Question 3 Consider the utility function of the form: U = x¨*x2² Given the budget constraint: P1X1 + P2X2 = M Show that the implied Marshallian demand curves are: a1 M X1 = (a1 + a2) P1 a2 X1 = (a1 + a2) P2
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Question 3
Consider the utility function of the form: ?=?1?1?2?2
Given the budget constraint: ?1?1+?2?2=?
Show that the implied Marshallian demand
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- A consumer can consume two goods, A and B, and has the utility function U-15A1/281/2, The consumer's budget is $900, the price of good A is $15 per unit and the price of good B is $45 per unit. (Assume A is the horizontal axis good and B is the vertical axis good. Both goods are infinitely divisible, but round numerical answers to 2 decimal places as necessary) What is the consumer's marginal rate of substitution? What is the consumer's marginal rate of transformation? What is the formula for the consumer's budget constraint? What is the consumer's utility-maximizing bundle given the utility function and budget constraint? A B H1. Given the Cobb-Douglas utility function: U = AX Y¹-a (1) Derive the Marshallian demand function for good XPankti consumes two goods, x and y. Her utility function is given byU(x, y) = ln(xy).(a) Suppose when Pankti’s income is 12, her optimal bundle consists of 2 units of x and 6units of good y. Without solving for Pankti’s Marshallian demands for x and y,determine how her consumption of x and y would change if her income doubled(holding constant the prices of the goods). Justify your answer as well as you are able.(b) Find an expression for Pankti’s indirect utility function, V (px, py, m), using themethod of Lagrange multipliers. Confirm your answer to part (b) using theMarshallian demands you derive in the process of solving the optimization process.(c) Suppose the price of good x is 2 and the price of good y is 2. Find Pankti’s utilitywhen her income is 24. Now suppose the price of good x doubles to 4. How much extraincome does Pankti need to obtain the same level of utility she had prior to the priceincrease?
- ii) Utility function: U = xy + 3x + y x = 15 [Ans. x - 15. y - 72 = = 7 3,2 = 1 ] Income constraint: I 212, Py = 12, Px = 8 U = Q;Q2 » Pi = !, 2 = 4 1=120 iii) Utility function: Constraint function: Qi + 4Q2 = 120 [ Ans. O = 60, Q2 =15, 1 = 15] %3!Suppose you have the following indirect utility function: V(Pa, Py, I) = In PxPy What are marshallian demands for x and y? I (a) (9x9y) = (22) (b) (9,9y) = (In, In 2) (c) (9, 9y) = (exp(2p/py), exp(2ppy)) I (d) (9x, gy) = (2pr+py' px+2py) What is the expenditure function for the associated expenditure minimization problem? (a) E(pa, Py, U) = (P + Py) ln(U) (b) E(pa, Py, U) = √exp(U)Papy (c) E(pa, Py, U)= (p²+p²) In(U) (d) E(pa, Py, U) = exp(U)²papy What are the individual's Hicksian demands for goods x and y? (a) (h₂, hy) = ((BU)¹/², (PU) ¹/²) (b) (ha, hy) = (RU, DU) (c) (ha, hy) = ((2 exp(U))¹/², (exp(U))¹/²) -1/2 (d) (hx, hy) = ((P₂PzU)−¹/², (P₂PzU)-¹/2) Are x and y complements or substitutes?Principles of Economics 1 | S1 21/22 Time left 0:5) When the price of one good increases and the price of the other good and income are held constant, the budget line Select one: shifts parallel to the original budget line so that the new budget line is farther from the origin O b. shifts parallel to the original budget line so that the new budget line is closer to the origin rotates so that the intercept is farther from the origin on the axis representing the good that has experienced an increase in price O a. O c. O d. rotates so that the intercept is closer to the origin on the axis representing the good that has experienced an increase in price Next page
- 10:55 Sun 8 Aug 30% Screenshot_20210808-073803_Office.jpg a) Choose X, and X, to max the utility fxn, UX,X)=chX;+(l-a)hX, st: M = P, X, + P¿X ; a) Consider a price decrease for good 1 from P toP" . Write down the Demand functions for good 1 when prices are P, and when prices drop from R toP" b) Obtain the SE and IE associated with the price changeAssume that the prices of good X, Y and Z are as follows R5,R1 and R4 respectively, and the Judith has an income of R37 to spend. HOW much of each good will judith consume in order to maximise her utility? What will be her total utility and marginal utility of the last rand spent on each good? Show all the calculationsMorgan has the following utility function: u(x, y) = 5 ln(x) + 3y. Her income is given by I = 15 and the prices originally are pr = 2 and py = 3. = (a) What are Morgan's Marshallian demands? (b) How much of each good is Morgan currently consuming? (c) What is the utility level that Morgan can achieve? (d) Assume the price of x increases to p = 4, find Morgan's new levels of consumption. X X (e) Find the total, substitution and income effects for good x caused by the price change. Consider this price change a "large" price change (Apz = Pz - Px=4-2=2).
- Consider the following consumer’s problem: max u(c1 ) + βu(c2) c1c2 subject to c1 + s1 = y1 c2 = y2 + (1 + r)s1 u(c)={1/(1-1/σ}c1-1/σ a) Describe all equations using economic terminology.b) Derive the intertemporal budget constraint for this consumer.c) Set the Langrangean for this problem.d) Find the first order conditions for a maximum.e) What is the effect of an increase in the interest rate on savings? Show in equation and explain.19:32 Fri 6 Aug 91% Screenshot_20210806-193021_Office.jpg a) Choose X, and X, to max the utility fxn, UX,„X,)=chX;+(l-a}hX; st: M - P, X + P,X ; a) Write down the demand functions for goods 1 and 2 b) Consider a price decrease for good 1 from P, toP" . Write down the demand functions for good 1 when prices are P, and when prices drop from R toP c) What is the compensating income, M associated with the price change? Obtain the SE and IE associated with the price change 24a. Determine the demand functions of x and y in the case of a Cobb-Douglas type utility function, in the following cases: α=0.40;β=0.60 Graph the demand functions of the two goods (price as a function of quantity) assuming the individual's income is $500 - Determine what is the quantity demanded of x and y, if the price of good x is USD 1, the price of good y is USD 4, and income is USD 500 - Now, explain what happens to the quantity demanded if the prices of the goods are doubles holding income constant.