1γt−1γtMt−1θβθst−1−1−θ(rte)1−θ(1−1t)1−θ+1

(41)<st−1+1θst−1

(42)=(1+1θ)st−1.

The lemma thus follows. □

Finally, we note that the result in Lemma 4 leads necessarily to the conclusion in Proposition 2.

C Learning to forecast only

Figure 8 presents simulation results on stability of steady state under learning to forecast only.

Figure 8:

Stability of steady states under learning to forecast only.

Upper panel: the economy converges to the low-income steady if initial capital is lower than the threshold. Lower panel: the economy converges to the high-income steady if initial capital is higher than the threshold.

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Published Online: 2018-09-26

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https://doi.org/10.1515/bejm-2017-0182

Keywords for this article

big push policy; bounded rationality; learning; massive aid policies; poverty trap