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Question: the argument is certainly a tempting one and makes some...

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The argument is certainly a tempting one and makes some sense. The question is, does it work in practice? To get a handle on the answer, we propose a simple model consisting of one equation that relates tax revenue, income, and tax rate: TR t-Y. where TR is tax revenue, t is tax rate, and Y is aggregate income, e.g. GDP. Suppose, the tax rate t is reduced by fraction c, where 0<c<1. For example, if c-0.05, then the new tax rate is t,-t(1-c)-0.95t, i.e., the rate is reduced by 5% and the new rate is 95% of the old one. Then the proposition of increased tax revenues as a result of tax cut takes definite mathematical form. For the new tax revenue to be greater than before we have: where 1-c is the new reduced tax rate, and g is the rate of increase in income Y that results from the tax cut. Thern t(1-c) Y(1+g) > tY (1-c-g >1 Thus, if the new tax revenue to exceed the old one, the increase g in income Y has to be greater than tax reduction fraction c. Now, lets consider how likely this to happen. When the Trump administration implemented tax it reduced top marginal rate from approximately 40% to 37%, or (37%-40%) 40% -0.075 a reduction of 7.5%. To compensate for this reduction in tax rate, our model requires that g > c 0.075. But we know that mature advanced economies, like that of the U.S., do not grow at such high rates. So, it is highly unlikely that the tax cut will lead to higher tax revenues and much more likely will lead to increase in budget deficit. Now, the simple model we introduced is, perhaps, overly simple and misses some crucial aspects of reality we try to modl. In particular, when the tax cut was implemented by the Trump administration, it is the top marginal rates have been substantially reduced, not all the rates. Thus, we need a model with more than one tax rate, and then have a tax cut which affects only the top rate. To that effect, let ti and t2.0 < tl < t2 < 1, be two tax rates at which all of the national income is tax, and P1 and P2 are proportions of income which which are taxed at ti and t2, respectively (natural bounds on P and P2 are 0 < P1 < 1 and P2 = 1-Pl). Then tax revenue is TR = ti (YP) + t2 (Yp2). The condition TR > TR leads to Your assignment is simplify this last inequality as much as possible with an eye to obtaining a nice relationship between rate of income increase g and tax cut fraction c. Then interpret, or restate, the found condition in plain English, e., what does it mean? And, drawing on your general knowledge of the U.S. economy and its recent history, determine the likelihood of the relationship being true in reality (and, by extension, whether we can expect the tax cut pay for itself or budget deficit to increase). This is what we have done with the simpler model during the first class.
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