Macroeconomic Models

Macroeconomic Models
 * Ct = c + dYt-1 c,d>0
 * It = i + v(Qt-1 - Qt-2) i,v>0
 * Yt = Ct + It and Qt = Yt
 * Yt = (c + dYt-1) + (i + v(Qt-1 - Qt-2))
 * Yt = c + dYt-1 + i + v(Yt-1 - Yt-2)
 * Yt = (c + i) + (d + v)Yt-1 - Yt-2
 * Yt - (d + v)Yt-1 + Yt-2 = c+i

Non-homogenous Equations
 * in yt + byt-1 + cyt-2 = k there is a particular solution
 * sub y* into all the yt, yt-1 and yt-2 and so on y* + by* + cy* = k
 * y* = k/(1+b+c) is the particular solution
 * General Solution for non homogenous equations is Particular Solution + General Solution
 * General Solution is above equation solved as if k was 0 and substituting yt, yt-1 and yt-2 with lamba^2, lambda and zero.
 * Solution to non homogeous equations is Particular Solution + General Solution

No real roots in auxiliary equation?
 * when lambda^2 + b*lambda + c = 0 has no real solutions as b^2 - 4ac < 0, the solution is of the form:
 * r^t*(Ecos(theta*t) + Fsin(theta*t))
 * r = sqrt(c)
 * Find out theta using -b/2r. cos theta = -b/2r. Find our which value of theta gives -b/2r.

Behaviour of Solutions
 * For solutions of the form similar to Yt = a^t ± b^t, the behaviour of the solution can be figured out easily
 * Yt = a^t(1 ± (b/a)^t)
 * Depending on the size of a and b comparatively and sign, the behaviour of solution can be figured out
 * For r^t*(Ecos(theta*t) + Fsin(theta*t)), the solution will always have a oscillatory behaviour due to cos always being between -1 and 1
 * Oscillations will become greater in magnitude if r > 1 that is if c > 1 and if c < 1 the magnitude will decrease

Business Cycles
 * Can apply behaviour of solutions to Macroeconomic model solutions to figure out the business cycle