Rosie has U =C^2R where R = leisure and C = all consumption goods. She has 90 ho
ID: 1166691 • Letter: R
Question
Rosie has U =C^2R where R = leisure and C = all consumption goods. She has 90 hours available for labor or leisure per week and a $60 allowance. Her wage is $10/hr.
a) find her optimal bundle of consumption and leisure and her labor supply.
b) if the government places a 50% tax on labor income (so her wage is 50% lower) but gives her an additional allowance of $290 find her new optimal bundle of consumption and leisure and her labor supply. Does she work more or less hours?
. Mr. Cobb and Mr. Leontieff work for the same firm. Mr. Cobb's utility is U = C^2 R^4 . Mr. Leontieff's utility is U = min {C, 3R}, where C = consumption and R = hours of leisure. They both have 15 hours per day to allocate to either leisure or labor. They both get a wage of $6 per hour and neither has any other income.
a) Find their respective demands for consumption goods, leisure and labor supply.
b) Suppose their employer offered paid them $6 per hour for the first 5 hours of work and $12 per hour for every hour worked above five. Find their new respective demands for consumption goods, leisure and labor supply. Does this overtime wage schedule extract more work from them?
c) If the employer were to offer a wage of $12 for every hour of work would this induce Mr. Cobb and Mr. Leontieff to work more hours than originally?
Explanation / Answer
Solution:
1) For Rosie, utility, U = C2R, w = $10/hr, A = $60, if L is the labor hours supplied by Rosie, then L+R = 90. Since, nothing is mentioned about the price of consumption good, p, we take the good as a numeraire good, so p = 1
a) Finding optimum bundle:
Budget line for Rosie : p*C + w*R = A + w*(R+L)
1*C + 10*R = 60 + 10*90
C + 10*R = 960
Slope of this budget line = w/p = 10/1= 10
With given utility, slope of the utility function (indifference curve) = marginal rate of substitution (MRS)
MRS = Marginal utility of leisure (MU(R))/ Marginal utility of consumption (MU(C))
MU(R) = C2, MU(C) = 2*C*R
So, MRS = C2/2*C*R = C/2R
Optimality occurs where slope of indifference curve = slope of budget line
So, C/2R = 10 giving us C = 20R
Then, using the budget line: C + 10*R = 960
20R + 10R = 960, so R = 32
C = 20*R = 20*32 = 640
L = 90 - R = 90 - 32 = 58
So, Rosie's optimal consumption = 640 units, leisure time = 32 hours, and labor supply = 58 hours
b) Now Rosie's wage is 50% lower, so, new wage rate, w' = 0.5*10 = $5 per hr
So, Rosie's budget line becomes: 1*C + 5*R = 290 + 5*90
C + 5*R = 740
Slope of this budget line = w/p = 5/1 = 5 (< 10, thus this budget is flatter than previous budget line)
MRS still remains the same, MRS = C/2R
Using optimality condition, C/2R = 5 so, C = 10R
Using the new budget line then, C + 5*R = 740
10R + 5R = 740 giving us R = 49.333 (approx)
C = 5*49.333 = 246.667
L = 90 - 49.333 = 40.667 < 58. So, clearly now Rosie works for less hours.
2) For Mr. Cobb, U = C2*R4, for Mr. Leontiff, U = min{C,3R}, for both, A = 0, w = $6/hr, L + R = 15
a) Finding the optimum for both: Since both earn same wage income, have same time allocation, and face same price for consumption good, p=1, so budget line for both, Mr. Cobb and Mr. Leontieff is also same, which is:
p*C + w*R = w*(L+R) since A = 0
1*C + 6*R = 6*15
C + 6R = 90
And slope of this budget line = w/p = 6/1 = 6
Now, for Mr. Cobb, MRS (as calculated above) = MU(R)/MU(C)
MU(R) = 4*C2*R3; MU(C) = 2*C*R4
So, MRS = 4*C2*R3/2*C*R4 = 2C/R
Then optimal bundle for Mr. Cobb: 2C/R = 6 giving us, C = 3R
Using the budget line then, C + 6R = 90
3R + 6R = 90, so we have R = 10
So, C = 3*R = 3*10 = 30
L = 15 - R = 15 - 10 = 5. So, Mr. Cobb's demand for consumption good = 30 units, leisure = 10 hours, labor = 5 hours
For Mr. Leontieff, we have a min utility function, so we cannot calculate MRS by partial derivatives, but we know that optimality occurs as kink. So, for Mr. Leontieff, at optimality, C = 3R
Notice, that for Mr. Cobb we had the same optimality condition, and with same budget line, optimal bundle for Mr. Leontieff will be same as that of Mr. Cobb, i.e, demand for consumption good = 30 units, leisure = 10 hours, labor = 5 hours
b) Now with such differentiated wages, we have a kinked budget line for both.
As per the question, new budget line: C + 6*R = 6*15 for L < =5, or equivalently, R > 10
So, C + 6R = 90 in this case
For L > 5, or R<=10, budget line becomes C + 12*R = 5*6 + 10*12, i.e, C + 12R = 150
Now, For Mr. Cobb, when L<=5, slope of budget line = 6, as calculated in part (a) then R = 10, L=5 hours, C = 30 units. Then Mr. Cobbs's U = (30)2*(10)4 = 9,000,000
when L > 5, slope of budget line = 12, optimality condition becomes, 2C/R = 12 or C = 6R
So, optimal bundle, using budget line C + 12R = 150, we get, 6R + 12R = 150, R = 8.333 hours
C = 6*(150/18) = 50, L = 15 - 8.333 = 6.667 hours. So U in this case = (50)2*(8..333)4 = 12,054,398.3
Since, 12,054,398.3 > 9,000,000, Mr. Cobb chooses the latter bundle, so new demand for consumption = 50 units, leisure = 8.333 hours, labor = 6.667 hours (> 5 hours, so yes, more work is extracted from Mr. Cobb)
For Mr. Leontieff, optimality condition is still C = 3R
Now, when L <= 5 , using budget line C + 6R = 90, we obtained optimal same as Mr. Cobb, i.e, consumption, C = 30 units, leisure = 10 hours, labor = 5 hours
When L > 5, using budget line C + 12R = 150, 3R + 12R = 150, so, R = 10. So optimal bundle for Mr. Leontiff is consumption, C = 30 units, leisure = 10 hours, labor = 5 hours (same as obtained in part (a), so no, more work isn't extracted by Mr. Leontieff)
c) Now, w = 12
So, budget line for both = 1*C + 12*R = 12*15
C + 12R = 180
Now, slope of budget line = 12/1 = 12 (so steeper budget line as compared to (a) part)
MRS remains same as obtained, so now,
For Mr. Cobb, optimality condition becomes: 2C/R = 12 or C = 6R
For Mr. Leontieff, optimality condition remains same as above, i.e, C = 3R
Optimal bundle for Mr. Cobb, 6R + 12R = 180 , so, R = 10
Then C = 6*10 = 60, L = 15 - 10 = 5. So, Mr. Cobb's optimal consumption is 60 units, leisure is 10 hours and labor is 5 hours (so, works for same no. of hours as compared to part (a))
Optimal bundle for Mr. Leontieff, 3R + 12R = 180, so, R = 12
Then, C = 3*12 = 36, L = 15 - 12 = 3. So, Mr. Leontieff's optimal consumption is 36 units, leisure is 12 hours and labor is 3 hours (so, works for less no. of hours as compared to part (a))
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