By Professor Dr. Pierre N. V. Tu (auth.)
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Additional resources for Dynamical Systems: An Introduction with Applications in Economics and Biology
28) is Xt = 1 + (AI + A2t)( _2)t = 1 + (1 + 2t)( _2)t where (At, A 2 ) = (1,2) as from initial conditions. 4. 56. 56t). At t = 0, Xo = 2 = 1 + Bl => Bl = 1. 9975) B2 ~ 1/3. 56t). 16 as before. 4. e. (n > 2) is of the form + alXt-1 + ... 31) with n given initial conditions. Without loss of generality, we can set ao = 1 (and of course an =i' 0) and g(t) = 0, and obtain an exact parallel of the treatment of ODE in Chapter 2. t-l + ... n-l + ... n-l + ... + an, an nth order polynomial whose solution involves n roots which may be all real and distinct, or complex, or repeated on lines (multiplicity m, see Chapter 4).
Some Economic Applications 1. The IS-LM Model of the Economy. e. Y = h(D - S) and r = m(L(Y) - M). e. C = cY (0 < c < 1) and I = -ar where c, a > o. e. S = Y. e. money is demanded for transactions purposes only, in the spirit of Fisher's quantity theory of money which was subsequently developed by the Cambridge School as represented by Marshall and Pigou. Money supply (M) is assumed to be undertaken by the Central Bank. e. h = 1 = m for simplicity, we can describe the economy as Y = h(D - Y) = -sY - ar (h r = m[L(Y) - 1\1] = kY - M = 1, s,d > 0) (m = 1, k > 0).
This selection is rather difficult since ODE are commonly used in almost every field of Economics. It is hoped that this review of ODE provides a useful background and a handy reference for use in later chapters. 1. Introduction In Chapter 2, we discussed ODE which involve a variable xCt) and its derivatives X, X, x which give continuous rates of change. ) involving a variable Xt and its differences AXt, A 2Xt etc .... The variable in question varies discretely, or more correctly, although it changes continuously, the observations of these changes are made and recorded only at intervals.
Dynamical Systems: An Introduction with Applications in Economics and Biology by Professor Dr. Pierre N. V. Tu (auth.)