Jerry has the utility function we used in class where c is c

Jerry has the utility function we used in class where c is consumption, l leisure and the parameter ? > 0 indicates the relative preference for leisure. Jerry has no profit income, -0, but he pays a tax rate E [0,1] proportional to his (real) wage rate w. This makes Jerry\'s budget constraint (a) [3 points | Solve for Jerry\'s optimal choice of consumption, leisure and labour supply. (b) [2 points ] What effect does tax ? have on labor supply? (c) [5 points Find the optimal r that maximizes governments tax revenue. (d) 15 points | Can tax revenue decrease as r increases? Briefly discuss the effects at play and illustrate.

Solution

a).

Consider the given problem here the utility function of the consumer is given by.

=> U = C + a*lnL, => MUL = a/L and MUC = 1, => MRS = MUL/MUC = a/L.

Now, the budget line is given by, C = (1-t)*w*(H-L), => C + (1-t)*w*L = (1-t)*w*H.

So, the absolute slope of the budget line is given by, “(1-t)*w”, => at the equilibrium “MRS” must be equal to the absolute slope of the budget line.

=> a/L = (1-t)*w, => L = a/[w*(1-t)].        

So, the optimum labor supply is given by, Ls = H-L = H-a/[w*(1-t)] = Ls.

Now, the optimum consumption is given by, “C = (1-t)*w*H - (1-t)*w*L”.

=> C = (1-t)*w*H - (1-t)*w*a/[w(1-t)] = (1-t)*w*H – a = C.

b).

the labor supply is given by “Ls = H – a/[w(1-t)]”. Now, as “t” increases, => w(1-t) decreases,

=> a/[w(1-t)] increases, => “Ls = H – a/[w(1-t)]” decreases. So, as the tax rate increases implied the labor supply decreases.

c).

Now, given the tax rate and the labor supply the tax revenue is given by.

=> T = t*Ls = t*{ H – a/[w(1-t)]} = t*H – (a/w)*[t/(1-t)].

=> at the optimum “dT/dt = 0”.

=> H – (a/w)*[1/(1-t) + t(-1)*(1-t)^-2*(-1)] = H – (a/w)*[1/(1-t) + t/(1-t)^2] = 0.

=> H = (a/w)*[(1-t + t)/(1-t)^2], => H = (a/w)*[1/(1-t)^2], => (1-t)^2 = (a/wH).

=> (1-t) = (a/wH)^0.5, => t = 1 - (a/wH)^0.5.

So, the above expression represent the optimum tax rate here.

d).

So, here the tax revenue is given by, T = t*Ls. Now, as “t” increases, => “Ls” decreases but “T” is not decreasing for all values of “t”. Initially for a small “t” as “t” increases implied increase in “T” and then reached at the optimum where “T” is maximum, then it starts falling as “t” increases. So, initially “T” is rising as “t” increases then it starts falling.