Q1. Consider a Cobb-Duglas utility function: log U = log A +a log X1 +b log X2 Given the prices of commodities X1, X2 as P1 and P2.respectively and money income M a. Derive the marshallian demand function of X1 and X2 b. Find the expression for the budget shares c. Show that the demand functions satisfy the properties of homogeneity and additive.

Q1. Consider a Cobb-Duglas utility function: log U = log A +a log X1 +b log X2 Given the prices of commodities X1, X2 as P1 and P2.respectively and money income M a. Derive the marshallian demand function of X1 and X2 b. Find the expression for the budget shares c. Show that the demand functions satisfy the properties of homogeneity and additive.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.3P

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A utility function is called separable if it can be written as U(x, y) = U (x) + U,(y), where U > 0, U" < 0, and U1, U, need not be the same function. a. What does separability assume about the cross-partial derivative U„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.
I need asnwers of f,g Assume there is consumer, his utility function is u(x,y) =8 * x0.5+y , and his budget constraint is px*x +y = m, which implies py = 1. a.Please derive the Marshallian demand function of x. b.Please derive the indirect utility function. c. Please derive the expenditure function If originally m = 40, px=2. d. What is his original highest utility level? Now px has decreased to 1, m and py do not change. e. What is his new maximum utility level? f. Based on (c) (d) and (e), what is his compensating variation? g.Based on (c) (d) and (e), what is his equivalent variation?
I need asnwers of e,f,g Assume there is consumer, his utility function is u(x,y) =8 * x0.5+y , and his budget constraint is px*x +y = m, which implies py = 1. a.Please derive the Marshallian demand function of x. b.Please derive the indirect utility function. c. Please derive the expenditure function If originally m = 40, px=2. d. What is his original highest utility level? Now px has decreased to 1, m and py do not change. e. What is his new maximum utility level? f. Based on (c) (d) and (e), what is his compensating variation? g.Based on (c) (d) and (e), what is his equivalent variation?
This question will let you examine/explore a more interesting utility func- tion than the simple example discussed in class as there will be both cross-price elas- ticity and an inferior good. Suppose you are told a consumer has the following utility function: U (9x9₂) = 9x + √√9₂ +9₂ You should assume income is Y, the price of good z is Pr, and the price of good z is P₂. This question will ask about several concepts discussed in lecture. (a) What is the Marshallian demand for goods x and z? I.e. find (9) for both interior solutions and corner solutions. Note: the outcome is "ugly" for the interior section and both corner should include constraints, i.e. limits using Y relative to f(P2, P₂). Hint: if solving using MRS = MRT, can use this information again in part (d).
A consumer has the utility function U=x0.2 y0.8 The price of xis $2, price of y is $4 and she a budget of $100 to spend on the two commodities. a. Solve the constrained maximization problem for the consumer (use the Lagrange approach) b. Establish that the solution is a true maximum
Could you please help me only with the last part-money metric utility function? Thank you!
is a key additional assump- The assumption that preferences are tion to deduce that the area under the curve shown in Figure 2 below is a measure of a consumer's utility. A x:(Pi, Pj, y) Ра Pb Price of Good i Figure 2. Marshallian Demand for Good i Qty of Good i
1. Given the utility function U = (x + 1)y and the constraint xPx + yPy = B a. Find the optimal levels of purchase x* and y*. b. Is the second order condition for maximum is satisfied? c. Calculate the comparative statistics derivatives interpret the results. ax ax ax ay ay ay Əв 'ƏPx' OPу Əв 'Əpx' apy > and 1. Given the utility function U = (x + a)(y + b) and the comstraint xPx +yPy = B a. Find the optimal levels of purchase x* and y*. b. Is the second-order sufficient conditions for maximum satisfied? ax ax ax ay ay ay c. Calculate the comparative statistic derivatives ав'aP['aPy'B'др'дру the results. and interpret
A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x₁, x2) = x³x² where x₁ and ⁄2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. Derive the consumer's Marshallian demand functions. Derive the consumer's Hicksian demand functions. Derive the consumer's expenditure function. Let m = = 120, p₁ = 2, and p2 = 1. Suppose that the price of good 1 drops to p₁ = 1. Find the following Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV. Let m = 120, P₁ P₁ = 2. Find the following = 1, and p2 = 1. Suppose that the price of good 1 increases to Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV.
fonsumer's demand xy for two different and the consumer's in come is $72. different Consumer's demand xy for chosen to maximize two the శ్రిందికి goods are x² lefility function The per respectively unit of , and the Consumer's in come is $7 x and y are $3and $4. 22,2 © telrite out the fagrangean for the Constrained maximization problem © Find the utility maximizing demands for both goods and he Larange multiplier, 2 o What is the change in consumer's ufility if income Changes from $72 to $75
(2) Consider the utility function u(x, y) := x'/³y²/3 for all nonnegative x and y. Given the budget constraint p„r + PyY = m, show that the Slutsky equation which is given by m Əm 3p (3) Recall the proof in question (2). Given that all prevailing conditions remain constant except the price of good y, Py, show further that the total effect of a price change in y results in the equality ду ie %3D
What is the logarithmic transform of the utility function U= 52, given the budget constraint - M. Select one: O a. In U = a ln a +Bin 2 +y In os %3D O b. In U = In a + In + In e In U = a ln a -Bln a -yln es Od. InU=n 2? + In e? + In a