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Problem Set 1 (Growth Models)

ECON 6002

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 popula- tion growth n.

(a) Show that f′ (k) > 0, f′′ (k) < 0, and the Inada conditions limk→0 f′ (k) = < and limk→/ f′ (k) = 0 are satisﬁed.

(b) What are the steady-state values of k* , y* and c* ? Show your workings.

Now consider an economy with α = 0.3, saving rate s = 30%, population growth n = 3%, technology growth g = 2%, and depreciation δ = 10%. Assume labour and capital are paid their marginal products and that the country is on its balanced growth path.

(d) Solve for the numerical vales of k* , y* and c* ? Show your workings. (e) What is the growth rate of capital K˙ /K along the balanced growth path? (f) What are the growth rates of wages w˙ /w and return to capital ˙r/r? (g) 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) = 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 = ka . 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˙ (1 _ τ )f′ (k) _ ρ _ θg

c θ

(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 τ ) and the elasticity of saving rate with respect to the capital income tax (∂ ln s* /∂ ln τ ).

Now consider the speciﬁc numerical values α = 0.3, the discount rate ρ = 2%, population growth n = 2%, 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%.

(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 τ = 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.