1. Production Function of the Solow Growth Model (20 points) Which of the following functions are neoclassical production functions? Why/why not? Show all your work for your conclusions. i. Y=2/0.3/0.6 ii. Y=5K iii. Y=K+2L iv. Y=5/0.5/0.5 + 4/0.410.6 2. Dynamics of the Solow Growth Model (40 points) Consider the Solow model with constant population growth, in which the population growth rate equals n=2X%, where X denotes the last digit of your student ID number. For instance, if your student ID number is 20189992050, the population growth rate you should use is n=20%. Suppose the saving rate in the economy is s=27%, where Z denotes the second last digit of your student ID number. For instance, if your student ID number is 20189992050, then the saving rate you should use is s=25%. Suppose that physical capital depreciates at the constant rate =1/8=12.5%. Further, the aggregate production technology of this economy is given by Y=8/0.5105, where K refers to aggregate physical capital and L refers to aggregate labor(=population). i) Using the information and values above, write down the law of motion for aggregate physical capital and per- capita physical capital. Plug in the above numerical values when you write down the law of motions below. i) You are further provided with the information that today's physical capital is K2020=400, and labor(=population) is L2020=100. Using this information, fill in the table below (on which k refers to per-capita physical capital in year t, yt refers to per-capita income, ct refers to per-capita consumption and it refers to per- capita investment). 1 t (year) kt yt Ct it 2020 2021 2022 ii) Calculate (analytically) the steady-state per-capita values of the economy, k', y', c', i".Consider the following extension to the Solow model with constant population growth, in which the population growth rate equals n. Suppose the saving rate in the economy is constant at s, and capital depreciates at the rate 5. The aggregate production technology is given by Y,=F(K,,L=) where Kr refers to aggregate physical capital at time t and l.E refers to aggregate labor(=population) at time t. Suppose that the government makes some unproductive (wasteful) government purchases, and in order to finance it, the government imposes a re 17 n l fit: in m x at the rate, Q. In other words, if households have an income of Y. units, b fraction of it, i.e., bYt, is paid to the government, and the government spends all of this tax revenue to nance government spending of G.=bK during the same period. Note that the above saving rate applies to est-tax after-tax dis usable income level i.e., income net of taxes paid to the government. 50, households are left with a post-tax income level of Yd=YibY=uibJY and they save 5 fraction of Yd, not Y. i) Using all the information and values above, write down the new aggregate resource constraint (which was K=Ct+it before when there was no government}. ii} Derive the law of motion for aggregate physical capital AND law of motion for per-capita physical capital. m not have any explicit G: in [our Law of motions. Hint: You need b to show up in your law of motions. iii) Suppose that the production technology is of the Cobb-Douglas form with A=l. Derive and calculate the steady-state levels of per-capita physical capital, per-capita output, per-capita consumption, per-capita government spending and per-capita investment in term of para meters ( s,b,ct,n, 5,...). Show all your work. Productive Government Consider the following modification to the previous environment: Government purchases are no longer wasteful but they are used in building productive infrastructure, which contributes to aggregate production. Suppose that the aggregate production function is now ofthe following modified Cobb-Douglas form: r.=F(K.,G, a}: Know-\"'3 The government spending is still financed by the proportional income taxation at the rate, I]. So, if the households have an income of Y, units, b fraction of it, i.e. M, is paid to the government, and the government spends all of this revenue that period in the form of government spending, i.e. 6;be Hint: You can define g. as government spending per person, which is to be used in your solutions. iv} Derive the law of motion for W. Hint: Note that there no G/g should show up explicitly in your law of motion. v) Derive and calculate the steady-state levels of per-capita physical capital, per-capita output, per-capita consumption, per-ca pita government spending and per-capita investment in term of parameters (s,b,a,[3,n, 5,...).Show all yourwork