Question

4. Suppose you have the following Cobb-Douglas Utility Function: And $200 to spend. a. Use the method of Lagrangian Multipliers, to maximize this consumer's utility and derive demand equations for both goods. Sketch their respective demand curves. Show all work. (5 pts) b. If Px = Py = $1, how much utility will the consumer enjoy? Show work/explain. (2.5 pts) c. Does this allocation satisfy the rule of equal marginal utility per dollar spent? Explain/show work. (2.5 pts)

Answer 1

cobb-Douglas W suppose we have the following Utility function Us x Š y uno de S Now in come M = $200 Suppose price of x = Px. .. and price of y = Pya hence budget constraint ist M= xpx + y pj . -) 200= xPx typy . a.). Now lagrangian function for utility manimising - Pioblem will be? .. .L= x ² y + 1200- XPx – ypy Now first order condition will be contabx - $ yt. Mar so = 2/2 x 3/ 4 / = ) ---) Pris : B = 32y - . Py = 0 3 x 25 y - 25 => 50 =) -- 4") cs Scanned with CamScanner

I Now, dividing equation () by il we get's , - % у % Г — ?) x - хvs и 215 - зя Р = ypy = 3 x px ? - 3x Px 3x Pr - и 2 Ря е) 2 2) p. 2 бр - Ұfх- УРУ = 0 Wow, we at:- 2 х/х 2PY whichid uting : 8 2P8 2 . -) а) 2 0 - xP - 3x fx . P = D 2.0 - x fx - 3xfх во - ( x Pя + 3x fx ) = - 200 С БҮР 200 2 CS Scanneal Jiith CamScanner

2) 54 px - 400 : 210 hence, demand functions are's :: K= 80 and 120 and y= g= py Now, x = 8 Now do y = - P LO Hince demand for a will be negatively sloped. Now y = 120 hence demand for y will be negatively sloped. . Now we can sketch their respective demaid comuned for good X and Y as follows. with : CamScanner

. Demand fory Demand for (6) Now it Px = Py = $.1, now demand for :.X = 80 - 90 - 80 demand for yo. 120 120 - 120 ... Now Substituting these values in the utility function We getr U= x y = (80)% (120 35 = 102.03 Huce the consues enjoys 102.03 utils of Utility Scanned with CamScanner

A (e) Now no 80 = 80 28 = 420 ys 1.20 Now U= x 2 y 3 Maginal utility of x, muxa a la 1. - š you Now Marginal utility of Y, muy = 2 2 x - %%s y/ ΜΟΥ na PX1 NOW MUX mux a at X = 80 and Y= 120, § (80 -% (120 ) %5 PX Sa 0.5 cs Scanned with Scanned with CamScanner

at at x = X - 80 and y=120 weget? MUY x 35 y-25 muy .= al (80) 3 (120)=45 = Ois MUY at x = 80 and Y=120 EPY Huce as Mox - MOY at yao satis is the rule of pn dollas spent. hence equal this allocation marginal Utility cs Scanned with CamScanner.