Question

Solve the following Utility Maximization Problem for x* and y* that Max U(x,y)= ln(x) subject to Pxx + pyy = I ----.-.(2) where In denotes the natural logarithm (base e) and x and y>0. a) (25 points) by Substitution and show that your values of x* and y* max U (x*,y*).

Answer 1

U to lity manimisation problem ts Maa U (X,Y) = en (Jag) subject to Pa X + PyY = I we start with the Constraint, aa Now PxX + Ry y = I I- Px X PyY => I-Px X PY Subs tituting thie value of y in the utili ty buction subs fit ut'on method we get! by Max U(X) = en( JX/I-PIX Now, The tise or des Condition dur). w'll bc fe(I=! PY PxX) PY "I-PxX- X PxX FO I- 2XX PY ק-1)יא (T- २%X Par 2X (I-PAx) =0 =) CS Scanned with CamScanner

> I- 2 PxX =0 х* =) 2 PX I– PxX* Py Also we Know, y= we get Sübstteting Value of PX 2PX I- y*= Py 2I-I 2. PY multi plin Now by the Lagnauge method probleim f b), we Coun sct up the dagranyian for ublity as fo llow št. mani mi sation evay) + (P I- Px X- Py y where 1 is t he lagsrainge multiplin. Scanned with CS CamScanner

Now, the first ordn Condition Bor utility mani misation Problein will ben (n4)"?y -x Bx .y-APx =0 2 Px X =) 7x- 1Py = 0 Alse, aL 7. ry 2 my АРУ =) -(3 ) ই) 2 Py Y value f 1 brom equation (2) Now, equating get, wC and (3) 2Py Y 2 Py Y 2 Bx X 2 Px X = =) - -14) PX I- Px X - Py y0 Now, aL- Substituting value of x brome eq uation (4) ien ration we get, get , wit Camscanner WC

I- Px BY) - By y=O -> > I- 2Bjy =0 = 2 Py Y =) 2PY Also, we know, Py y* PX X* Py 2PY. PX x* = Ta PX Scanned with CS CamScanner