Macroeconomics AECO601 I Solution to Final Exam (1) The rst 3 parts were straight from class notes. The equations for the system are: ct = f (kt ) + (1 u (ct ) = f 0 (kt ) + (1 u0 (ct+1 )
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Fall 2005
)kt )
kt+1
In the 4th part of the question there is an unforeseen decline in the discount rate. Recall that is the discount factor and that if is used for the 1 discount rate then = 1+ and in steady-state the second equation becomes f 0 (k ) = +
A decline in therefore leads to an increase in k because f (:) is concave. (I did not penalize any one for mistaking the discount rate for the discount factor. If that mistake was made then k would fall.) The main diagram for part 4 is Figure 1. For simplicity we start from an initial steady-state, point A. We know that eventually the economy will reach point C. Other than that we must always be in equilibrium and there are no anticipated jumps in c: When the unanticipated fall in happens the economy jumps from A to B immediately (i.e. on Jan 1). After that the economy converges over time (asymptotically) to point C. Figure 2 shows at the case when the change in 1). In this case we know that we have to be on the saddle-path through point C on Jan 1 but only the news in Nov 1 can cause a jump in c: On Nov 1 the economy jumps to point B1 between Nov 1 and Jan 1 it is on the path from B1 to B2. The economy reaches B2 exactly on Jan 1. Thereafter it tracks toward point C. is anticipated (on Nov
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c A C
B
k*c
k*b
k
Figure 1: Unanticipated decrease in :
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c A B1 B2 C
k*c
k*b
k
Figure Anticipated 2: decrease in :
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(2.1)There are 3 employment states: unemployed, value Vu ; employed without promotion, Ve and promoted, Vp : (The wage is another dimension to the state.) Asset value equations. (Ignoring the possibility of both events happening in any period means that an employed worker gets promoted with probability ; laid-o with probability and remains employed with probability 1 :) rVu = b + Ew [max fVe (w) Vu ; 0g] rVe (w) = w + (Vu Ve (w)) + (Vp (w) Ve (w)) rVp (w) = w + (Vu Vp (w)) To show that promotion is always a good thing we evaluate Vp (w) r (Vp (w) So Vp (w) Ve (w) = Ve (w)) = ( 1)w (Vp (w) Ve (w)) (Vp (w) Ve (w): Ve (w))
( 1)w >0 r+ +
(2.2) The 3 unknowns in population are nu ; ne and np : Figure 3 shows the diagram. ow Equations: nu + ne + np ne + ne ne ne + np = 1 = (1 = np = (1 F (w ))nu F (w ))nu
One of these linear equations is super uous. (2.3) This i...