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Problem Set 1 (Growth Models)

ECON 6002

Due date: Monday, 21 March, 6pm

NOTE: To receive full marks, it is crucial to show all your workings and not just

provide a final algebraic or numerical answer. Numerical answers should be rounded

to two decimal places at most. It is also important that you provide answers in your

own words. Any quotations from the textbook or other sources must be in quotation

marks and attributed to the original source.

1. Consider the Solow-Swan model with the Cobb-Douglas aggregate production function,

y = k ? , constant savings rate s, depreciation rate ?, productivity growth g and population growth n.

(a) Show that f 0 (k) > 0, f 00 (k) < 0, and the Inada conditions limk?0 f 0 (k) = ? and
limk?? f 0 (k) = 0 are satisfied.
(b) What are the steady-state values of k ? , y ? and c? ? Show your workings.
(c) Why is the steady state unique?
Now consider an economy with ? = 0.3, saving rate s = 30%, population growth n = 3%,
technology growth g = 2%, and depreciation ? = 10%. Assume labour and capital are paid
their marginal products and that the country is on its balanced growth path.
(d)
(e)
(f)
(g)
Solve for the numerical vales of k ? , y ? and c? ? Show your workings.
What is the growth rate of capital K?/K along the balanced growth path?
What are the growth rates of wages w?/w and return to capital r?/r?
Could the economy achieve a higher c? than for s = 30%? Why or why not?
Now assume a meteorite destroys half the capital stock such that the new capital stock at
t = 0 is k(0) = 21 k ? .
(h) What is the growth rate of capital K?/K at t = 0?
(i) What are the growth rates of wages w?/w and return to capital r?/r at t = 0?
(j) Compare the growth rates of capital, wages, and returns to capital before and after
the meteorite hit. What do the results predict about growth in an economy after a
war in which a lot of the capital stock is destroyed? Are the results consistent with
what happened in, say, Japan after World War II?
2. Consider the Ramsey model with the Cobb-Douglas aggregate production function, y = k ? .
Suppose that capital income is taxed at a constant rate 0 ? ? < 1 . This implies that
the real interest rate that households face is now given by r(t) = (1 ? ? )f 0 (k(t)). Assume
that the government returns the revenue it collects from this tax through lump-sum transfers. With the introduction of capital income tax, the only change in the model is the Euler
equation, which implies the modified law of motion for consumption:
c?
(1 ? ? )f 0 (k) ? ? ? ?g
=
c
?
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(a) Derive y ? and c? as functions of the model parameters.
(b) Find an expression for the saving rate s? = (y ? ? c? )/y ? on the balanced growth path.
(c) Derive expressions for the elasticity of capital with respect to the capital income tax
(? ln k ? /? ln ? ) and the elasticity of saving rate with respect to the capital income tax
(? ln s? /? ln ? ).
Now consider the specific numerical values ? = 0.3, the discount rate ? = 2%, population
growth n = 2%, technology growth g = 2%, the coefficient of relative risk aversion ? = 3.
Assume initially that the economy is in steady state with no capital taxation, i.e. ? = 0%.
(d) Determine the numerical values of y ? and c? and s? in the steady state.
(e) Suppose that the government increases the capital income tax to ? = 20% and that
this change in tax policy is unanticipated. Compute the new steady-state values of
y ? and c? and s? . How does the new steady state compare to the situation without
taxation?
(f) Describe how the introduction of the capital income tax affects each of c? = 0 curve
and k? = 0 curve?
(g) Draw the transition for the economy given the introduction of the capital income tax
using the phase diagram for the Ramsey model.
3. Jones (2005) Growth and Ideas discusses how ideas are different from other economic
goods in that they are non-rivalrous. Explain in words why the non-rivalrousness of ideas
means that there can be ongoing endogenous economic growth. Why does this non-rivalrousness
also mean that economic growth would not occur under perfect competition? Keep your
answer to less than 200 words.
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T1 Solow-Swan Model
T3 Endogenous Growth
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Tags:
solow model
Capital accumulation
Growth Models
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