巴罗宏观经济学pdf课件barro_ch11

• Instead we assume that saving is a constant fraction of income, denoted s, and that aggregate saving equals aggregate investment, as in the Chapter 9 model:
VeriÞcation in the Data
• When the data correspond to different US states or different regions within Europe, the observation that poor regions (with low Y /L in some initial observation period) tend to grow faster (they have faster growth of Y /L in some subsequent period of observation). See Figure 3(a).
5. The model predicts that K/L and Y /L will converge towards constantly moving target paths that correspond to the changing values of (K/L)∗ and (Y /L)∗ = f [(K/L)∗, 1].
Allowing for Population Growth • New Assumptions:
— Now we will assume that the quantity of labor grows at the same rate as the population. If we assume that the population growth rate is a constant n, then Lt = (1 + n)Lt−1.
2
Allowing for Technological Progress
• Our last modiÞcation of the model assumes that there is steady technological progress that steadily shifts up the production technology, f (k, l). As a result, the steady state for K/L illustrated in Figure 2 keeps moving upward over time as the f -curve shifts.
and (2):
Kt = sf (Kt−1, Lt) + (1 − δ)Kt−1.
(4)
This implies that the change in the capital stock (net investment) is given by
∆K = Kt − Kt−1 = sf (Kt−1, Lt) − δKt−1.
Barro, Chapter 11 Craig Burnside Economics 302
University of Virginia
The Solow Growth Model
• To understand long-term growth we step aside from the model of Chapter 9 in the sense that we do not use the same model of what determines investment. It turns out that much of what we will say about long-term growth would be the same if we used the Chapter 9 model, but the simpliÞcations of Chapter 11 make the math easier.
1
• The Solow model can be interpreted as a model of economic growth. Notice that if the quantity of labor is constant, the gross growth rate of GDP is
Empirical Validation of the Model
Model Properties
• The model has the following properties:
1. The lower K/L is, the lower Y/L is, since Y /L = f (K, L)/L = f (K/L, 1). 2. If K/L is below (K/L)∗ then K/L grows (and, given the previous property, so does
1
!1
"
δ
(K/L)t − 1 + n(K/L)t−1 = sf 1 + n (K/L)t−1, 1 − 1 + n (K/L)t−1.
Solving for ∆(K/L) we get
!1
" δ+n
(K/L)t − (K/L)t−1 = sf 1 + n (K/L)t−1, 1 − 1 + n (K/L)t−1
• Notice that this would allow us to draw a diagram like Figure 2 with the x-axis now
representing the capital labor ratio. The point where the two curves cross would now be a steady state in the capital-labor ratio, (K/L)∗.2
• The model does not work as well when the individual regions are countries that include vastly different countries, such as developing, middle-income and industrialized countries. In this case, the data suggest no association of initial Y /L and subsequent growth.
Yt+1/Yt = f (Kt, L)/f (Kt−1, L).
In this basic version of the model growth happens because capital is accumulated if a country starts out with less capital than the steady state amount.
1Be sure you understand the distinction between constant returns to scale and diminishing returns to the inputs k and l.
2It turns out that the faster the population growth rate, n, the lower the steady state value (K/L)∗ becomes.
• In this setting there are two components to the growth process: (i) technological progress that shifts the steady-state or target level of K/L upward over time, and (ii) accumulation of capital in an attempt to catch up with the moving target.
(5)
Barro simpliÞes this by using the notationБайду номын сангаас∆K = sf (K, L) − δK. • Assumptions:
— L is constant
— MP K diminishes in K (we assumed this in Chapter 9). We also assume that MP K → 0 as K → ∞ (this is a slightly stronger assumption).
• The capital stock can now grow, even in the long-run, as long as the population grows. The problem is that industrialized economies seem to have continuously growing values of (K/L). In other words, population growth can only explain part of the process of capital accumulation we have observed.
It = sYt.
(1)
• We still use two components of the Chapter 9 model:
Yt = f (Kt−1, Lt)
(2)
Kt = It + (1 − δ)Kt−1.
(3)
• Notice that we can rewrite (3) by substituting out the expressions for It and Yt in (1)
Y /L) 3. It also turns out that the lower K/L is the faster the growth of K/L.3
4. As a consequence of the previous point, the lower Y /L is the faster the growth of Y /L should be.
• These assumptions imply that sf (Kt−1, L) and δKt−1 look like the graphs in Figure 1 and cross at some steady state K∗ where ∆K = 0.
• So,
— if Kt−1 = K∗, it follows that Kt = K∗. — if Kt−1 < K∗, it follows that Kt > Kt−1 and — if Kt−1 > K∗, it follows that Kt < Kt−1.
— We assume that the technology displays constant returns to scale; i.e. f (ak, al) = af (k, l) for any a > 0.1
• Now we want to look at how the capital-labor ratio, Kt/Lt changes over time. Notice that from (5), if we divide through by Lt, we have
Kt Lt
−
Lt−1 Kt−1 Lt Lt−1
=
1 Lt
sf
(Kt−1,
Lt
)
−
δ
Lt−1 Lt
Kt−1 . Lt−1
If we use the constant returns to scale assumption, and the fact that Lt = (1 + n)Lt−1,
we get
• A problem with the model is that it is inconsistent with the apparent ability of industrialized economies (such as the US) to keep accumulating more and more capital, apparently with no convergence towards a steady state K∗.