Transcript Trade and Growth

Trade and Growth
A Brief Tour
Suggested Reading
Aghion, Phillipe and Howitt, Peter (1998),
Endogenous Growth Theory, MIT Press,
Chapter 11
Note in particular that this chapter discusses
Ventura (1997) which I have neglected
The Solow-Swan Model
• We have already seen (in Chapter 4) that
the rate of growth of per capita income is
given by the formula:
(dy/dt)/y = ζ[sA{f[k,1]/k} -n ]
(15)
Trade (openness) could plausibly increase:
• S
• A
Growth is not Efficiency
The chief reason why trade does not
necessarily increase growth is that growth
is not a simple measure of good economic
performance.
In the original von-Neumann growth model
maximal steady-state growth is the
objective
The von-Neumann Growth Model
Theorem 2: If two or more von-Neumann growth models
are allowed to trade with each other, either at
predetermined relative prices (when there may be
quantity rationing), or with voluntarily-negotiated
quantitative trade agreements, the maximal balanced
growth rate of no country can fall, and often maximal
balanced growth rates will increase.
Proof: The possibility of international trade is equivalent
to adding extra activities to the input and output matrices
of the model. As these additional activities need not be
used, the maximal growth rate cannot fall. As the
additional activities will be useful in many cases of
interest, maximal growth rates may well rise.□
The Ramsey von-Neumann Model
Max Σ∞t=1U[ct]δt-1 0 < δ < 1 (6)
Subject to:
Bxt-1 ≥ Axt + ct
(7)
Assume that consumption is proportional to
c0 ≥ 0
although c0 will have many zeros.
The level of consumption is measured by ct
and utility can be written U[ct]
Ramsey von-Neumann Model II
The Lagrangean is
Σ∞t=1U[ct]δt-1 + pt.[Bxt-1 - Axt – ctc0] (1)
Maximization requires:
U1[ct]δt-1- pt.c0 = 0
(2)
pt.B – pt-1.A ≤ 0 (3)
Where the inequalities (3) are complementary to
xt-1 ≥ 0
Such a solution is called a Price Equilibrium
CES Utility Function
[1/(1-η)]ct1-η
(8)
Max:
Σ∞t=1 [1/(1-η)]ct1-ηδt-1 (9)
Bxt-1--Axt - ct c0 ≥ 0 (10)
Lagrangean:
Σ∞t=1 [1/(1-η)]ct1-ηδt-1 + pt{Bxt-1--Axt - ct c0}
(11)
The Steady-State Growth Menu
ct-ηδt-1 - ptc0 = 0
(12)
Pt+1B – pA ≤ 0
(13)
In a Steady State:
xt = (1+γ)t-1x0
(14)
ct = (1+γ)t-1c0
(15)
The Steady-State Growth Menu is the set of
values c0 and γ that satisfy (14) and (15)
Theorem 3
If a von-Neumann model with Ramsey optimization is allowed to trade
with another economy growing at the same balanced growth rate,
this will expand (strictly cannot contract) the said economy's steadystate growth menu. The consequence of this menu expansion for
the choice of γ is ambiguous.
Informal Argument: Note again that the possibility of international
trade is equivalent to adding extra activities to the input and output
matrices of the model. The benefit of having these additional
activities need not be taken out as higher growth, because the
growth rate is not being maximized. All that is certain is that the
value of the objective function (6) can only increase as additional
activities are made available.□
A Simple Model of Endogenous
Growth
The Production Functions
Oi = ni[liβi - αi] (16)
Maxnl
n[lβ-α] + λ[L - nl]
(17)
O is:
α{(β-1)/β}βL(1/(1-β))-((1+β)/β) (24)
Output is linear in L
Oi = μiLi
(25)
Production equations
Log consumption is:
lnC = T + Oc (26)
Total labour is 1.
Lc is labour producing consumption
The dynamic equation for T is:
dT/dt = aμr(1 - Lc) (27)
Maximization
The Planner maximizes:
∫∞0U[T + μLc]e-δtdt
(28)
where U is the utility of the log of consumption
Hamiltonian with p0 = 1
U[T + μLc]e-δt + p1aμr(1 - Lc) (29)
U1[T + μLc]e-δt - p1aμr = 0
(30)
The co-state variable condition
dp1/dt = - U1[T + μLc]e-δt
(31)
An Optimal Condition
Differentiating (30) totally with respect to
time and taking into account (31)
-(du/dt)/u = a(μr)/(μc) – δ (32)
where u = U1
(32) Is like a Ramsey necessary condition
but NB U is the utility of the log of
consumption
Translating the Optimal Condition
Note that:
dU[lnc]/dc = u/c
(33)
-[d/dt(dU[lnc]/dc)]/dU[lnc]/dc
= (1/c)(dc/dt) + a(μr)/(μc) – δ
(36)
Consumption growth has the same effect as
a reduction in the discount rate
Try a Special Case
U[lnc] = {1/(1-η)}e(1-η)lnc (37)
This is the standard constant-elasticity
function with its argument lnc
Now (36) becomes:
(1/c)(dc/dt) = (a(μr)/(μc) - δ)/η (38)
The solution is always a steady-state
Theorem 4
Theorem 4: Trade increases the steady state
growth rate of the economy iff it increases the
ratio μr/μc.
Proof: By inspection of equation (23).□ To
provide the intuition of this result it is only
necessary to note that for trade to increase the
growth rate it has to raise R&D efficiency relative
to the efficiency of delivering current
consumption.
A similar point is made by Grossman and Helpman
(1991), Innovation and Growth in the World
Economy