=»

o <

y <

1; ay > 1

Possibility i, where both and b2 are positive fractions, duly satisfies condition (17.17) and hence conforms to the model specification 0 < y < 1. The product of the two roots must also be a positive fraction under this possibility, and this, by (17.15'), implies that ay < 1. In contrast, the next three possibilities all violate condition (17.17) and result in inadmissible y values (see Exercise 17.2-3). Hence they must be ruled out. But Possibility v is again acceptable. With both 6, and b2 greater than one, (17.17) is again satisfied, although this time we have ay > 1 (rather than < 1) from (17.15'). The upshot is that there are only two admissible subcases under Case 1. The first—Possibility i—involves fractional roots bx and b2, and therefore yields a convergent time path of Y. The other subcase—Possibility v—features roots greater than one, and thus produces a divergent time path. As far as the values of a and y are concerned, however, the question of convergence and divergence only hinges on whether ay < 1 or ay > 1. This information is summarized in the top part of Table 17.1, where the convergent subcase is labeled 1C, and the divergent subcase ID.

The analysis of Case 2, with repeated roots, is similar in nature. The roots are now b = y(l + a)/2, with a positive sign because a and y are positive. Thus, there is again no oscillation. This time we may classify the value of b into three possibilities only:

(vi) 0 < b < 1 => y < 1; ay < 1 ( vii ) b = 1 =» y = 1

Under Possibility vi, b (= bx = b2) is a positive fraction, thus the implications regarding a and y are entirely identical with those of Possibility i under Case 1. In an analogous manner, Possibility viii, with b ( = b{ = b2) greater than one, yields the same results as Possibility v. On the other hand, Possibility vii violates (17.17) and must be ruled out. Thus there are again only two admissible subcases. The first—Possibility vi—yields a convergent time path, whereas the other—Possibility viii—gives a divergent one. In terms of a and y. the convergent and divergent subcases are again associated, respectively, with ay < 1 and «y > 1. These results are listed in the middle part of Table 17.1, where the two subcases are labeled 2C (convergent) and 2D (divergent).

Finally, in Case 3, with complex roots, we have stepped fluctuation, and hence endogenous business cycles. In this case, we should look to the absolute value R = fa 2 [see (17.8)] for the clue to convergence and divergence, where a2 is the coefficient of the yt term in the difference equation (17.1). In the present

Case |
Subcase |
Values of a and y |
Time path Y, | |

1 Distinct real roots 4a y > (1 + a)' |
1C: ID: |
0 < b2 < b] < 1 1 < b2 < bt |
ay > 1 |
Nonoscillatory and nonfluctuating |

2 Repeated real roots 4a Y ~ 2 (1 + a) |
2C: 2D: |
0 < b < 1 b > 1 |
ay < 1 ay > 1 |
Nonoscillatory and nonfluctuating |

3 Complex roots 4 a y< i (i +ay |
3C: R < 1 3D: R > 1 |
ay < 1 ay > 1 |
With stepped fluctuation |

model, we have R = y/ay, which gives rise to the following three possibilities:

(ix) |
R < 1 |
=> ay < |

(x) |
R = 1 |
=> ay = |

(xi) |
R > 1 |
=> ay > |

Even though all of these happen to be admissible (see Exercise 17.2-4), only the R < 1 possibility entails a convergent time path and qualifies as Subcase 3C in Table 17.1. The other two are thus collectively labeled as Subcase 3D.

In sum, we may conclude from Table 17.1 that a convergent time path can obtain if and only if ay < 1.

The above analysis has resulted in a somewhat complex classification of cases and subcases. It would help to have a visual representation of the classificatory scheme. This is supplied in Fig. 17.2.

The set of all admissible (a, y) pairs in the model is shown in Fig. 17.2 by the variously shaded rectangular area. Since the values of y = 0 and y = 1 are excluded, as is the value a = 0, the shaded area is a sort of rectangle without sides. We have already graphed the equation y = 4a/(l 4- a)2 to mark off the three major cases of Table 17.1: The points on that curve pertain to Case 2; the points lying to the north of the curve (representing higher y values) belong to Case 1; those lying to the south (with lower y values) are of Case 3. To distinguish between the convergent and divergent subcases, we now add the graph of ay = 1 (a rectangular hyperbola) as another demarcation line. The points lying to the north of this rectangular hyperbola satisfy the inequality ay > 1. whereas those located below it correspond to ay < 1. It is then possible to mark off the subcases easily. Under Case 1, the broken-line shaded region, being below the hyperbola, corresponds to Subcase 1C, but the solid-line shaded region is associated with Subcase ID. Under Case 2, which relates to the points lying on the curve y = 4a/(l + a)2. Subcase 2C covers the upward-sloping portion of that curve, and Subcase 2D, the downward-sloping portion. Finally, for Case 3, the rectangular hyperbola serves to separate the dot-shaded region (Subcase 3C) from the pebble-shaded region (Subcase 3D). The latter, you should note, also includes the points located on the rectangular hyperbola itself, because of the weak inequality in the specification ay > 1.

Since Fig. 17.2 is the repository of all the qualitative conclusions of the model, given any ordered pair (a. y). we can always find the correct subcase graphically by plotting the ordered pair in the diagram.

Example 1 If the accelerator is 0.8 and the marginal propensity to consume is 0.7, what kind of interaction time path will result? The ordered pair (0.8,0.7) is located in the dot-shaded region, Subcase 3C; thus the time path is characterized by damped stepped fluctuation.

Example 2 What kind of interaction is implied by a = 2 and y = 0.5? The ordered pair (2,0.5) lies exactly on the rectangular hyperbola, under Subcase 3D. The time path of Y will again display stepped fluctuation, but it will be neither explosive nor damped. By analogy to the cases of uniform oscillation and uniform fluctuation, we may term this situation as "uniform stepped fluctuation." However, the uniformity feature in this latter case cannot in general be expected to be a perfect one, because, similarly to what was done in Fig. 17.1, we can only accept those points on a sine or cosine curve that correspond to integer values of /, but these values of t may hit an entirely different set of points on the curve in each period of fluctuation.

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