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IH1_RQ - Review Questions Innite Horizon Models in Discrete...

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Review Questions: In°nite Horizon Models in Discrete Time Prof. Lutz Hendricks. August 6, 2009 1 Taxes and government spending Consider the following growth economy, extended to include government spending and a tax on output at rate ° t . A representative household chooses f c t ; k t +1 g 1 t =0 to solve the problem max X 1 t =0 ± t u ( c t ) subject to c t + k t +1 = (1 ° ° t ) f ( k t ) k 0 given. Capital depreciates completely each period. The government °nances exogenous spending, g t , each period by levying taxes at rate ° t . Government spending does not a/ect private utility or production possibilities. The government budget constraint is ° t f ( k t ) = g t You may °nd it convenient to de°ne government spending as a share of output by introducing the variable s g t = g t =f ( k t ) Further, note the two budget constraints imply the aggregate resource constraint c t + g t + k t +1 = f ( k t ) or c t + k t +1 = (1 ° s g t ) f ( k t ) The household takes the time paths of the policy variables as given. Assume the functional forms: u ( c ) = ln( c ) and f ( k ) = k ° , 0 < ² < 1. (a) Show that there is a maximum sustainable capital stock, k max , for this economy. (b) Assuming that k 0 2 (0 ; k max ) °nd the steady state level of the capital stock. Assume that ° is constant over time. Note that there are no °rms; households produce and consume. 1 (c) Write down the Bellman equation for the household. (d) Solve for the equilibrium consumption and investment decision rules as functions of the current state. Hint: What are reasonable guesses given log utility? (e) Why doesn±t expected future policy a/ect current consumption and investment decisions? Consider the same economic environment as above, but now allow the government to sell real one-period bonds to °nance any discrepancies between spending and revenues. Let b t +1 denote bonds sold in period t at price q t , which pay one unit of consumption goods in period t +1 . Agents enter each period t with capital, k t , and bonds, b t . The representative household now chooses f c t ; k t +1 ; b t +1 g 1 t =0 1 You can convince yourself, however, that it would not make a di/erence if °rms were added to the model. 1
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to maximize utility subject to c t + k t +1 + q t b t +1 = (1 ° ° t ) f ( k t ) + b t k 0 given. Assume the same functional forms before. The household takes as given the time paths of policy variables. The government chooses paths for f ° t ; b t +1 g 1 t =0 to °nance s g t according to: q t b t +1 + ° t f ( k t ) = g t + b t b 0 given. (f) De°ne the state of the economy and write down Bellman±s equation for the household. (g) Assuming that in a steady state the tax rate and government spending share are given and constant, derive expressions for steady state consumption, capital, and government debt. 1.1 Answer: Taxes and government spending (a) It su¢ ces to show that k is bounded, even if c and g both equal zero forever. Since k t +1 = k ° t the corresponding steady state has k = 1 . If k t > 1 then k t +1 < k t .
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