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# Econ 222 Assignment 3

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Econ 222 Assignment 3

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Econ 222
Macroeconomic Theory I

Assignment #3

Question 1: The Solow growth model
Consider the following production function:
Y (t ) = F (K (t ),A(t )N (t )),

(1)

where Y (t ) denotes output, K (t ) denotes total capital input, A(t ) denotes “knowledge”,
and N (t ) denotes total labour input. F (•, •) is the aggregate production function. Note that
A(t )N (t ) is commonly referred as effective labour, and technological progress that enters
the production function in this fashion is known as labour-augmenting or Harrod-neutral.
a) Derive the intense form (per unit of effective labour) of the production function, using
the abstract production function above.
b) We know the actual production function is
Y (t ) = K (t )α [A(t )N (t )]1−α .

(2)

Denote the population growth rate as n, the depreciation rate as d, the growth rate of
knowledge as д, and the saving rate as s. First derive the steady state condition for this
economy (involving investment per unit of effective labour), and then solve for the steady
state level of capital per unit of effective labour, k ∗ (t ).
c) Given the result in b), solve for the steady state level of output per worker y ∗ (t ), and
consumption per worker c ∗ (t ).

1

d) Solve for the Golden rule level of capital per worker, kG . If the government can choose
a saving rate for the economy, what saving rate should the government choose, assuming
that it wants to achieve kG in the steady state.
e) Suppose the economy is at a steady state, and there is a decrease in the saving rate
s. Explain the impact of this shock regarding steady state variables, with the help of a