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11 Dec 2019

4. Pareto Optimal Allocation and Competitive Equilibrium with Proportional Income Tax . Consider an economy with the representative consumer, representative firm, and government. For simplicity, suppose that G = 0. The consumer’s preferences over the consumption good and leisure are given by the utility function U(C, l) = 2 ln C + ln l. Let h = 18. The firm’s production function is Y = zK1/2N1/2 , where z = 2 and K = 1.

(a) Write down the social planner’s maximization problem and the social planner’s optimality condition. (Recall: MRSl,C = ∂U/∂l ∂U/∂C and MPN = ∂zF(K, N)/∂N.)

(b) Derive the Pareto optimal allocation for leisure, labor, output, and consumption.

(c) Suppose the government imposes a proportional income tax t = .5 on the representative consumer, and distributes the revenue back to them as a lump-sum transfer (subsidy). That is, the consumer’s budget constraint is c = w(1 − t)(h − l) + π + S, where S is the subsidy. Compute the amount of labor in the competitive equilibrium, and show how it compares to the one you found in part (b).

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Related questions

Suppose that the government imposes a lump-sum tax on the representative consumer’s labor income, whose amount is T. Further suppose that the governemnt increases the labor income tax amount. What are the effects of an increase in T on consumer’s consumption (c) and leisure (l)? Explain why.

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 β discount parameter as follows,

U(c1, c2) = u(c1) + βu(c2) = ln(c1) + β 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+τtc)ct + xt =(1−τtI)(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, τtc the tax rate imposed by the government in consumption goods in every period t, and τtI 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−δ)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 =ztKtαNt1−α, 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

τtcct +τtI(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 τtc = 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 τtI = 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? 


Country A is located on a small island that is isolated from the outside world. The country
has one representative consumer, whose preference is represented by:
U(c, l) = ln(c) + ln(l)
There is one representative fi rm in the economy which is owned by the consumer, it produces
one type of good that can be used for consumption or government expenditure using capital
and labour as inputs. The fi rm owns the capital it uses and it's production technology is
represented by
F(K;N) = zK^aN^1-a
where z is the total factor productivity, and a is the capital share of income. The fi rm pays
all of its profit back to consumer as dividend.
The government of country A spend a predetermined G amount of goods to provide public
services, and it taxes consumer through a lump-sum tax t to finance the expenditure. the
government's budget is balanced:
G = t
Now, suppose that the firm cannot change its capital stock, or improve its production technology in the short period, such that z and K are given.
(a) When describing this economy as an macroeconomic model, what is the set of exogenous
variables? What is the set of endogenous variables that can be determined given the set
of exogenous variables using the concept of Competitive Equilibrium?
(b) Defi ne the competitive equilibrium for this economy, be specific about what it is, what
conditions have to be satisfied who solves what problem and etc.
(c) List the set of equations that will be used to determine all of the endogenous variables.
Which four of these endogenous variables are essentials, such that once you know these
four, all other endogenous variables can be obtained easily?
(d) Write down the four equations that can be used to solve for these four essential endogenous variables, and describe where do they come from. Describe in details the steps of
how would you solve for the competitive equilibrium (without actually solving for it ).
(e) Setup the Social planner's problem for this economy. What is the marginal condition
that pins down the planner's choices.
(f) Now, describe what would happen to c, l, and w if the firm had more capital, such
that Knew > Kold.

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