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Unformatted text preview: Economics 2023, Problem_Set 2
maurice Obstfeld 1. (A iineaﬁiaed dynamic model) Consider a case with no population growth and a 100% per period capital depreciation rate, so that the planner maximizes
t .
{:05 11(3) subject to km = f(kt) w ct. Let u(c) = logm], m) = k“, at. < 1. (a) Show that the dynamics along the model's optimal path are described by the pair of nonlinear equationsr a
c w c = {k — c —
t+1 t °#3 t t) t t’ n+1 t t t t “do o k ~k=1<ck. (*) (b) Compute the'steadystate values E and E as functions of a and B. [In an
endogenousgrowth context you might want to return to this problem and think hard about what happens as a e 1.) (c) The multivariate version of Taylor’s theorem states that if the function gtx.y} is smooth enough, its linear approximation near (§,§) 15
Six Y} “ 8(E §) + 25f; §3Ix  E] + §§{§ §)[y _ ‘3
D I ‘ I , y . Use this result to show that the system (*3 you derived in part (a) has the. matrix linear approximation c —'6 on+[1a:]/ct.6 —(1—u)(1—qu)/aﬁz c —E
t . ("J k ~E «1 1/5 kt“? near its steady state. (d) Show that the characteristic roots of the matrix M in (**) (that is. the solutions 7 to the equation detih » 71) a 0, where I is the identity matrix) are 1/118 and 0:. Define the matrix 1‘ = [13% (e) Find any 2 x 2 matrix X such that MK a X?. iHint: Columns of X are called eigenvectors belonging to i/aB and a; they are rays rather than welldefined vectors but you can tie them down by assuming that x21 x22 m 1, then solving for x11 and x12 only.] . I
(f) Why do we need such an X? Define the transformed variables c’, k' by [;,] = Xﬂ1[;]' Show that multiplying (‘*) through by Xl gives the equation *1 ct+{— C ct+1 * C, 1 1 ct _ c ct — c
x — = ﬂ = x a = I“ _ . k m k k' ' k' k — k k'  k’
t+1 t+1 t t Note with satisfaction that F (conveniently and not accidentally) is diagonal. {3} Show that all solutions to system (t) take the form: ‘ I w": __ t": 1 w l I “"11: by! I __"I
ot+1 c — (i/aﬁ) (oo o ), kt+1 k a (k0 k I, where t = 0 is an initial date. (h) Now we want to use these simple solutions to retrieve o and k. Find c w E and kM « E by doing the inverse transformation: if is] wt] m [a Soive for co by showing that only if oo = E + [% « at)“:C — i} will the system reach its steady state. Solve for ct w E and kt — k along the saddlepath. Show that the planner should set con5umption [near the steady state) by the rule a
“ (lmaﬁ) 1a
of _ m—§—— kt + (a8) (1~aB){1~a). 2. (Zero discounting) In his famous 1928 Economic Journal article on optimal saving, Frank P. Ramsey argued that it is morally indefensible to discount the welfare of future generations. He therefore argued that a benevolent economic planner should: maximize V = u{c(t)}dt cat—as subject to ltEt) = {mm — cut), mo; given. (Ramsey assumed Zero population growth.) You can see right away the problem
with this formuation: any path that approaches a constant steadystate'
consumption level will yield an infinite value of V. Thus, it is not clear how
to compare such paths and identify one as "optimal." Ramsey finessed the problem in the following way. He defined 3 to be the
"bliss" or maximal steadyvstate consumption level {the existence of which
presupposes that f(k) either eventually becomes decreasing in k or asymptotes to a finite maximum as k goes to m}. He then redefined his problem as that of minimizing I:{u{E) w u[c{t)}}dt, society’s cumulative distance from “bliss”,
subject to the above constraints. Note that this integral can be finite if c{t} e E as t e m (and if it isn’t, it’s not the optimum we seek in any case]. {a} Use the Maximum Principle to derive necessary conditions for a solution to the Ramsey problem. (You can assume a depreciation rate of O for capital. The
resulting Euler condition is sometimes called the KeynesRamsey rule because
J. M. Keynes, a friend of Ramsey’s and editor of the Economic Journal, helped
him to interpret it intuitively.) Show the economy indeed should converge to
"bliss" (also known as the "golden rule" in growth theory.) Interpret the
model's intertemporal Euler condition. You can do so by adressing the
foliowing question: Suppose the economy starts with k(0) < g. Since Ramsey
believed in intergenerational equality, why isn’t it optimal in his view for each generation simply to consume f{k(0)]? (b) Let {GNU}:lo denote the Ramsey consumption path starting from an initial
capital stock Rio), and let {C(tl}:;° be any other consumption path. Show that the Ramsey path overtakes any other feasible consumption path starting from k(0), in the following sense: there exists a finite time 5 such that for all r > r, Iguzcumdt > Igutcmldt. 3 (c) Ramsey states the KeynesRamsey rule as: “rate of saving multiplied by marginal utility of consumption should always equal bliss minus actual rate of utility enjoyed." {By "bliss" he meant u{E).} Can you derive this rule? ...
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 Fall '07
 AKERLOF

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