There is a small open economy with two types of agents, high-skilled and low-skilled. The utility function is type-independent and of the form
U (c,i) = cg1g
c is consumption and i is leisure. Agents have one unit of time. Perfectly competitive rms produce the consumption good according to the function
Y = (l + k)a h1a ,
where l and h are low- and high-skilled labor. k is the amount of capital in the economy, which is supplied by foreigners such that the rate of return is equal to r . To guarantee that the high-skilled earn more and that capital is utilized, assume that r < a and r < aa (1 a)1a . The price of consumption is one and goods can be traded at no cost with the rest of the world.
Find labor supply. What is consumption as a function of the wage? What are wages wl and wh and capital supply k as a function of parameters only( i.e as functions , a and r*)
Demand for a commodity x is D(q) with a decreases in q = p + t
Supply for commodity x is S(p) with an increases in p
Equilibrium is satisfied under the condition: Q = S(p) = D(p + t)
Begin from t = 0 and S(p) = D(p). So as to characterize dp/dt: the result of a tax increase on price, which regulates the load of tax:
Adjust dt to causes change in dp so that equilibrium holds:
S(p + dp) = D(p + dp + dt)
S(p) + S(p)dp = D(p) + D(p)(dp + dt)
S(p)dp = D(p)(dp + dt)
dp /dt = D(p) /S(p) D(p)
If the utility function is:
U(c, l) = c g ` 1g .
Y = (l + k) ah 1a ,
The government introduces a small tax on capital at rate t k > 0, so that the price of capital for producers is now r (1 + t k). Who gains and loses from this tax? What could explain this?
If the government introduces a small tax on the capital at a rate t k > 0, then the price for producers now becomes
r (1 + t k)
If the tax introduced t k > 0, then even though the consumers pay all the prices of a particular good formally, the producers still find themselves bearing part of the tax burden. As a result, both the consumers and producers lose by paying tax as the government gains double. This situation is best explained by the fact that the consumer price is a function of the tax rate p (t).
The government tries to maximize a utilitarian welfare function, the sum of the utilities of both types of agent. It can use linear taxes on each type of labor and capital. What characterizes the optimal tax system? [Hint: There is only one equation that needs to be satised. Justify why.] What is the trade-o between eciency and redistribution?
For the government to ensure they maximize the utilitarian welfare function, they have to make decisions that are optimal as far as the consumption rate are concerned. Also, they have to ensure that the labor and leisure issues more specifically are decided upon on an optimal level.
The optimal tax system must be characterized by solving the equation of the utility and balancing it.
From the function of utility which is:
U(c, i) = c g ` 1g
Where r + i = 1 and the other condition to be met is i > 0 and c = y.
Therefore the satisfied equation becomes
U(c, 1- r) = y g ` 1g
One can make a conclusion that when one tries to pursue equality, then the efficiency, better known as the total output produced with given resources can drastically reduce. Also, it becomes clear that an equal distribution of incomes causes a reduction of work and investments incentives. In addition, trying to do redistribution through minimum wages and tax code sometimes prove to be costly.
What is your biggest concern as to what makes this model ill-suited for the analysis of tax policy? What would you change to make it reasonable? Be specic about which functions have to change in what way.
The model is ill suited as it brings about the excess burden of taxation or the deadweight loss. This condition is usually as a result of price distortion. Taxes can distort prices and even drive wedges between supplier and consumer and as a result we have the excess burden of a tax.
To make this model reasonable, one has to do the following:
The government must collect its revenue in bulk to avoid excess burden of taxation.
The goods that are consumed in the economy also need to be taxed at the same rate so as not to change the relative prices hence no excess burden.
Ramsey Taxation with an Untaxable Factor
Consider a standard Ramsey taxation problem as shown in class, but assume the following production function f (k, z, n) satisfying CRS and the Inada conditions. All cross-derivatives are positive, so fzk > 0. z is another factor that is supplied by the owner (the representative household), lets say it is land. The owner incurs a loss to utility from providing land to the market, since they could instead use it for their own enjoyment, so uz (c, , z) < 0. The return to land is denoted by s and taxes on it are t z .
Derive the private sectors rst-order conditions.
If we consider all goods including land that is given out ad people use it for leisure. Then leisure will be considered as a non-taxable good. The first order condition for the good will be as follows:
- ix'u a - 0
The first order condition for the tax on k is then written as
dVdki-l(xi- jtj dxjdki)=0By applying the envelope theorem, w eobatin that dVdki= - axi hence the condition can be expressed as:
(l- a)xi+ ljtj dxjdki)=0Derive the implementability constraint, set up the government Lagrangean using this, and briey explain why optimal steady-state capital taxes are still zero.
The implementability constraint can be derived and expressed as follows:
Assume we have two consumers i=1,2
Their preferences will be given by:
t=0infinitybtUi(citkit)The implementability constraint for the two consumers becomes
t=0infinitybtUctiCit+ Uktikit= Uc0iRK0k0iThe budget constrain becomes
0yTyhydy=RIn Lagrangian, it becomes
L=0y(u(y-T(y)+l[Ty-R])h(y)dy0=dLdT(y)= -u'y-Ty+ lh(y)0=u'(y-Ty)= ly-Ty=c=l c=y-R The optimal steady-state capital taxes are zero because there is no new investment capital that is being generated and as a result the capital stock depreciates without replacement. The steady state hence ends up being unsustainable. There is an exception at the zero output, which also implies that the consumption level is zero.
Assume now that the government cannot tax the new factor z. What additional constraint does this impose on the government?
If the government cannot tax the new factor z, this means that the government will collect less tax and as a result must introduce the budget constrain that is written as:
TYL+TYHRwhere R is the revenue that the government needs in addition to the money it may redistribute
With this additional constraint, show that optimal steady-state capital taxes are positive. What could explain it?
The additional constraint that the government is forced to consider is mainly compulsory that it be added in the governments revenue when doing redistribution. The budget constraint is given by the following formula:
0infinityntct+ kt+1= t=0infinitypt[1- ttwthtk1t+ Rktkt]Where
RKT=1+(1- tht)(r- dt) The best explanation as to why the optimal steady state capital taxes are positive is the additional constraint introduces increment in tax. The increasing tax will be optimal because all of the general equilibrium expenditure elasticities are not constant over time. The steady state hence is always equivalent to an ever-increasing tax on consumption.
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