C = Yss – sYss = 0.8 * 8 = 6.4

I = 0.1kss = 0.1* 16 =  1.6

Now assume that the savings rate is not fixed.

f)   Solve for the Golden-Rule level of capital per worker and the associated consumption level. (3pts)

C = 2(k)0.5 – 0. 1k

First order condition: (k)-0.5 – 0.1 = 0 , k = (0. 1)-2 , kGR = 100

Thus C = 20- 10 = 10

g)  Solve for the savings rate that allows for the golden rule level of capital in the steady state. (5pts)

In the steady state: 2s(k)0.5 = 0. 1k, sGR = 0.05(k)0.5 = 0.05(100)0.5 = 0.05*10 = 0.5

Question 2 Ramsey-Cass-Koopmans Model (25 pts)

Consider an economy that follows the assumptions of the Ramsey-Cass-Koopmans model.

a)  Write down the k ̂˙ = 0 and ĉ˙ = 0 equations and graph the phase diagram. 5 pts

k ̂˙(t) = f(k ̂(t)) − (g + n + δ)k ̂(t) – ĉ(t)

c˙(t)/c(t) = 1/9 (f’(k ̂(t)) - δ - ρ - 9g)

k ̂˙ = 0: 6 = f(k^( - )2 + n + 6(k^

c˙ = 0: t’)k^) = 6 + q + 9g

ǩ˙=0

b)  An increase in international oil prices permanently decreases the production function f(k ̂), which means that for any given k̂, f(k ̂) and f’(k ̂) are lower than before. How does 6 and k^ react? Explain and graph on the phase diagram from a). 10 pts

A decrease the production function directly affects the k ̂˙ = 0 locus. As f(k ̂) decreases, consumption will be lower for any level of k̂, so the curve shifts down to the green curve.

At the same time, the decrease of f’(k ̂) will unbalance the equation for c˙ = 0. For the equation to remain balanced, k ̂ must decrease, shifting to the new green line to the left.

As a result, both 6* and k^* will decrease to a new equilibrium.

c)  How does ĉ and react to a demographic change that reduces the population growth? Explain and graph on the phase diagram from a). 10pts

The reduction of n does not affect the ĉ˙= 0 locus, so * will remain the same. For any level of capital, the reduction of n leads to a higher associated ĉ , shifting the ˙=0 locus upward to the red curve. The result is the same * and a higher level of ĉ*.

Question 3 Overlapping Generations Model (25 points)

Consider an economy where 2 groups coexist, those young moved to the region to work, and those retired moved-in in the past and now live from their passive income. The utility function for someone born at time t is:

The growth rate of the population and technology are Lt =  (1 + n)Lt−1 and At =  (1 + g)At−1, respectively. Markets are competitive and there is capital depreciation.

(a) Write down the capital accumulation equation. 5pts

Kt+1 = Lt (wtAt − C1,t ) − 6Kt

(b) Specify the budget constraint and the Lagrangian. 5pts

The budget constraint is

C1,t +             = (At wt)

The Lagrangian is

L = + + 入[At wt − C1,t − ]

(c) Derive the Euler equation.  5pts

The First order conditions are

Ct(e) = 入

Ct(e)+1 1

1 + p      1 + rt+1