HE3002 Take-home Quiz 1
2022 September
Instructions: 1) Do the following 4 questions as a take-home assignment. 2)
Discussions around peers are strictly NOT allowed. 3) In every page of your submission, please specify your matric number and your name. 4) Working is not
compulsory in your submission, but they may be useful if your final answer is
wrong, then I can still offer you some credits from your workings. 5) Each question is 20 marks.
1. In the “Endogenous economic growth and R&D model” in our lecture notes,
when we allow both capital and knowledge to depreciate at rate δK > 0 and
δA > 0, respectively, so that the capital and knowledge are accumulated according
to
K 0 (t) = I(t) − δK K(t)
A0 (t) = B[αk K(t)]β [αl L(t)]γ A(t)θ − δA (t)A(t).
Please show whether there is a BGP. If yes, characterize it. If no, prove your
argument.
2. We introduce government spending into the Solow model. The growth accounting equation now becomes: Y (t) = C(t) + I(t) + G(t). Production function still
takes the standard Cobb-Douglas form:
Y (t) = AK(t)α L(t)1−α
where A is a constant and total population grows at rate n. Assume government
spending is given by G(t) = σY (t).
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(a) If government spending is fully financed through investment so that investment becomes I(t) = I 0 (t) − G(t), where I 0 (t) denotes the investment in the
case of no government spending. Derive the physical capital accumulation
equation. Characterize the steady-state of the economy. Is it possible to
have multiple steady-state equilibrium? (Hint: I 0 (t) is essentially sY (t)).
(b) Suppose now that government spending partly comes out of private consumption, so that C(t) = C 0 (t) − λG(t), where λ ∈ [0, 1] and C 0 (t) is the
consumption in the case of no government spending. The remaining (1 − λ)
of G(t) is still financed by investment: I(t) = I 0 (t) − (1 − λ)G(t). Discuss
how the value of σ affects your answer to part (a)?
(c) Now suppose that a fraction φ of G(t) is invested in the capital stock, so that
total investment at t is given by:
I(t) = (s − (1 − λ)σ + φσ)Y (t)
show that if φ is sufficiently high, the steady-state level of capital-labor ratio
will increase as a result of higher σ.
3. Consider the following production function:
Y = A[αK 1/2 + (1 − α)L1/2 ]2 , α ∈ (0, 1)
where K and L are capital and labor inputs respectively. A denotes total factor
productivity.
(a) Prove that the production function (1) has positive marginal product of capital (MPK) and labor (MPL). (2) MPK is decreasing in K and MPL is
decreasing in L. (3) constant return to scale in K and L.
(b) When time is continous and infinite, assume population growth rate is n and
capital depreciation rate is δ > 0. Characterize the steady-state of a Solow
economy.
(c) When utility function takes form u(c) =
c1−σ −1
1−σ ,
characterize the balanced
growth path in a neoclassical growth economy. (σ > 0)
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4. We introduce the notion of subsistence consumption into both Solow and neoclassical growth economy. The idea is that “normal” consumption and investment
decision will be made when subsistence consumption is met. Denote c to be the
subsistence consumption per-individual. Therefore, in any time t normal consumption (c(t)) and investment (I(t)) satisfy
c(t)L(t) + I(t) = Y (t) − cL(t),
where L(t) is population size in t. Assume production function take form Y (t) =
AK(t)α L(t)1−α and capital depreciation rate is δ > 0. In addition, assume population growth rate is n > 0.
(a) Discuss steady-state and balanced growth path in a Solow economy.
(c−c)1−σ −1
(b) When utility function takes form u(c) =
, discuss steady-state and
1−σ
balanced growth path in a neoclassical growth economy.
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