Question

There's a country with 1 company and 1 person.

A company's total production, which is country's real GDP, is Y=B* N+b

A company's labor demand by matching MPN is w=B which is perfectly elastic

total time h=N+l N for work time and l for leisure time

C is consumption.

to maximize a person's utility, U(C,l) = C^(1-a)l^a

desired C is (1-a)(w*h+b)

desired l is (a/w)(w*h+b)

labor supply curve is (1-a)*h-(a*b)/w

labor demand curve is B

if there's a tax rate t and its tax revenue twN is a government revenue.

Show how this new tax will influence real gross domestic product and a person's desired consumption

Answer 1

company's total production, which is country's real GDP, is Y=B* N+b

A company's labor demand by matching MPN is w=B which is perfectly elastic

total time h=N+l, N for work time and l for leisure time

C is consumption.

to maximize a person's utility,

The budget constraint of the person is

C=Y

or, C=Nw+b

or, C=(h-l).w+b (as h=N+l)

or, C+w.l = w.h+b.........BL

This is the budget line of the person. Now he will maximize his utility U(C,l) subject to the budget comstraint BL. The outcomes are given in the question.

desired C is C*=(1-a)(w.h+b)

desired l is l*= (a/w)(w.h+b)

labor supply curve is Ls= (1-a)h-(a.b)/w

labor demand curve is B=w.

Now we have to analyze the effect of the taxation on his Real GDP and desired Consumption. Let's do this here.

The tax rate is given as t and the tax revenue collected from this taxation is t.w.N.

If the wage rate is w and tax rate is t, then after taxation the wage rate becomes w(1-t). Now the person will be paid wage rate = w' = w(1-t).

Hence his Real GDP will be affected by this taxation.

New Real GDP = Y' say.

Y' = w(1-t).N+b

or, Y' = wN+b-w.t.N

or, Y' = Y - twN.........(1)

Here the taxation will decrease the Real GDP as it has decreases by twN units.

Hence, the new tax will decrease the real GDP by twN units and the new GDP will be

Y' = wN+b-twN.

Now, we will also check the effect on the desired consumption of the person.

The new Budget Line becomes

C=Y'=N.w(1-t)+b

or, C=(h-l).w(1-t)+b

or, C+w(1-t).l = h.w.(1-t)+b..........BL1.

This is the new budget line. Now we will maximize the utility U(C,l) subject to the new budget line i.e. BL1. The calculations are shown below.

Hence we get,

w(1-t).l = C.a/(1-a).........(2)

Now we will put the value of w(1-t).l in the budget line to determine C**.

C+w(1-t).l = h.w.(1-t)+b

or, C+C.a/(1-a) = h.w.(1-t)+b

or, C** = (1-a).{h.w.(1-t)+b}

or, C** = (1-a)(h.w+b) - (1-a).h.w.t

or, C** = C* - (1-a).h.w.t

Hence, we can see the desired consumption will decrease by (1-a).h.w.t units. This is how his desired consumption will be affected.

Hence, the person's desired consumption will decrease by (1-a).h.w.t and it will be

C**=(1-a){h.w.(1-t)+b}.

Hope the solution is clear to you my friend.