Answer to Question #158109 in Economics for george

Question

Consider the following model. There is a representative household whose utility function is: ∑ ∞ t=0 β t log(Ct), where Ct is consumption and β ∈ (0,1) and K0 = K¯ 0 > 0. Unlike the model discussed in a class, households can buy and sell a bond issued by the government Bt . They buy the government bond at price qt and sell it at price 1 in the next period. So, for each period, the household budget constraint is written as Ct +qtBt+1 +Kt+1 ≤ (1−τ)rtKt +Bt , where rt denote the rental rate of capital and τ is capital income tax. We assume that capital depreciation is 100% (δ = 1), so that Kt+1 is investment here. Firms production function is Y = AKt , where A > 1 β is productivity level. Lastly, the government does not consume any goods, but it has some initial debt outstanding, B¯ 0 > 0, which it needs to deal with. It can raise revenue by taxing capital income at a constant rate τ and by issuing a bond. Tax revenue in period t is then τrtKt . Answer following questions. (a) Solve the households problem and derive the optimal conditions. (b) Solve the firms problem and derive the optimal conditions. (c) Define a competitive equilibrium. [Tips: It might be helpful to refer to a description of a competitive equilibrium in Chen, Imrohoroglu, Imrohoroglu (2006) or Hayashi and Prescott (2002).] (d) Calculate a competitive equilibrium of {Ct ,Kt+1} ∞ t=0 as functions of the model primitives β,A,K¯ 0,and B¯ 0. [Hint: The recommended approach here is to first make a good guess and then verify it. A good guess here is that equilibrium consumption Ct and investment Kt+1 are constant multiples of the current capital stock Kt .] (e) There is empirical evidence which suggests that countries that start out with high government debt to GDP ratios tend to grow slower than others. Is the model consistent with this fact? 

Answer 1

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