3.    (FromLecture Note) Let K and P be the stocks of capital and pollution, respectively. Output isproduced with the production function Y = Kα (α ∈ (0, 1)). In this economy, the market clearing condition isthat savings, which is a fixed proportion of output, is always equals investment. A given stock of capital generates Kβ (β > 0) units of pollution. Capital andpollution decay at the rate of δ > 0 and γ > 0, respectively. We also have initial conditions K(0) = K0 ≥ 0 and P (0) = P0 ≥ 0. Thelaws of motion for this economy are:

To characterizethe dynamic graphicallym we firstplot the loci of points(in (K, P ) space), such that = 0 and   = 0, with their intersection determining the interior steady state.

and = 0 ⇒ P = (1/γ)Kβ. The = O consists of the P axis and a vertical line at K= K*, while the P=0 locus is a monotonicallyincreasing function of K. There are two steady states.  i.e. points where the two loci meet; the origin and (K∗, P ∗), where P ∗= (1/γ)(K∗)β. The phase space is divided into four regions. To determine the direction of movement withinany region consider the followingpartial derivatives:

Answer the following questions.

(a) Draw the phase diagram of thesystem above ((4) and (5)).

(b) Markthe stablesteady state (K∗, P∗)

NOTE: The slope of thetrajectories are given by dP/dK = 1, and in this case the trajectories never cross the = 0 locus,providing access to the equilibrium from above and below. The trajectories may move with the slope dP/dK =/= 0/=0.

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