Exercise 1 (Labor versus capital taxation) Consider an economy with three goods: a private good (say...

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Exercise 1 (Labor versus capital taxation) Consider an economy with three goods: a private good (say food), labor, and capital. We fix the price of the private good at 1 by convention (that is, the private good is the numeraire), the wage rate (i.e., the price of labor) is denoted w, and the rental price of capital is denoted r (for simplicity, capital can be rented but not sold). The firms: There are two kinds of firms. Labor-intensive firms purchase labor (their input) to produce the private good (their output). They all have the same linear technology: one unit of labor can produce one unit of the private good. They are competitive. That is, they take the wage rate w and the price of the private good-normalized to 1- -as given. So for each unit of labor it hires, a firm pays w to the worker and it receives an income of 1 from selling the product of that unit of labor. Capital-intensive firms rent capital (their input) to produce the private good (their output). They all have the same linear technology: the rental of one unit of capital can produce one unit of the private good. They are competitive. That is, they take the rental price of capital r and the price of the private good-normalized to 1-as given. So for each unit of capital it rents, a firm pays r to the capital owner and it receives an income of 1 from selling the product of that unit of capital. The individuals: Each citizen i EN is endowed with 1 unit of labour, and ki units of capital, where k 0. Citizens care only about the amount of private good they consume, so the utility function of each individual is simply the amount of private good she consumes nes. As a result, each consumer i inelastically sells all of her labor endowment and she inelastically rents out all of her capital endowment, and she spends all of her after-tax income on the private good. Note that this implies that the aggregate labor supply is N irrespective of the wage rate, and the aggregate capital supply is LEN ki = K irrespective of the rental price T. The government: The government needs a fixed amount of revenue T from taxes (suppose for instance that this is the revenue he needs to provide basic services such as a sewage system, a court, and some roads). It can raise revenue from two types of taxes: A labor tax. This tax is levied on every unit of labor exchanged in the economy. It is paid by the seller of the unit of labor (i.e., the worker). The labor tax rate is denoted by ti [0, 1]. A capital tax. This tax is levied on every unit of capital that is rented, and it is paid by the renter of the unit of capital (i.e., the individual who owns the unit of capital). The capital tax rate is denoted by tk [0, 1]. 1. The competitive equilibrium: (if you cannot demonstrates the formulas in the questions below, you can just assume them for the following questions). (a) Show that at the competitive equilibrium, the equilibrium wage rate is w = 1 and the equilibrium rental price of capital is r= 1 (if you cannot answer that question, just assume it for the rest of the homework). (b) For any tax labor tax rate ti [0,1] and any capital tax rate tk [0,1], determine the pre-tax income and the tax paid by each individual i in the competitive equilibrium. (c) Show that in the competitive equilibrium, each individual i consumes a quantity of private good given by CAT (ti, tk) = 1-ti + k (1 tk). (if you cannot answer that question, just assume it for the rest of the homework). (d) Show that the total revenue raised by the government as a function of the tax rates t and tk, the number of tax payers N, and the aggregate capital endowment K = EieN ki is given by R = Nti+ Ktk. Exercise 1 (Labor versus capital taxation) Consider an economy with three goods: a private good (say food), labor, and capital. We fix the price of the private good at 1 by convention (that is, the private good is the numeraire), the wage rate (i.e., the price of labor) is denoted w, and the rental price of capital is denoted r (for simplicity, capital can be rented but not sold). The firms: There are two kinds of firms. Labor-intensive firms purchase labor (their input) to produce the private good (their output). They all have the same linear technology: one unit of labor can produce one unit of the private good. They are competitive. That is, they take the wage rate w and the price of the private good-normalized to 1- -as given. So for each unit of labor it hires, a firm pays w to the worker and it receives an income of 1 from selling the product of that unit of labor. Capital-intensive firms rent capital (their input) to produce the private good (their output). They all have the same linear technology: the rental of one unit of capital can produce one unit of the private good. They are competitive. That is, they take the rental price of capital r and the price of the private good-normalized to 1-as given. So for each unit of capital it rents, a firm pays r to the capital owner and it receives an income of 1 from selling the product of that unit of capital. The individuals: Each citizen i EN is endowed with 1 unit of labour, and ki units of capital, where k 0. Citizens care only about the amount of private good they consume, so the utility function of each individual is simply the amount of private good she consumes nes. As a result, each consumer i inelastically sells all of her labor endowment and she inelastically rents out all of her capital endowment, and she spends all of her after-tax income on the private good. Note that this implies that the aggregate labor supply is N irrespective of the wage rate, and the aggregate capital supply is LEN ki = K irrespective of the rental price T. The government: The government needs a fixed amount of revenue T from taxes (suppose for instance that this is the revenue he needs to provide basic services such as a sewage system, a court, and some roads). It can raise revenue from two types of taxes: A labor tax. This tax is levied on every unit of labor exchanged in the economy. It is paid by the seller of the unit of labor (i.e., the worker). The labor tax rate is denoted by ti [0, 1]. A capital tax. This tax is levied on every unit of capital that is rented, and it is paid by the renter of the unit of capital (i.e., the individual who owns the unit of capital). The capital tax rate is denoted by tk [0, 1]. 1. The competitive equilibrium: (if you cannot demonstrates the formulas in the questions below, you can just assume them for the following questions). (a) Show that at the competitive equilibrium, the equilibrium wage rate is w = 1 and the equilibrium rental price of capital is r= 1 (if you cannot answer that question, just assume it for the rest of the homework). (b) For any tax labor tax rate ti [0,1] and any capital tax rate tk [0,1], determine the pre-tax income and the tax paid by each individual i in the competitive equilibrium. (c) Show that in the competitive equilibrium, each individual i consumes a quantity of private good given by CAT (ti, tk) = 1-ti + k (1 tk). (if you cannot answer that question, just assume it for the rest of the homework). (d) Show that the total revenue raised by the government as a function of the tax rates t and tk, the number of tax payers N, and the aggregate capital endowment K = EieN ki is given by R = Nti+ Ktk.