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- Consider the following indirect utility function:ʋ(P,y) = y(P1r + P2r)-1/r Wherer = ρ/(ρ-1, Pi are parametric prices, and y is the consumer’s budget a) Solve for the Marshallian demand functions xi (P, y) and verify that these functions are homogenous of degree zero (Hint: you can also use Roy’s Identity). b) Derive the Hicksian demand functions xih (P,u)Q 2. Xinyi has an income of I = 140, and faces prices px = 1 and py = 2. Her utility function is U(x,y) = xy + x. (a) (a) Find her optimal consumption bundle, using the Lagrangian. (Note: You are not required to do so here, but it would be good practice to also solve for Xinyi's demand function for x by leaving prices and income as variables.).A consumer is faced with the following utility function, U(x1 x2)=(xp1 1+xp2)1/p, where 0<p<1. The consumer also faces the prices p1 and p2 and has income level m. (A) Set up a Lagrangian optimisation function for the consumer and compute the optimal consumption bundle for the consumer B)the solution in (a) represents the mashallian demand function for x1and x2. usuing the solution in (a) compute the indirect utility function. C) derive the the corresponding expenditure function for the consumer and the hicksian demand function. D) using your knowledge on classical demand theory, explain under what circumstances the expenditure minimisation problem and the utility maximization problem would yield the same results.
- Brit-Brick is a company that produces bricks and cement in the UK. Their largest consumer is ConstrUK, a UK construction company. The manager of Brit-Brick has asked the research department to find out how sensitive ConstrUK’s demand for bricks is. The research department has estimated that ConstrUK’s preferences over bricks (x) and cement (y) can be described by the utility function U(X,Y)=X1/2 Y1/2 where x and y are measured in bags. The price for one bag of cement is equal to £1. It is estimated that ConstrUK’s budget is £10,000. Find the price-consumption curve for bricks and the corresponding demand curve.A consumer is faced with the following utility function, U(x1 x2)=(xp1 1+xp2)1/p, where 0<p<1. The consumer also faces the prices p1 and p2 and has income level m. (A) Set up a Lagrangian optimisation function for the consumer and compute the optimal consumption bundle for the consumer B)the solution in (a) represents the mashallian demand function for x1and x2. usuing the solution in (a) compute the indirect utility functionFurther, provide arguments for oragainst the following statement: “The optimal bundle chosen by a consumer with welldefined preferences (rational, continuous, strictly monotone and strictly convex) alwayscontains luxury goods.”
- Suppose the consumer solves the following UMP: max (x1)^2 + (x2)^2 , s.t. p1x1 + p2x2 ≤ w where p1,p2 > 0. a) Plot the indifference curves b) Find the Marshallian demand functions. Show graphically the utility maximizing choices.UniQ is a company that produces speaker HF drivers and woofers in the UK. Their largest consumer is KEF, a UK loudspeaker manufacturing company. The manager of KEF has asked the research department to find out how sensitive KEF’s demand for HF driver is. The research department has estimated that KEF’s preferences over HF drivers(x) and woofers(y) can be described by the utility function U(x,y) = x^1/2 y^1/2 The price for one unit of woofer is equal to £1. It is estimated that KEF’s budget is £10,000. Find the price-consumption curve for HF drivers and the corresponding demand curve.Suppose that an individual has a Utility function represented by a CES function. The utility function of the individual is given as: U(x,y) = x1/2 + y1/2 b. Calculate the own price elasticity using the "share elasticity" of any good. Let us assume that the prices of both goods are equal.
- Brit-Brick is a company that produces bricks and cement in the UK. Their largest consumer is ConstrUK, a UK construction company. The manager of Brit-Brick has asked the research department to find out how sensitive ConstrUK’s demand for bricks is. The research department has estimated that ConstrUK’s preferences over bricks (x) and cement (y) can be described by the utility function U(x,y)=x^(7/10)y^(1-7/10) The price for one bag of cement is equal to £1. It is estimated that ConstrUK’s budget is £10,000. Find the price-consumption curve for bricks and the corresponding demand curve.A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle(8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consumingbundle (4, 4). She is indifferent between these two bundles. Assuming that the utilityfunction is Cobb-Douglas of the form U(x,y)=xα yβ, where α and β are positiveconstants, what is the utility function for this consumer?Irene spends all of her income M on soda and chips. The price of soda is 2 per unit. Irene’s utility function is U (s, c) = 5 ln(1 + s) + ln(1 + c).Hint: throughout the question, you can assume M ≥ 50 and pc ≤ 8 (these assumptions just rule out the possibility of corner solutions, but do not affect the way we solve the problem). (a) Derive Irene’s demand function for chips (as a function of her income and the price of chips). (b) Are chips an inferior good for Irene? Is the magnitude of the (total) effect of a decrease in price of chips greater or smaller than the corresponding substitution effect. (c) Obtain Irene’s inverse demand function for chips when her income is M = 50. Show that this function is decreasing and convex. is irenes engel curve upward or downward sloping? explain