Question

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EXERCISE 3 Consider a consumer who consumes two goods and has utility function u(x1, x2) = x2 + VX1. Income is m, the price of good 2 is 1, and the price of good 1 changes from p to (1+t)p. Compute the compensating variation, the equivalent variation, and the change in consumer's surplus for a change in the price of good 1, holding income and the price of good 2 fixed.

Answer 1

is Guen: Ulility = u(2,22) = N₂ +55 Income om Þei bib - bi-Cr+t)b - _ Compensating variation is the amount we need to give to the consumer to keep the ouginal ofternal bundle affordable at the new prices i.co. ucp., P2, m) - ulpi, pz, m+ci) Budget constraint : 6.2, + d = m

We need to get optimal demand function is - max u Cd., ₂) set paith = m. dagrange function o L = wait H, td [m-pri-H2] differentiating writ, an de and , we get" the to t^ [+b] =0 - + A [-p ] =0 - Apeg þ1 = 1 - 2X T + [-1] =0

al. m-prev - X, 20 Substituting seqind - eq , we get 26 5xi = 0 452 from eq and (w we get one m-W | 4ks) - = 0 m . 42

thibleng uld, 12). V 17462 +/m- + m atm-14 24 m4b) - Line 45 ula, 2) = m(46) + 45. - mt1 чр 46 um, b, 1) = m + to Now, u (m, pup2) = u(mtcv, plitt), I) s (mun) + I 4$(17) Now, we know that ulm, p. 1) = u(mtcv, PLITE), ) h = m + C + 4p(14) By CV = - 1 46 4p(14) Cr= (1+t-1 = t

is equal to Therefore compensation vouatio t/4p(HE). Equivalent variation is the amount that we need to take away from the consumer to keep the new sutility at the ouginal Pada ham ulim, pis pu) - ulm-ev, kub2) Þ' is the new price , ev 8 equivalent Variation ulm kirke) - u(m, plitt), 1) - 4p(11) u Em-ev, B., ka) = (m-ev) + LWta : 4pi requating them, we get 4plit) = ev = 1 46 L 4b (14)

i et - Itt-1 4p(tt) ht 4p(tt) - equwalent variation is equal to t/4p (lt) Consumer Surplus é at puce p : x = 1 at place p (itt) : x = 1 . 48 (146) Consumer Surplus = X . P C at puce p) 45. Consumer Surplus = X op (ite) (at price plity I MCHU) Aplitje per aplitu change in CS = . . 4p(tt)

Consumer surplus decreases by t/46 (ltat) The C, Ev and cs are equal algebraically Because the Utility function is quasilinear!