ECONOMIC GROWTH. Lviv, September 2012. Growth in Finland. The Sources of Economic Growth. Production function Y = AF ( K , N ) Decompose into growth rate form: the growth accounting equation D Y / Y = D A / A + a K D K / K + a N D N / N
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Lviv, September 2012
Y = AF(K, N)
DY/Y =DA/A + aKDK/K + aNDN/N
Labor TFP Growth rate
Productivity of K/N
Labor productivity growth may exceed TFP growth because of faster growth of capital relative to growth of labor
where L*E is the number of effective workers.
Ct=Yt – It
(kt is also called the capital-labor ratio)
c=f(k) - (δ+n+g)kt
where s is the saving rate, which is between 0 and 1
sYt= (n+d + g)Kt
sf(k) = (n+ δ + g)k
consumption is c* = f(k*) – (n + d + g)k*
1. capital – physical or human – per worker
2. the efficiency of production (the height of the production function)
Y = AK
DK + dK = sAK
DK/K = sA – d
DY/Y = sA – d
Growth may accelerate also because substitution between labor and capital increases (the efficiency in production increases!)
The net investment is:
K’ = I – δK = Y – C – δK
Let the objective function to be:
max ∫ U(C)e– βt dt
Here we first move to labor intensive forms so that y = Y/L and k = K/L, and c = C/L.
Thus, the net investment equation can be written in the form:
K´= knL + Lk’, where n is the growth rate of labor force.
k´= y – c – (n+δ)k = φ(k) – c – (n+δ)k
Ĥ = U(c) + μ(φ(k) – c – (n+δ)k)
Here u is the control and k the state.
Now the necessary conditions are:
∂H/∂c = U´(c) – μ = 0
μ´ = – ∂Ĥ/∂k + βμ = – μ(φ’(k) – (n+δ + β))
k´= φ(k) – c – (n+δ)k
and k(0) = k0 and lim(T→∞)λ(T) ≥ 0,
lim(T→∞) k(T) ≥ 0 and lim(T→∞) λ(T)k(T) = 0
differentiating both sides
μ´ = U”(c)c’
so we can rewrite c’ = (U’/U”)( φ’(k) – (n+δ) – β)
U”/U’ is the so– called elasticity of marginal utility, to be denoted
Its inverse, η(c)– 1, in turn is called the instantaneous
elasticity of substitution.
c’/c = – (1/η(c))(φ’(k) – (n+δ) – r)
which says that consumption growth is proportional to the
difference between the net marginal produce of capital
accounting for population growth and depreciation, and
the rate of time preference.
Keynes– Ramsey rule.
In the steady state, k’ = 0, optimum consumption is given by:
c* = φ’(k*) – (n+δ)k*
If we maximize c* with respect to k* we get
the optimum consumption as:
φ’(k*) = (n+δ)
which is the golden rule. Now, we have the steady state, the difference equations give us the dynamic paths, then we have the draw the phase diagram and analyze the dynamics of k and c.
As usual, in this kind of models, the solution has the saddle point property. Thus, there is only one (stable) trajectory which leads to the steady state.