Question

Pierre values consuming goods (C) and enjoying leisure (L). Pierre has h = 1 units of time to divide between working and enjoying leisure. For each hour worked, he receives w = 1 units of the consumption good. Suppose that Pierre's preferences are described by the the utility function U(C,)=2/31/3 Pierre also owns shares in a factory which gives him an additional = 0.125 units of income. The government in this economy taxes labour income only and uses the proceeds to buy consump- tion goods that are given to the army. Pierre pays a lump sump tax equal to 0.35. 1. Write down Pierre's budget constraint. (05 points) 2. Is it optimal for Pierre to supply 0.75 units of labour? (10 points) 3. What is Pierre's optimal choice of consumption and leisure. Illustrate with a graph? (15 points) 4. Suppose the government increases the tax to 0.45. How are Pierre's optimal decisions affected by this change? (10 points 5. Suppose that w decreases to 0.8 with the taxes still being 0.35. How are Pierre's optimal decisions affected by this change. (10 points 6. Explain your results in Question 5) in terms on income and substitution effects. Which effect is the strongest in the present case? 10 points)

Answer 1

Here the utility function of Pierre is 'U = (21521/3, where 'C=consumption of goods and "L=leisure. She hash=i units' of time to divide between working (15) and enioying lesure (L). So, her budget constraint is given by. -> C=1775-A - t, where "A-adotional income outside of wage" and "t=lumo sum tax. => C= 7h-)-A-t => C-WL - W A -t, be the budget line of Pierre with absolute slope "W". 21. Here utility function is given by. => U = C*2/51741/3, => MRS = U_/UC => MRS = [11/3MTC/L)*2/31/162/3/TL/C)*1/5) = C/21. => MRS = C/24. => at the optimum MRS =W. => C/2L = W. => 6 = 2W2. The equation of budget line is given by -> C- W W -:-:=> 2W -W2=W -A-t => 3W7 = (Wan -À – t). => L = ( -4 - 11/5W, the optimum lesure. So, given the values of "W=?: h=1; A=0.125' and 't=0.35 the optimum le sure is given by. => L = ( w n -A - 11/5W = 11 -0.125 -0.351/3 = 0.25, => L = 0.25. So, the optimum labor supply is Ls = h-L = 1-0.25 = 0.75", -> Ls = 0.75. 3) From "MRS = Wwe have the following condition => C = 272 = 210.25 = 0.5, => C = 05. So, the optimum consumption and leisure are "C=0.5' and "L=0.25: Consider the following fig.

C = Consumption < u2 11 vo = L = Leisure 1°1 Here 41B1 is the budget line ander' be the equilibrium where the budget line is tangent to the utty function ul. So, the cosmum consumption and leisure are 'C7=0.5' and (?=0.25 respectively. Suppose the government increases tax to 't=0.45. => the optimum leisure is given by. => L = (win-A-11/5W, => L" = (1 -0.125 -0.45/3 = (1 -0.125 -0.451/3 = 0.225 < 1 = 0.25. => 6 = 262 = 230.225 = 0.45 < 2 = 0.5, and the working hour is 'L5 = h-L = 1 - 0.225 = 0.775 > L51. So, as the tax increase the she increases her working hour and decreases enjoying leisure and consumption of good.