Question

Sofia has utility function U = (wH)" (24 – H)1-a, where w is wage, H is hours worked per day, and Y is quantity of other goods. What is her labor supply function if a = ? We know that: H = 24 – N Y = WH + Y* , where wH is earned income and Y* is unearned income. We can set up an unconstrained maximization problem: max U = U(Y,N) = U(wH, 24 – H) = (wH)^(24 – H)1-a, H where N is number of hours of leisure. Using the chain rule with equations (1) and (2) gives us the following first-order condition: du dY 42U 2N = wUp - Uy = 0 (3) ay dh' an dH To find the values that maximize Sofia's utility, set the derivative of her utility function with respect to H equal to zero. Using (3), we have: wUy – Un = wa(wH)a-1(24 – H)1-a – (1 – a)(wH)" (24 – H)-a = 0. Simplifying, we find that H = 24a. Substituting a = }, she works 8 hours a day, regardless of what the wage is.

can someone show me step by step how to simplify to H=24a STEP BY STEP THANKS!

Answer 1

OH На. U=((H)(24-H) 1* Maximise utility wrt H. OH (WH) a (ld) (24-H) TW + (24 - H)ld. a WXHX-1=0 thus (WH)" (1-2) (24-H)4 = + (24-H)td. award- WA HA (1-4) (24 -H)4 = tawara-1 (24-H) 1-2 (1-4).(84-4) 4-***Hot (16) (24-H5'- Hoxt-g 80. 4 45 # = (1-4) H = 0 (24H) #put X=1/3, then (1-13) H = (24 H) (13) 2H=24-H 37=24, TH"=8 L Ang H-44 = 24.-12. thus H = 24 x). Poved (or