Let us have an economy that lives only for 2 periods. The representative household has a utility function that values consumption goods for any period as follows u(c) = ln(c). The valuation throughout both periods can be represented as a total utility function that discounts future utility with a discount parameter as follows, U(c1, c2) = u(c1) + u(c2) = ln(c1) + ln(c2), where ct represent consumption good in period t for every period t = 1, 2. The budget constraint for this person in every period is expressed as (1+tc)ct + xt =(1tI)(wtnt +rtkt), t=1,2. where xt represent investment in period t, wt the wages earned by hours worked, rt the return of capital in period t, nt the hours worked supplied by the household in every period t, kt the stock of capital that the household has in period t, tc the tax rate imposed by the government in consumption goods in every period t, and tI the tax rate imposed by the government to total household income in every period t. In every period, the total amount of hours that the representative household is of H = 1. Investment in every period can be defined by the following law of motion of capital kt+1 = xt +(1)kt, t=1,2. Acknowledge that k3 = 0 because there is no period 3 in this economy. The household starts with a given amount of capital k 1 > 0. The representative firm produces good by the following production function in every period t, Yt =ztKtNt1, t=1,2. There is a government that covers government expenditures in period 1 and period 2 by taxing capital rent and labor. Hence, their budget can be expressed as tcct +tI(wtnt +rtkt)=gt, t=1,2. a) Before writing any competitive equilibrium definition or the Social Planner Problem, can we modify the budget constraint in order to get rid a variable in every period? Explain. b) Define the Social Planner Problem. c) Solve the Social Planner Problem. This is, find the Euler equation in terms of exogenous variables and the only endogenous variable k2. d) Define the tax distorted competitive equilibrium of this economy. e) Solve the representative household problem. f) Solve the representative firm problem in period 1 and 2. g) Find the optimal Euler equation in terms of exogenous variables and the only endogenous variable k2. h) Imagine that the government can only implement income taxes. This is tc = 0 for every period t = 1, 2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem? i) Imagine that the government can only implement consumption taxes. This is tI = 0 for every period t = 1,2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem? Solution the problem related with the society and mainly concerned with the government\'s utilisation of funds.in short social planner is the decision maker of the society. 1 A Closed-Economy One-Period Macroeco-nomic Model We now want to take the microeconomic behavior of consumers and rms and combine.

1. Let us have an economy that lives only for 2 periods. The representative household has a utility
function that values consumption goods for any period as follows
u(c) = ln(c).
The valuation throughout both periods can be represented as a total utility function that discounts
future utility with a discount parameter as follows,
U(c1, c2) = u(c1) + u(c2) = ln(c1) + ln(c2),
where ct represent consumption good in period t for every period t = 1, 2. The budget constraint
for this person in every period is expressed as
(1+tc)ct + xt =(1tI)(wtnt +rtkt), t=1,2.
where xt represent investment in period t, wt the wages earned by hours worked, rt the return of
capital in period t, nt the hours worked supplied by the household in every period t, kt the stock
of capital that the household has in period t, tc the tax rate imposed by the government in
consumption goods in every period t, and tI the tax rate imposed by the government to total
household income in every period t. In every period, the total amount of hours that the
representative household is of H = 1. Investment in every period can be defined by the following
law of motion of capital
kt+1 = xt +(1)kt, t=1,2.
Acknowledge that k3 = 0 because there is no period 3 in this economy. The household starts with
a given amount of capital k 1 > 0. The representative firm produces good by the following
production function in every period t,
Yt =ztKtNt1, t=1,2.
There is a government that covers government expenditures in period 1 and period 2 by taxing
capital rent and labor. Hence, their budget can be expressed as
tcct +tI(wtnt +rtkt)=gt, t=1,2.
a) Before writing any competitive equilibrium definition or the Social Planner Problem, can we
modify the budget constraint in order to get rid a variable in every period? Explain.
b) Define the Social Planner Problem.
c) Solve the Social Planner Problem. This is, find the Euler equation in terms of exogenous
variables and the only endogenous variable k2.
d) Define the tax distorted competitive equilibrium of this economy.
e) Solve the representative household problem.
f) Solve the representative firm problem in period 1 and 2.
g) Find the optimal Euler equation in terms of exogenous variables and the only endogenous
variable k2.
h) Imagine that the government can only implement income taxes. This is tc = 0 for every period

2. t = 1, 2. Can the government implement a tax plan that achieves the Pareto allocation reached by
the Social Planner Problem?
i) Imagine that the government can only implement consumption taxes. This is tI = 0 for every
period t = 1,2. Can the government implement a tax plan that achieves the Pareto allocation
reached by the Social Planner Problem?
Solution
the problem related with the society and mainly concerned with the government's utilisation of
funds.in short social planner is the decision maker of the society.
1 A Closed-Economy One-Period Macroeco-nomic Model We now want to take the
microeconomic behavior of consumers and rms and combine them into a workable model of the
macroeconomy. As we build this model we will learn the meaning social eciency. Can also begin
some preliminary work on government policy analysis.Our model will begin by being structured
in a closed economy.
Denition 1 A closed economy is an economy that does not trade with the rest of the world.In a
closed economy all prices are determined by the balancing of domestic demand and domestic
supply. In an open economy those price are set by balancing global demand and global supply.
Our economy will have three economic agents consumers, rms, and a government. For
consumers and rms we simply utilize the information from the previous chapter. We need to now
address government behavior.
We begin with a very simple government which purchases a quantity of goods,G, and nances this
purchase with taxes on the consumers,T The government purchases are to be thought of a public
goods. Where all the consumers will receive some utility from the good based on the level of G
For now we will take the simple stance and treat taxes a lump sum and G as being spread equally
across the consumers. The important thing to note is the the government is using up some
resources in the economy. The output of the economy is produced by the rms in the private
sector and the government purchases some exogenous,G, portion of the aggregate economy.
Exogenous variables are those determined outside the model. Endogenous are variables
determined by the model. For now the government must satisfy the following budget constraint
G=T
For now will ignore government borrowing and and solely focus on how changes in the level of
G in uence the economy.
Applying the spending approach to calculating GDP, this economy would
have
Y=C+G2 in equilibrium.