Economic growth and economic development 65
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Introduction to Modern Economic Growth
output levels, consumption levels, wages and rental rates {K (t) , Y (t) , C (t) , w (t) , R (t)}∞
t=0
such that K (t) satisfies (2.11), Y (t) is given by (2.1), C (t) is given by (2.10), and
w (t) and R (t) are given by (2.5) and (2.6).
The most important point to note about Definition 2.2 is that an equilibrium is
defined as an entire path of allocations and prices. An economic equilibrium does
not refer to a static object; it specifies the entire path of behavior of the economy.
2.2.3. Equilibrium Without Population Growth and Technological Progress.
We can make more progress towards characterizing the equilibria by exploiting the
constant returns to scale nature of the production function. To do this, let us make
some further assumptions, which will be relaxed later in this chapter:
(1) There is no population growth; total population is constant at some level
L > 0. Moreover, since individuals supply labor inelastically, this implies
L (t) = L.
(2) There is no technological progress, so that A (t) = A.
Let us define the capital-labor ratio of the economy as
K (t)
,
(2.12)
k (t) ≡
L
which is a key object for the analysis. Now using the constant returns to scale
assumption, we can express output (income) per capita, y (t) ≡ Y (t) /L, as
∙
¸
K (t)
y (t) = F
, 1, A
L
(2.13)
≡ f (k (t)) .
In other words, with constant returns to scale output per capita is simply a function
of the capital-labor ratio. From Theorem 2.1, we can also express the marginal
products of capital and labor (and thus their rental prices) as
R (t) = f 0 (k (t)) > 0 and
(2.14)
w (t) = f (k (t)) − k (t) f 0 (k (t)) > 0.
The fact that both of these factor prices are positive follows from Assumption 1,
which imposed that the first derivatives of F with respect to capital and labor are
always positive.
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