Consider the following version of the closed-economy RBC model studied in class: The economy is populated...

   

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Consider the following version of the closed-economy RBC model studied in class: The economy is populated by a representative household, who lives for two periods. The house- hold chooses consumption in the first and second period C and C', respectively, and leisure I in the first period to maximise her utility U(C, 1, C') = log(C) + a log(1) + Blog(C') subject to her lifetime budget constraint C' C+- =w(h-1)(1-1) ++ 1+r where a > 0, ß € (0,1), h is the total number of hours available for productive work, w is the hourly wage, r is the real interest rate, л and л' are dividends in the first and second period, T is a proportional tax levied in the first period, and I' is a lump-sum tax levied in the second period. π'-T' 1+r 1. Show that the household chooses consumption and leisure according to the optimality condition: a = w(1-1). (2) 2. Suppose that output is produced by a representative firm. In the first period, the repre- sentative firm produces output Y by operating the Cobb-Douglas technology: F(K,N) = ZK (Nd)1-8, where 8 € (0, 1), z is total factor productivity, K is the capital stock, and Nd is labour de- manded in the first period. The firm chooses labour demand based on the optimality condition FN (K, Nd) = w. Show that this optimality condition boils down to w = (1-0)- Y Nd' 3. Show that equilibrium employment, N*, satisfies (1-0) h 1-0 + 1 N* = (3) (4) 4. Next, you will use equation (4) to quantify the impact of the income tax r on the level of employment predicted by the model. Follow these steps: • Let 0 = 0.3224, α = 1.54, and let h = 100.¹ This last value should be interpreted as the total number of productive hours per week. • Set C/Y = 0.7, which roughly matches the contribution of consumption to GDP in Australia. Using these values, plot the relationship between N* and 7 in Excel. Use a grid of values of T between [0,0.8] with a step size of 0.05. Consider the following version of the closed-economy RBC model studied in class: The economy is populated by a representative household, who lives for two periods. The house- hold chooses consumption in the first and second period C and C', respectively, and leisure I in the first period to maximise her utility U(C, 1, C') = log(C) + a log(1) + Blog(C') subject to her lifetime budget constraint C' C+- =w(h-1)(1-1) ++ 1+r where a > 0, ß € (0,1), h is the total number of hours available for productive work, w is the hourly wage, r is the real interest rate, л and л' are dividends in the first and second period, T is a proportional tax levied in the first period, and I' is a lump-sum tax levied in the second period. π'-T' 1+r 1. Show that the household chooses consumption and leisure according to the optimality condition: a = w(1-1). (2) 2. Suppose that output is produced by a representative firm. In the first period, the repre- sentative firm produces output Y by operating the Cobb-Douglas technology: F(K,N) = ZK (Nd)1-8, where 8 € (0, 1), z is total factor productivity, K is the capital stock, and Nd is labour de- manded in the first period. The firm chooses labour demand based on the optimality condition FN (K, Nd) = w. Show that this optimality condition boils down to w = (1-0)- Y Nd' 3. Show that equilibrium employment, N*, satisfies (1-0) h 1-0 + 1 N* = (3) (4) 4. Next, you will use equation (4) to quantify the impact of the income tax r on the level of employment predicted by the model. Follow these steps: • Let 0 = 0.3224, α = 1.54, and let h = 100.¹ This last value should be interpreted as the total number of productive hours per week. • Set C/Y = 0.7, which roughly matches the contribution of consumption to GDP in Australia. Using these values, plot the relationship between N* and 7 in Excel. Use a grid of values of T between [0,0.8] with a step size of 0.05.