ECON 711 Macroeconomic Theory and Policy Problem Set 2
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ECON 711 Macroeconomic Theory and Policy
Problem Set 2
Module 2-1: The Neoclassical Growth Model in Discrete Time
1. Ramsey- Cass-Koopmans model. model. Suppose the planner seeks to maximize the
intertemporal utility function
L(&)βtu ╱ 、 L
subject to the sequence of resource constraints
Ct + Kt|1 = F (Kt, AL) + (1 _ δ)Kt , 0 < δ < 1
given initial Kd > 0. Suppose for simplicity that the labor force L and the level of productivity A are constant. Let ct, kt, yt etc. denote consumption, capital, output etc in per worker units. Suppose that the period utility function and the production function have their usual properties.
(a) Derive optimality conditions that characterize the solution to the planner’s problem. Give intuition for those optimality conditions. Explain how these optimality conditions pin down the dynamics of ct and kt .
The planner’s problem is to maximize intertemporal utility (per worker)
L βtu (ct)
subject to the sequence of resource constraints
ct + kt|1 = F (kt, A) + (1 _ δ)kt ,
given the initial condition kd > 0. Notice that I have used constant returns to write the production function as
Y F (K, AL) K
Setting up the Lagrangian
c = L βtu (ct) _ L λt[ct + kt|1 _ F (kt, A) _ (1 _ δ)kt]
The key ﬁrst order conditions for this problem are, for consumption,
[ct] : βtu\ (ct) = λt
and for capital,
λt = λt|1 [Fk(kt|1, A) + 1 _ δ]
where Fk denotes the marginal product of capital per worker. Optimal allocations must also satisfy the transversality condition
lim βT u\ (cT ) kT|1 = 0
and the resource constraints
ct + kt|1 = F (kt, A) + (1 _ δ)kt .
Eliminating the multipliers λt gives the consumption Euler equation
u\ (ct) = βu\ (ct|1) [Fk(kt|1, A) + 1 _ δ]
To interpret this condition, let Rt|1 = Fk(kt|1, A) + 1 _ δ denote the gross return on capital. Then consumption is increasing, ct|1 > ct if and only if u\ (ct|1) < u\ (ct) [since u\ (c) is strictly decreasing] which happens if and only if
1 < βRt|1
In short, consumption in t + 1 will be high relative to consumption in t when the return on capital is high relative to time discounting. When the return on capital is relatively high, it makes sense to defer consumption today and invest in physical capital to secure more consumption tomorrow so that ct|1 > ct .
Rewriting the discount factor β = in terms of the discount rate ρ > 0 we can equiva- lently say that consumption is increasing if and only if
ρ + δ < Fk(kt|1, A)
that is, if and only if the marginal product of capital is relatively high. The consumption
Euler equation and the resource constraint are two nonlinear diﬀerence equations in ct, kt . To pin down the dynamics of ct, kt we also need two boundary conditions. One of these is the given initial condition kd > 0. The other is the transversality condition given above.
(b) Now suppose that the production function is Cobb-Douglas, Y = F (K, AL) = K← (AL)1一← with 0 < α < 1. Derive expressions for the steady state values k* , y* , c* in this economy. How do these steady state values depend on the period utility function? Explain.
If the production function is Cobb-Douglas F (K, AL) = K← (AL)1一← then y = F (k, A) = k← A1一← with marginal product Fk(k, A) = α (k/A)← 一1 . In a steady state with ct = ct|1 = c* the consumption Euler equation implies
1 = β[α (k* /A)← 一1 + 1 _ δ]
which can be solved to get steady state capital per worker
k* = ╱ 、1 _o A
which is proportional to the level of productivity A. Steady state output per worker is then found from the production function
which is also proportional to the level of productivity A. Note that the steady state capi- tal/output ratio is independent of productivity
y* ρ + δ
Using the resource constraint c* = y* _ δk* , the steady state consumption/output ratio is then
c* k* αδ ρ + (1 _ α) δ
y* y* ρ + δ ρ + δ
also independent of productivity. The level of steady state consumption per worker is then
c* = y* = ╱ 、 A
None of these expressions depend on the period utility function u(c) at all. This is an artefact of time-separable preferences (and the absence of trend growth). In this setup, the utility
function matters for the short-run dynamics but not for the long-run (steady state) values.
(c) What is the steady state savings rate in this economy? Explain how the steady state savings rate depends on the parameters α, β, δ and the level of productivity A. Give intuition for your answers.
Let the savings rate be st = St/Yt = It/Yt . In steady state, s* = δ so using our results from part (b) we have
y* ρ + δ
The steady state savings rate s* is increasing in capital’s share α, increasing in the depreciation rate δ, and decreasing in the rate of time preference ρ . Intuitively, when capital contributes more to output then savings is higher (other things equal). When the depreciation rate is higher, then in steady state the savings rate is higher to compensate. When the planner discounts the future more (ρ is higher, i.e., the planner is more impatient), then the savings rate is lower.
(d) Suppose the economy is initially in the steady state you found in (b). Suppose the economy becomes more patient with the discount factor increasing from β to β\ > β . Use a phase diagram to explain (i) how this change aﬀects the long-run values of consumption, capital and output, and (ii) how the economy transitions to these new long-run values. How would your answers diﬀer if the discount factor fell from β to β\ < β . ?
First note that since the discount factor is β = 1/(1 + ρ), an increase in the discount factor is equivalent to a decrease in the discount rate. For example, an increase in the discount factor from β = 0.98 to β\ = 0.99 is a decrease in the discount rate from ρ s 0.02 to ρ\ s 0.01 . Then using the expressions in part (b) above, it is clear that a decrease in the discount rate from ρ to ρ\ < ρ increases steady state capital from k* to k*\ , say, and hence increases steady state output from y* to y*\ and steady state consumption from from c* to c*\ .
Then relative to these new steady state levels the economy ‘begins’ with initial capital per worker kd = k* < k*\ . On ‘impact’ the level of consumption immediately jumps down to cd < c* on the new stable arm going through the new steady state. Capital does not jump on impact because it is predetermined and hence output does not change on impact either. The level of consumption jumps down because the economy is more patient and is saving more. This increase in saving/investment is what allows the economy to build up a new higher level
of capital in the long run. As the economy transitions to its new long run, output rises as more capital is accumulated and this allows consumption to rise too. As time passes, consumption keeps increasing, passing the old steady state c* , and converging in the long run to the new steady state c*\ > c* .
If instead there is an increase in the discount rate from ρ to ρ\ > ρ then steady state capital decreases to k*\ < k* and hence steady state output decreases to y*\ < y* and hence steady state consumption decreases to c*\ < c* . Then relative to these new steady state levels the economy‘begins’ with initial capital per worker kd = k* > k*\ . On ‘impact’the level of consumption immediately jumps up to cd < c* on the new stable arm going through the new steady state. Capital does not jump on impact because it is predetermined and hence output does not change on impact either. The level of consumption jumps up because the economy is less patient and is saving less. This decrease in saving/investment leads to a long-run decline in the level of capital. As the economy transitions to its new long run, output falls as the capital stock falls and this means consumption falls too. As time passes, consumption keeps decreasing, passing the old steady state c* , and converging in the long run to the new steady state c*\ < c* .
2. Isoelastic utility function. Consider the utility function
c1一。 _ 1
u (c) = 1 _ σ , σ > 0
(a) Show that u(c) is strictly increasing and strictly concave.
1 _ σ
u\\ (c) = _σc一。一1 < 0
Hence u(c) is strictly increasing and strictly concave.
(b) Show that the relative curvature of the utility function
u\\ (c) c
is independent of the level of consumption.
Using these derivatives, the relative curvature of the utility function is
u\\ (c) c _σc一。一1 c
_ u\ (c) = _ c一。 = σ > 0
The elasticity of marginal utility u\ (c) with respect to consumption is the constant σ .
(c) Consider the case σ = 1. Show that this corresponds to u(c) = log c. [Hint : what is the antiderivative of c一1 ?]
From l’Hoˆpital’s rule we have that for any c > 0 the limit
。(l) c1一1 = ┌ ┐ = 。(l) dd。╱dd。c1(一1
lim。→ 1‘[c1一。log c┐ (_1)、 [log c] (_1)