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Homework answers / question archive / Solow model with a government [20 marks] Consider a version of the basic Solow model which is the same as that discussed in the lec- tures except there is now a government
Solow model with a government [20 marks] Consider a version of the basic Solow model which is the same as that discussed in the lec- tures except there is now a government. Suppose that a government purchases goods in the amount of g per worker every year; with N? workers in year t, total government purchases are gNt. The government has a balanced budget so that its tax revenue in year t, Tt, equals total government purchases. Total national saving St is St = s(Y4 – Tt), where Yt is total output and s is the national saving rate. (a) Derive the equation for the dynamics of capital per worker in this economy. Illustrate the steady-state for this economy on a diagram. (b) Suppose that the government permanently increases its purchases per worker and con- tinues to balance its budget. What are the effects on the steady-state levels of capital per worker, output per worker, and consumption per worker? Illustrate with a diagram. (c) Describe how the economy moves from the original steady state to the new one. What happens to the total capital stock during this process? (d) Now suppose that the government receives annual foreign aid that exactly covers the cost of its purchases. How will this affect the steady state levels of capital, output and consumption per worker?
(a)
In the Solow model, the steady state of capital per worker and the output per worker is given by the equation as follows:
K* = ( sa/ g + d ) 1/1-b
First, let us define the capital-output ratio as
xt = kt / yt ...(i)
So, the production function can be expressed as
Yt = At (xtYt)^ α Lt ^1−α .... (ii)
Here, we are using the fact that
Kt = xt Yt ...(iii)
Dividing both sides of this expression by Y α t , we get
Yt^ 1−α = At xt^α Lt^ 1−α
Taking both sides of the equation to the power of 1/ 1−α we arrive at
Yt = At^ (1/ 1−α) xt^ (α/ 1−α ) lt
So, output per worker is
Yt/ Lt = At^ (1/ 1−α ) xt^ (α /1−α)
(b)
If the government permanently increases purchases per worker, the s[f(k) – g] curve shifts down from s[f(k) – g 1 ] to s[f(k) – g 2 ] in the diagram represented as below. In the steady-state equilibrium, the K/L ratio is lower. The optimal level of purchases by the government is not 0 (zero) because it depends on the benefits /profits and cost of the purchases.
please see the attached file for the complet solution.
