ECON 6002 Problem Set 1 Answer Key
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Problem Set 1 Answer Key
ECON 6002
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.
(a) Show that f/ (k) > 0, f// (k) < 0, and the Inada conditions limk二0f/ (k) = o and limk二o f/ (k) = 0 are satisfied.
limk二0α = o given → o as k → 0 .
limk二o α = 0 given → 0 as k → o.
(b) What are the steady-state values of k* , y* and c* ? Show your workings.
k* = ╱ 、1/(1_α)
y* = ╱ 、α/(1_α)
c* = (1 - s) ╱ 、α/(1_α)
(c) Why is the steady state unique?
Correct answer requires discussion of single crossing given Inada conditions.
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. Using the numerical values provided,
k* s 2.69
y* s 1.35
c* s 0.95
(e) What is the growth rate of capital K˙ /K along the balanced growth path?
= + = n + g = 3% + 2% = 5%
(f) What are the growth rates of wages w˙ /w and return to capital ˙r/r?
w˙
(g) Could the economy achieve a higher c* than for s = 30%? Why or why not?
Some discussion that golden rule is achieved when s = α = 0.3 in this case is required.
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?
The economy has k(0) = 1.35 at t = 0 and so
k˙ sf (k) - (n + g + δ)k (0.3)(1.35)0.3 - (0.03 + 0.02 + 0.1)(0.5)
k k 1.35
Thus, the growth rate of capital at t = 0 is
K˙ k˙ A˙ L˙
K k A L
(i) What are the growth rates of wages w˙ /w and return to capital ˙r/r at t = 0?
At t = 0, we have k = 1.35, f (k) s 1.09, f/ (k) s 0.24, f// (k) s -0.13 and k˙ s 0.13 . Thus, the growth rate of wages and return to capital are
w˙
w r˙ r
= g + s 0.05 = 5%
f// (k)k˙
f/ (k) - δ
(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?
Some discussion about how model predicts high growth along the transition path needed. Some discussion also needed about how this is broadly consistent with what happened in the postwar years for Japan. Japan grew faster than other countries that had less destruction of their capital stocks. Once the transition was complete, Japan’s growth was no longer faster and real interest rates became quite low.
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 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 - τ )f/ (k) - ρ - θg
=
c θ
(a) Derive y* and c* as functions of the model parameters.
y* = ╱ 、α/(1_α)
c* = ╱ 、α/(1_α) - (n + g) ╱ 、1/(1_α)
(b) Find an expression for the saving rate s* = (y* - c* )/y* on the balanced growth path.
s* = (n + g)k*1_α = (n + g) ╱ 、
(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 ).
ln s* = ln(n + g) + ln(1 - τ ) + ln α - ln(ρ + θg)
÷
∂ ln s* τ
∂ ln τ 1 - τ
ln k* = (ln(1 - τ ) + ln α - ln(ρ + θg))
∂ ln k* τ
∂ ln τ (1 - α)(1 - τ )
ln s* = ln(n + g) + ln(1 - τ ) + ln α - ln(ρ + θg)
∂ ln s* τ
∂ ln τ 1 - 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. Using the numerical values provided,
k* s 6.61
y* s 1.76
c* s 1.50
s* = 0.15
(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?
With the introduction of capital income tax, the new steady-state values become
k* s 4.80
y* s 1.60
c* s 1.41
s* = 0.12
The introduction of capital income tax reduced households’ incentive to save and so the steady-state values of k* , y* , c* and s* are lower in the new steady state.
(f) Describe how the introduction of the capital income tax affects each of c˙ = 0 curve and k˙ = 0 curve?
Some discussion is needed that the after-tax rate of return must equal ρ + θg . Com- pared to the case without a tax on capital, f/ (k), the pre-tax rate of return on capital, must be higher and thus k must be lower for c˙ = 0. Thus the c˙ = 0 locus shifts to the left.
For a given k, the level of c that implies k˙ = 0 is given by c = f (k) - (n + g)k. Because the tax is rebated to households in the form of lump-sum transfers, this k˙ = 0 locus is unaffected.
(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.
Answers should discuss the replication principle such that also doubling technology/ideas in addition to capital and labour would more than double output. Answers should note that perfect competition would mean that producers of ideas will receive no payment for them as a factor of production. That is, constant returns to scale and perfect competition implies rivalrous inputs will be paid marginal products and there will be no economic profits left over to pay for ideas. Thus the development of ideas must be generated by non-economic motivations and/or imperfect competition that allows owners of ideas to earn rents for their use.
2022-05-28