Economics Dynamics Problems 59

Economics Dynamics Problems 59 - so there are turning...

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Continuous dynamic systems 43 Figure 2.7. not possible to consider all points in the ( x , y )-plane. One procedure is to con- sider the points in the ( x , y )-plane associated with a fxed slope. IF m denotes a fxed slope, then f ( x , y ) = m denotes all combinations oF x and y For which the slope is equal to m . f ( x , y ) = m is reFerred to as an isocline . The purpose oF constructing these isoclines is so that a more accurate sketch oF f ( x ) can be obtained. ±or dy / dx = ax by = m the isoclines are the curves (lines) ax by = m or y = ax b m b These are shown in fgure 2.7. OF course, the slope oF f ( x , y ) at each point along an isocline is simply the value oF m . Thus, along y = ax / b the slope is zero or inclination arctan0 = 0 . Along y = ( ax / b ) (1 / b ) the slope is unity or inclination arc tan1 = 45 ; while along y = ( ax / b ) (2 / b ) the slope is 2 or inclination arc tan2 = 63 . Hence, For values oF m rising the slope rises towards infnity (but never reaching it). We have already established that along y = ax / b the slope is zero and
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Unformatted text preview: so there are turning points all along this isocline. ±or m negative and increasing, the slope becomes greater in absolute terms. Consider fnally m = a / b . Then the isocline is y = ax b − ± a b ²± 1 b ² = ax b − ± a b 2 ² with intercept − a / b 2 . Then along this isocline the slope oF the directional feld is identical to the slope oF the isocline. Hence, the direction felds look quite diFFerent either side oF this isocline. Above it the solution approaches this isocline asymptotically From above. Hence, the Function f ( x ) takes the shape oF the heavy curve in fgure 2.7. In general we do not know the intercept or the turning point. In this instance we consider the approximate integral curves , which are the continuous lines drawn in fgure 2.7. Such integral curves can take a variety oF shapes....
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