Solow growth model 索洛增长模型

Kt+1 = It + (1 − δ)Kt
hence
kt+1
=
AtLt At+1Lt+1
[it
+
(1
−
δ)kt]
=
it 1
+ (1 − δ)kt + g)(1 + n)
(2)
where it ≡ It/AtLt is investment per unit of effective labor. So, if kt+1 = kt we get
2Homegeneous of degree one functions have homogenous of degree 0 partial derivatives.
1
2 Steady State
An interesting question is: What would the level of investment need to be in order for the capital stock to remain constant as a fraction of effective labor; i.e. for kt+1 to equal kt? Notice that (1) implies that
or
g + n + gn + δ = f 0[k∗(s)].
(7)
The condition (7) deÞnes what is known as the golden rule level of the capital stock, which is the steady state level implied by the “optimal” savings policy (optimal in the sense that it maximizes c∗).
It ≡ Kt+1 − Kt + δKt = sYt
(1)
where δ is the rate of depreciation of the capital stock.
1If the production function were written Yt = AtF (Kt, Lt) then technological progress would be Hicksneutral.
kt+1
=
h(kt)
≡
sf (kt) (1 +
+ (1 − δ)kt . g)(1 + n)
(6)
Notice that
h0(k) = sf 0(k) + 1 − δ (1 + g)(1 + n)
so that h0(k) > 0, h00(k) < 0, limk→0 h0(k) = ∞ and limk→∞ h0(k) = (1−δ)/[(1+g)(1+n)] < 1. This allows us to draw the diagram in Figure 2. Importantly, h(kt) is increasing for all k∗, lies above the 45 degree line for kt < k∗, h(k∗) = k∗ is a unique crossing point, and h(kt) lies below the 45 degree line for kt > k∗. These properties imply that when kt < k∗, kt < kt+1 < k∗ so that the capital stock increases steadily as it converges towards k∗. Similarly, when kt > k∗, k∗ < kt+1 < kt so that the capital stock decreases steadily as it converges towards k∗.
I.e. F (cK, cAL) = cF (K, AL) for all c > 0. This means that we can write
yt
≡
Yt AtLt
=
F (Kt, AtLt) AtLt
=
F
µ
Kt
¶
,1
AtLt
≡
f (kt),
where kt ≡ Kt/(AtLt). Here we are expressing output per unit of effective labor as a function of capital per unit of effective labor.
dk∗(s)
f [k∗(s)]
ds = g + n + gn + δ − sf 0[k∗(s)]
Hence, the value of s that maximizes c∗, satisÞes the Þrst order condition
f [k∗(s)]
=
(1
−
s)f 0[k∗(s)] dk∗(s) ds
We will assume that f (0) = 0, f 0(k) > 0, f 00(k) < 0. Notice that f 0(k) = FK(K/AL, 1) = FK(K, AL) from the properties of constant returns to scale functions.2 We also assume that the Inada conditions, limk→0 f 0(k) = ∞ and limk→∞ f 0(k) = 0, are satisÞed.
and L denotes labor. When technology enters multiplicatively with labor it is referred to as labor-augmenting or Harrod-neutral.1
Suppose, in addition, we assume that F is a constant returns to scale production function.
minus investment. Hence
Ct ≡ Yt − It ct = yt − it.
Notice that in the steady state consumption per unit of effective labor is given by c∗ = (1 − s)f (k∗). Suppose we set s so as to maximize c∗ (this may or may not make sense, since we have not made any statements about the preferences of agents in the economy). Notice that (5) implicitly deÞnes k∗ in terms of s, so that totally differentiating we see
2
4 Balanced Growth
At the steady state, we have kt = k∗ for all t. Notice that this means Kt = k∗AtLt = k∗A0L0Gt, where G = (1 + g)(1 + n). Similarly, It = sf (k∗)A0L0Gt and Yt = f (k∗)A0L0Gt. So capital, investment and output all grow at the constant rate G − 1. In per worker terms, they all grow at the net rate g. So, the Solow growth model implies that regardless of the economy’s initial stock of capital, the economy eventually converges towards a steady state in which the variables have these properties, which together, are referred to as a balanced growth path.
Suppose labor and technology evolve according to
Lt+1 = (1 + n)Lt At+1 = (1 + g)At
and that gross investment, It, is a constant fraction, s, of output. I.e.
it = (g + n + gn + δ)kt.
(3)
The level of investment given by (3) is referred to as the break-even level of investment. On the other hand the actual level of investment is given by
kt1sfk?1?k?1g1nk?sf0k?1?1g1nsf0k?1?1g1nkt?k?kt?k?k?kt?k?thismeanskt?k?tk0?k?
The Solow Growth Model Notes Based on Romer’s Advanced Macroeconomics Chapter 1
Craig Burnside Economics 702 University of Virginia
1 The Model
The production function is given by
Yt = F (Kt, AtLt)
where t denotes time, Y denotes output, K denotes capital, A denotes the level of technology,
it = syt = sf (kt).
(4)
Notice that if the capital stock equals k∗, the value where the line in (3) and the function
in (4) coincide,
(g + n + gn + δ)k∗ = sf (k∗)
5 The Golden Rule
We have not talked about consumption, which is not really made explicit in the formulation of the Solow growth model. However, we might imagine deÞning consumption as output
(5)
then the stock of capital per unit of effective labor will remain constant forever (see Figure 1).
3 Dynamics
Another interesting diagram that we can create plots kt+1 versus kt, or kt+1 − kt versus kt. Notice, from (2) and (4), that
Notice that, as is pointed out in the Romer text, the growth rate of output per worker in this economy is determined purely by the rate of technological progress. If one imagined an exogenous change in the economy’s savings rate, s, for example, this would induce a change in the transitory dynamics towards a different value of k∗ (higher if s were higher). But this would have no effect on the economy’s long-run growth rate.