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nal exam Econ 415 (Due: June 11 Tuesday 4:30pm) 1. Study and review the model in Section 18.4, which uses the simple Key- nesian cross model in Chapter 5 to illustrate the main idea about the concept of automatic
scal stabilizers. Suppose we instead use the IS-LM model in Chap- ters 6-7, and assume that the governments income taxation policy is also given by equation (18.4) on p. 349, answer the following questions: (a) Derive the autonomous expenditure multiplier within this IS-LM model. (b) Can we still say in this model that the income tax policy equation (18.4) works as an auto- matic stabilizer? Explain. (c) Whats the e¤ect of an increase in t0 on income Y ? That is,
nd ��Y�t0 : (d) Whats the e¤ect of an increase in t1 on Y ? That is,
nd ��Y�t1 . 2. (La¤er curve) In the model studied in class, we modify the household utility function, and suppose it is given by insteady by U = C � L 3 3 The production funciton remains the same and is given Y = L The wage tax rate is � . Assume that the houseold has only wage income, and after paying the government tax, the rest is all used for consumption. (a) Write the household budget constraint, and solve the households utility maximization problem. (b) For
rms, from the pro
t maximization problem, marginal product of labor (MPL) = real wage. Find the real wage. (c) Find the peak of the La¤er curve. 3. Suppose that the government budget constraint is given by Bt �Bt�1 = rBt�1 +Gt � Tt where the notations for the variables are discussed in class. From the gov- ernment budget constraint, derive the approximate equation: Bt Yt � Bt�1 Yt�1 = (r � g)Bt�1 Yt�1 + Gt � Tt Yt where g is the constant growth rate of GDP. Using the second equation, explain why it is dangerous for a government to be a very high debt-GDP ratio. 4. In the Solow model, suppose that the production function is given by Yt = K 1 3 t L 2 3 t . Assume that the labor forces growth rate is n > 0, the capital 1 depreciation rate is � > 0, and the saving rate is s > 0. Answer the following questions: (a) Write the _Kt equation. (b) De
ne kt = KtLt . From the equation in (a), derive the _kt equation. (c) From (b),
nd the steady-state value for kt. (d) Find the steady-state value for yt (yt = YtLt ). At the steady state, what is the relationship between the saving rate s and y? (e) Does Solow model predict a poor country will grow faster than a rich country? Explain. (f) From (b), solve the di¤erential equation for kt, and shows that the econ- omy always converges to the steady state. We now include technological progress so that the production is given by Yt = K 1 3 t (AtLt) 2 3 , where the growth rate of At (technological progress) is g > 0. De
ne k̂t = KtAtLt : (g) Derive the di¤erential equation for k̂. (h) Find the steady-state value for k̂t. (i) For this economy with technological progress, at the steady state,
nd the growth rates for kt, yt, and Yt (note: kt = KtLt , yt = Yt Lt ). 2