4. Suppose that Clint works 8 hours a day for a wage of $15/hour. His utility fu

Question

4. Suppose that Clint works 8 hours a day for a wage of $15/hour. His utility function is given by U(c,w of Clint's wealth currently comes from his working wage. What is the minimum lump sum bonus that Clint's boss must give him in order to induce him to work two extra hours per day so that their company can prepare for the world cup? Assume that Clint has a total endowment of 24 hours. 0.21 0.8 with C, being Clint's consumption and leisure respectively. Assume that all The following questions are practice problems and are NOT graded. Do not turn them in. Try to

Explanation / Answer

Solution: Say I denote cosumption by C, labour by L, leisure by l, Total endowment (L+l) = 24 hours. Wage, w =$15/hour. Assuming that good 'consumption' is a numeraire good, i.e, price of consumption good,p = $1

Now, the budget constraint for Clint is: p*C + w*l = w*(L + l)

Why? Since, maximum income that Clint can have is by working for entire time he has, which is total endowment, and earning wage w per hour. So maximum income = w*(L + l).

By consuming C units of consumption good, Clint spends total of p*C on consumption. For the hours worked (labour hours, L), Clint earns wage w for each hour, then for the hours not worked (leisure hours, l), Clint loses w per hour. We can also say this as each leisure hour costs Clint wage w, hence total expenditure by not working (or leisure) for l hours = w*l.There is a trade off between leisure and labour, hence a trade off between leisure and consumption (since consumption increases as the income increases, which further increase with the labour hours)

So, budget line of Clint: 1*C + 15*l = 15*24

C + 15*l = 360

Utility function: U(C,l) = C0.2l0.8

We are given that for Clint, currently L = 8 hours, so l = 24 - 8 = 16 hours

Substituting this in the budget line, we get, C + 15*16 = 360 implying C = 120

Then Utility derived = (120)0.2(16)0.8 = 2.60517108*9.18958684 = 23.94 (approx)

But Clint's boss wants Clint to work for 10 hours a day, i.e, he wants L = 10, So, l = 24 - 10 = 14. Clint will agree to work for 10 hours if utility derived is atleast the same as before (when he worked for 8 hours)

Budget line (with minimum bonus, b) : C + 15*l = 360 + b

C + 15*14 = 360 + b, so C = 150 + b

Then utility derived = (150 + b)0.2*(14)0.8 and this should be equal to 23.94 (as obtained above)

So, (150 + b)0.2 = 2.8988

150 + b = 204.6956

b = 204.6956 - 150 = 54.6956 or 54.7 (approx)

So, minimum lumpsum bonus should be = $54.7

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