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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 = ka, 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) = k* .

(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 = ka . Suppose that capital income is taxed at a constant rate 0  s  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˙ (1 _ τ )f\ (k) _ ρ _ θg

=

c 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 ).

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.