Question

Does someone know how to solve this please been answered before but been wrong and confusing can you please help thanks!! PART B AND PART D please

1. Consider the utility function u(x, y) = ¼ In x + ½ In y for goods x and y. (a) At an income level of M, and market prices for x and y of px and p, respectively, set out the consumer optimisation problem and form the Lagrangian. Using this, find the Marshallian demand functions for goods x and y. [Note: d (4 In x)/ dx = 1/ (4 x)]. [30% mark] (b) For good x, find the own-price, cross-price and income elasticities of demand. Interpret these, indicating what they reveal about the nature of good x. [30% mark] (c) Sketch the Marshallian demand and Engel curves for good x in separate diagrams, carefully labelling the axes. Evaluate the demand for x at parameter values M = £70, px = £1 and py = £4, and sketch this in the same diagrams. [20% mark] %3! (d) What is a Giffen good? Explain why the Engel curve alone reveals that good x is not a Giffen good. More generally, using an appropriate diagram, and by identifying the income and substitution effects, show how Giffen behaviour arises. [20% mark]

Answer 1

2. (9) Consumer's objective is to maximize ulary) = 2 ln xttlny, subject to budget constraint n.Pa typy=m. Hence, the optimi- zation preoblen becomes: Max alatt my st. x P x + Y Pry zm. The Lagrangiam can be expressed as: L = matt my +2 [m- x Px -4 Py] First order conditions : voleno ang -o. o #4-7x=0>1- P 0 22 - 3 - 1 1 - 2 Py = 0 =) A = zyping 22 = M-4 x. Px -Y Py = 0 = x Px + O Pg = M 6. Second order conditions : 8 - 30 sept- }yz → Secondarder conditions are satisfied for maximization Now, from equation and Ⓡ, 1 - 4x Pa = 2y po => y = me Using equation & in equation ③ we get : x Px tape Py =M >x . Px + 4 Prix = M => 3 x P x + 4 x Pm Zm 7xPx = 3M ->> La Marshallom demand for an. - © ► yang bisa your mon |_ Marshallian demand for y~ 0. (b) From equation ♡ we have 2 - as the demand for a The elasticities can be defined as follows: -Own price elasticity: Exx = ane. Pre - cross price elasticity : Eray - open - encome elasticity: Exma Now Example EMC comes 7 - mai putin >> Exx = TEM(7.P. - -. ) *xx' => fxz2\-)|6xx=-|| Exy = 8 (3). P = 0 x P = 0.) Eny = ol Ezm lam cerut M/102]- za May P CM) 7 =>€rm = 1 | Lam Pu 23m/7Px 7PM 3M 3M) 7Px (3M) (7.22

Since own price elasticity of demand (Enn) - good a is havina unitary elastic, where a increases / decreases in same pre with a preoportionate decrease /increase in Post We have seen that Exy zo nie. consumption of n is not enflucenced by change in Py. It is neither a substitute nor a complement of y. On the other hand, Exm = 1 and axo . Hence, good is a nevernal good. Since Exm $1, it is not a levurey good. either (6) when M=70, Px = 1 and Pry= 4 , we can reeplace these values in equation and derive the demand for x as: x = 3x70 -2 7 X1 - 30 price and quantity Now, the demand curere can be shown in plane, where as Engel curve represents income omd olemand combinatica and can be shown an income and demand plan as follows: Fig. 2 Demand curve e Engel curene 2- => 겨족분 > M=7 x Upwow sloping curve from origin with slope 7. *** 10 20 30 0 10 20 30 when demand for a good increases with an ahcrease in its price or demand decreases with a decrease in its price, then the good is called Gebben good. The case of Gipt in good can be shown with okagram 3 as follows. Let a consumer initially is in equili- nyt Teg.31 bricon at E Cre, when budget line to AB is tangent AN. to the inditterence curve bo). of pride of a falls, let the budget line shitte to reight, say ABl. Due to nature of the good , suppose the consumer happens to be in equilibrium at Ei, where consumption of a Cire. oxi) is less than the initial conscumption (oxo o xxo x B 2 in the present case, the substitution effect - *.*s which in determined and with budget lene Alb which is langent to v ie. the income required to maintain the initial utility). And the income effect = •XX; . The income effect in this case outweighs Lü substitution effect. We'a keduction in fe price ob , the real income C ruises which shifts the budget line. On the other hand, the slope of lü esEngel curve may help to identity such nature of goods. In the present cas, Pal Nominal M= Real M. Due to a fall in Bu, real Minerceases and consumption of increase since Engel awtve a positively sloped sticaught lene. If xwere a Giften goods, a would have reduced with fall in Poxor increas in which is not the case herce. Hence, x is not a Giften good in the picesent case. EUD