Problem Set 1 (Neoclassical Growth Models)
ECON 6002
Due date: Monday, 20 March, 6pm
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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 = kα, constant savings rate s, depreciation rate δ, productivity growth g and population
growth n. Suppose α = 0.4, saving rate s = 25%, population growth n = 3%, technology
growth g = 2%, and depreciation δ = 0% (i.e., no depreciation). Assume labour and capital
are paid their marginal products and that the country is on its balanced growth path.
(a) Solve for the numerical vales of k∗, y∗ and c∗ to two decimal places? Show your work-
ings.
(b) What is the growth rate of capital K˙/K along the balanced growth path?
(c) What are the growth rates of wages w˙/w and return to capital r˙/r?
(d) Could the economy achieve a higher c∗ than for s = 25%? Why or why not?
Now assume a meteorite takes out 75% of the capital stock such that the new capital stock
at t = 0 is k(0) = 14k
∗.
(e) What is the growth rate of capital K˙/K at t = 0?
(f) What are the growth rates of wages w˙/w and return to capital r˙/r at t = 0?
(g) 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 = kα.
Suppose that capital income is taxed at a constant rate 0 ≤ τ < 1 . This implies that
the real interest rate that households face is now given by r(t) = (1 − τ)f ′(k(t)). Assume
that the government returns the revenue it collects from this tax through lump-sum trans-
fers. 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˙
c
=
(1− τ)f ′(k)− ρ− θg
θ
(a) Find an expression for the saving rate s∗ = (y∗ − c∗)/y∗ on the balanced growth path.
(b) Derive an expression for the elasticity of saving rate with respect to the capital income
tax (∂ ln s∗/∂ ln τ).
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Now consider the specific numerical values α = 0.4, the discount rate ρ = 2%, population
growth n = 3%, technology growth g = 2%, the coefficient of relative risk aversion θ = 3.
Assume initially that the economy is in steady state with no capital taxation, i.e. τ = 0%.
(c) Determine the numerical values of k∗, y∗, c∗, and s∗ in the steady state.
(d) If the saving rate were instead fixed at the Golden rule level, would households be bet-
ter or worse off in steady state? Explain your answer.
(e) Suppose that the government increases the capital income tax to τ = 20% and that
this change in tax policy is unanticipated. Compute the new steady-state value of s∗.
How does the new steady state compare to the situation without taxation?
(f) Draw the transition for the economy given the introduction of the capital income tax
using the phase diagram for the Ramsey model.
(g) What would be different and what would be the same if instead of imposing a capital
income tax of τ = 20%, the government mandated a fixed saving rate of 20%? Explain
your answer.
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