Question

?(?, ?) = ?? + ?
and the budget constraint ??? + ??? ≤ ?. Assume throughout that all prices and quantities arepositive and infinitely divisible.

Derive the consumer’s indirect utility function ?(∙). Then, derive the consumer’s expenditure function, e(∙), directly from ?(∙). Finally, derive the consumer’s Hicksian/compensated demand functions (denoted ? and ? , respectively) from e(∙).

Answer 1

no- du 9+1 2 С Given that :- О: 19+х Let budget constraint be px*+py Y=M du MU, = da MUY dy MRS = = MUx Px моу Py C+1) Ру Px X-Py 9+1 Р. (ov) - Py Putting in Budget constraint M= Px (G+D py М: Рx + Р Р - Ря) РЧ M = Px xt prx-py Н= Р - Ру + Pч 9 Мt P = 2 px x Mt Py * м- Ру 2px 2Р 22 рх Ру 2 Pe (Foto Py) + РЧ Ч М- P9 = 2 Pч 8

2px v (px, py,M) - ** (**+0) MAPY (M-Py + Martha 2РЧ М+РЯ 'M- Pytapy 2px 2 py = SMH py) (M+ py (2 px) (2py) Indirect utility MxpY² 4 px Py V When Px= Py = 1 and M=10 V = (10+)² 4X1X1 4 121 30.25 M=10 when Px=1 Py = 2 (10+2)² 4X2X1 2 doo 12 8 = 144 8 = 18

change in utility = V₂ V, = 18-30,25 = - 12.25 between x and y Then we First we differentiated The U with Respect to X and Y and then wring MRs we found a relation x we substitude it in budget constraint optimal X* y* and then derived indirect utility function wing them as marked in the bores, Then with the value given we found V, and v2 and subtracted them for finding the change. and found our OuY