Economics Dynamics Problems 139

Economics Dynamics Problems 139 - G for all t E t = C t I t...

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Discrete dynamic systems 123 is highly probable. Even with ak < 1, two possibilities arise: (i) if a < 0 then ak < 2 (ii) if 0 < a < 1 then 0 < ak < 2 with various paths for y t . This should not be surprising because we have already established that the discrete logistic equation has a variety of paths and possible cycles. 3.10 The multiplier–accelerator model A good example that illustrates the use of recursive equations, and the variety of solution paths for income in an economy, is that of the multiplier–accelerator model first outlined by Samuelson (1939). Consumption is related to lagged income while investment at time t is related to the difference between income at time t 1 and income at time t 2. 11 In our formulation we shall treat government spending as constant, and equal to G in all periods. The model is then C t = a + bY t 1 I t = v ( Y t 1 Y t 2 ) G t
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Unformatted text preview: G for all t E t = C t + I t + G t Y t = E t which on straight substitution gives rise to the second-order nonhomogeneous recursive equation Y t − ( b + v ) Y t − 1 + vY t − 2 = a + G The particular solution is found by letting Y t = Y ∗ for all t . Hence Y ∗ − ( b + v ) Y ∗ + vY ∗ = a + G i.e. Y ∗ = a + G 1 − b In other words, in equilibrium, income equals the simple multiplier result. The complementary result, Y c , is obtained by solving the homogeneous compo-nent Y t − ( b + v ) Y t − 1 + vY t − 2 = which has the characteristic equation x 2 − ( b + v ) x + v = with solutions r , s = ( b + v ) ± ± ( b + v ) 2 − 4 v 2 11 Samuelson originally related investment to lagged consumption rather than lagged income....
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