## Other Economic Applications

17.28. Derive the formula for the value Pt of an initial amount of money P0 deposited at i interest for t years when compounded annually.

When interest is compounded annually,

Moving the time periods back one to conform with (77.3), pt = (1 + 0^-1

Using (17.4) since i * 0, Pt = (P0 + 0)(1 4- i)[ + 0 = P0(l + 0'

17.29. Assume that Qdt = c + zPt, Qst = g + hPt, and

i.e., price is no longer determined by a market-clearing mechanism but by the level of inventory Qst Qdt- Assume, too, that a>0 since a buildup in inventory (Qst> Qdt) will tend to reduce price and a depletion of inventory (Qst< Qdt) will cause prices to rise, (a) Find the price Pt for any period and (b) comment on the stability conditions of the time path.

a) Substituting Qsl and Qdt in (17.26),

= [1 - a(h - z)] Pt ~ a(g -c) = [ 1 + a(z ~ h)] P( - a(g - c)

Shifting the time periods back 1 to conform to (17.3) and using (17.4), p,= \p0 + T J [1 + a(z - h)Y- - C)

Substituting as m (17. Ma).

b) The stability of the tune path depends on b - 1 + a(z - h). Since a>0 and under normal conditions t < 0 and h > 0. a{i h) < 0. Thua.

If 0<|«(r -A)|<1. If a(z — h) - -1, If -2<a(t -/»)< -1, If a(z - h) = -2. If a(z - h)< -2.

0< b < I; P, converge» and ts nono»cillalory.

h = -I; uniform oscillation takes place.

17.30. Given the following data, (a) find the price P, for any time period: (b) check the answer, using r = 0 and t — 1; and (c) comment on the stability conditions.

0ft-120-0JP, Q,,^ -30 + 0.3/®, P,*1 = P,-02(ya-Qj,) A) = 200

,7) Substituting. Plrl - P, - 0.2(- 30 + 0.3P, - 120 + 0.5P,) 0.84P, + 30

Shifting tune periods back 1 and using (17.4).

M P0 « 200; Pi = 198. Substituting in the first equation of the solution, 198 « 200 - 0.2(-30 ■*■

0.3(200) - 120 + 0.5(200)) = 198. c) With b = 0.84. P, coovcrges without oscillation toward 187.5.