Question

1. Charlotte loves apples and hates bananas. Her utility function is U (a,b) a-b2/4, where a is the number of apples she consumes and b is the num- ber of bananas she consumes. Assume that Charlotte's income is y • What are the demand functions for Charlotte ? • What are the Engel curves for Charlotte? 2. Wilbur likes both apples and bananas. His utility function is U(a,b) = ab1/2. Assume Wilbur's budget is m, the price of apple is Pa and price of bananas is po, then what is the optimal demand of Wilbur on each good? 3. Nalan's passion is fine wine. When the prices of all other goods are fixed at current levels, Nalan's demand function for high quality wine is 2 0.02m - 2p, where m is her income, p is the price of wine, and is the number of bottles of wine that she demands. Nalan's income is 7500 pounds, and the price of a bottle of wine is 30 pounds. • How many bottles of wine does Nalan purchase? • If the price of wine goes up to 40 pounds, what will be the substi- tution effect and income effect? 4. When prices are (4, 6, 2), Goldie chooses the bundle (6, 6, 8), and when prices are (6, 3, 4), she chooses the bundle (10, 0, 4). When prices are (5,5,5) she chooses (8,2,6) • Construct three by three revealed preference matrix by matching three price-quantity pairs • Does Goldie violate WARP? Explain • Does Goldie violate SARP? Explain

Answer 1

kns. I Utility function. u (a, b) a a-6/ let the price of apple Pa. and the price of banana =P Total income - y : budget constraent * y pa at pb. b Hence the problem, Mase u (a, b). 1 st. y = pq a t Po b. Setting Lagrangian L = a-B/4 + x(y- Paa - Pob) Ja = 1 - Apa 06/06 = - 26/4 -XPp. =-% -XP6. Od/ox = y paa - Pob FO.C. fou utility monimisation reequires = 1 = xla -0 - 2400 = 0 = 1 x Pa = 0 Ox/ab=0= - b/2 ->P6 = 0 3. do/ax = 0 ) y= paa + Pob. dividing @ by , we get 97-6/2. - out · The demand functions are, er, b =- 2P0 / Pa Now, y = Pa a tPob. y = Pa. at Po (- 2/6) or y = Pa a apo or Paa = y + 2Pb/Pas. a. + 24 ) au +2 (alle) ь - .

Enges ewree for Apple for good as we +2 er da lis = a slope is constant Hence, the slope of the engel curve is constant at 1/ Pa and it depicts that the Engel curve will be upward sloping straight line yg for good b, b= -2Polpa dod=0 demand for b is independent of income. hence, the slope of the Engel curve is o the good is zero income elastic So, the Engel ewere will be horizontal • Engel were for banana. to *- axis.