Question

(5) Consider how a consumer decides how much of two goods, x and y, to consume but races amb constraint. The consumer maximizes the following utility function s.t. the budget constraint U(x, y) = (x - rayß subject to Pxx + p,Y = 1 Where x and y stand for the two goods, Px, and Py stand for the prices of the goods X and Y, respectively, and I stands for Income (a) Write down the Lagrangian and solve for the first-order conditions (FOC) for x, y, and the Lagrange multiplier. (b) Using the FOC, solve for the marginal rate of substitution between x and y

Answer 1

Now, the ability fuction / U(x, y) = (1-r) y and budget constraint? - Px xt Pyy = I Where X Prices of X and Y and Y are two goals. and I is the Px and Py are level of income. Now, Яk ал огам (ът uk'я малим 29 Problem can be writtmas! L = (x-2) Kyptx [I - Px X - Pq Y Where X is the lagrange multiplin . for utility Now, the first order Condition manimisation will be ! 2x - 21 3 x (x-7) *y p _ A PREO 2) & (x-7) - 14 B - A - -(1) and 21 , P (RN) Ry A7 - App = 0 a P (x-rx y B-1 - - (2) Now equating vaurs of a I (2) we get from equations (1) and Scinned with Calm Scanner

2 (x-7) 22 y - Plz=73&4 ph, sa - Py 1 =) < (x-2] *-1 yp Py = B(u-ray B-1 PR R-n =) apy y = 8 px (0-2) =) y B. Pa (X-7.) or apy an Now, 21- IPix X-Pyy = 0 substituting value of y we get i ha dua It PxH - Py B Px (x-2) = 0 & Py 2. I- PxX - A Px(x-7) = 0 = a = BPx (237) – PXX - I 3) BPx (X) = 2 (Pax-I) | =) (x-7) = (P4 X-I) a) x 2 (Pir-I) + S P Pn PP anned with JamScanner

and we know y = BPx (x-7) set substituting value of a here we get' y = BPx & (Px H-I) trn apy L BPM =) (Pxx-I) . I and 1! A from equation 2 (4-2) 2-1 yß - substituting values of X and Y la te & (Pul-I) +8-81 Px X-IT A PP px 2 & ppu to p y K (Pull-I) TRI/PRX-İ18 step px [py ) s Scained with Cari canner bo

Huce valus of X and Y and s will be to x = a (Pre X-I 17 PLB Pro Dan Pn R-I - y = py and A Pur-I Ta CBX-IT L B PR en Py TCS scanned with JamScanner

Now, U(x, y) : (x-) Р Maginal utility of u je. mux = a u(x,y) әх . «(х-2) R-149 asd oyim, Fe, or a fe. MUу ләр(17) 24 = P (к-т)<y - and marginal rate of substitution betwan X and y I't. MRS = М DX (х-2) - чP муу (кг) чр-н - Now from foc. lic. from equation (3) WL yet? (х-2)-1 P PP (х-треур-) — «P4 «(-) <- чP . -Р (x-1) = P-— cs Scanned with CamScanner истината,

_or, MAS - Px py Huce the marginal rate of betwan xaud y from substitution foc will be pu py P ITrq Scanned with CamScanner