Question

3. Suppose an individual has a utility function U=U(M,X)=10 MX^2, where U is her

utility, M is her(daily) money income and x is her(daily) leisure hours. Each

day, the individual needs 6 hours for sleeping and other essential personal matters

3. Suppose an individual has a utility function U = U(M,X) = 10 MX, where U is her utility, M is her (daily) money income and X is her (daily) leisure hours. Each day, the individual needs 6 hours for sleeping and other essential personal matters. The rest of her time (18 hours a day) can be spent either working, earning a daily income (M) at a wage rate of $5 per hour, or taking leisure (X). i) Write down the daily time constraint this individual faces in terms of X and M. (Hint: Daily income equals the wage rate times the number of hours worked. Does this allow you to write an equation for the number of hours worked in term of M?) The individual wants to maximize her utility subject to her daily time constraint. Set up a Lagrangian function to find her optimum leisure time and income. iii) Verify that the second-order condition is satisfied.

Answer 1

Answer 3 (i) It is given that he has to spend on working (W) Leisure Cx) 18 hours and So x + W = 18 => W = 18- W M = = Income = Wage rate X M = SW = 5/18-x) M + 5x = 90 - We have to : Subiect to : Maximize : U=10MX M+SX -90 (1) Legrange is given by : 2= 10mx + 490-M-5x) first order condition: 22 20 = 10x = a (2)... aM al = 0 = 2000 =5h (3) 2x Dividing (2) from (3) we get! 1 11-

Putting this in (1) we get! M+5X=90 = 2058 +5X = 9 0 => x = 12 hrs. Optimal keisure Time => M=2:5() - 2'5 X 12 => M = $30 - Optimal Income (از Second Hessian Order Condition: is given by :

H = L x x L xm la thux tum Lma thax LAM for Maximum H will be negative definate i.e. Hiyo and Yr is a Symmetric Matrix when hab = ah abda using above Information we have 20M 200 -51 TH1= 120° 0 -1 | = -20M + 100X 1.-5 - 0 As M=30 sx = 12 t1oox => IMI = -20% 30 :100712 t 1oox. - IML = 18000 e M is symmetric This Hiil post negative definite and hence, It i ų a maximum c. u = 30 & X = 12 is a maximum.