Consider the following utility function, u (x1, x2) = min [√x1, √√αx2], where a 0 (a) [15 points] Derive the Marshallian demand functions. (Explain your derivation in details.) Does the Marshallian demand increase with price? Are the two consumption goods normal goods? (b) [15 points] Show two different ways to derive the Hicksian demand functions. Does the Hicksian demand increase with price?
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![Consider the following utility function,
u (x1, x2) = min [√x1, √√αx2], where a 0
(a) [15 points] Derive the Marshallian demand functions. (Explain your derivation
in details.) Does the Marshallian demand increase with price? Are the two
consumption goods normal goods?
(b) [15 points] Show two different ways to derive the Hicksian demand functions.
Does the Hicksian demand increase with price?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf58e778-209d-4051-b758-0c57e3f8187c%2F6296910f-45a2-4bc5-af9f-f950ca8815bf%2Fv7lw9ve_processed.png&w=3840&q=75)
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- Suppose that we can represent Joyce's preferences for cans of pop (the x-good) and pizza slices (y-good) with the utility function min[4x,5y]. a) Find her Marshallian Demand Functions. b) Find her Hicksian Demand FunctionsDerive Ryan's demand function for q₁, given his utility function is where o = = (9₁) P + (9₂)P, 1 1-p The demand curve for q₁ as a function of P₁, P2, and Y is (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ 9₁ = character.) U= Let the price of q₁ be p₁, let the price of q2 be p2, and let income be Y.If the utility function for a consumer is defined by U=6X^0.6Y^0.7 . Given that the consumer's income is 300 currency units and unit price of goods X and Y are 12 and 15 currency units respectively, calculate the equilibrium quantity of both goods. Compute the price elasticity of demand for both goods and interpret your results. If income and prices of the two goods increase by 50%, calculate the equilibrium quantities of both goods
- Suppose that i's preferences over goods x and y are represented by the following utility function U₁(x, y)=x45¹5. Let m denote the consumer's income, p denote the price of good x and let the price of good y equal 1. A) Find the Marshallian demand functions for goods x and y. B) Show how each of the demand function is affected by a change in the price of good x. C) Which of the goods is an inferior good?The individual's utility is given by the following function: ?(?,?)=−1/?−1/? a) Calculate the Marshallian demand functions for both goods.b) Determine whether x and y are gross substitutes or gross complements. Is there any asymmetry in the gross definitions?c) Without actually doing so, determine whether x and y are net substitutes or net complements. Please explain.5. Suppose that we can represent Pauline's preferences for cans of pop (the x-good) and pizza slices (y-good) with the utility function u(x, y) = min[4x,5y]. a) Find her Marshallian Demand Functions. b) Find her Hicksian Demand Functions
- Let's say that the demand side of the market for Blue Soda is comprised of 3 leading agents/individuals: Anthony, Brad, and Claire. Let P be the price of 1 liter of Blue Soda, and Qd be the quantity demanded of Blue Soda in liters. Here are the key points to the problem: - Anthony buys only one liter of Blue Soda if the price of it falls below his choke price of $10. - Brad's demand for Blue Soda is defined by QdB = 5 - P/2 - Claire buys 2 liters if the price is below $5, 1 liter if the price is between $5 and $10, and nothing if the price is above $10. Using this information, please sketch the individual demands and the market demand by aggregating the three agents/individuals. Label the graph clearly. Please make sure to sketch the individual demands first and then sketch the market demand.1. Assume you spend your entire income on two goods X & Y with prices given as PX & PY, respectively. Prices and income (I) are exogenous and positive. Given that U = X2 + Y2 , derive the Marshallian demand function for good Y and evaluate the type of good. 2. Assume you spend your entire income on two goods X & Y with prices given as PX & PY, respectively. Prices and income (I) are exogenous and positive. Given that U= X2Y2 , derive the Hicksian demand function for good Y.3. Suppose that initially PX = 2, PY = 8, I = 96 and the Marshallian demand function for good Y is given by Y∗ = (0.5I/ PY)+(0.5PX/PY)− 0.5. Calculate the own price & income elasticities of demand for good Y. Interpret your computed values and say something about the type of good.4. Suppose the economy has 100 units each of goods X and Y and the utility functions of the (only) 2 individuals are: UA (XA,YA) = X0.25Y0.75, UB (XB,YB) = X0.75Y 0.25Show that pareto-improvement is possible if,…Suppose the demand curves for goods A, B, and C have the following functional forms, where Q denotes quantity demanded, P denotes price, and M denotes income: QA = 120 - 3.5PA - 6PB + 14M QB = 100 - 2PB + 3PC + 1.1M Qc = 1500 -0.5Pc - 300M. Based on these demand curves, which of the following goods are known to be normal goods? A, B and C B A and B only C A
- A) Suppose a quasilinear utility function is given by u(x₁, x₂) = 2√√x₁ + 4x₂. 1. For this utility derive the demand functions xi (P₁, P2, 1) and x₂ (P₁, P2, 1). 2. For this utility derive the Engle function I = 1(x₂, P₁, P₂) for good 2. B) Do the same for the Cobb-Douglas utility function u(x₁, x₂) = x₁x². That is, 1. For this utility derive the demand functions x₁ (P₁, P2, 1) and x₂ (P₁, P2, 1). 2. For this utility derive the Engle functions I = 1(x₁1, P₁, P₂) and I = = 1(x₂, P1, P2).1/2 1/2 2. Cynthia has preferences represented by the utility function u(x) = x¹/² + x¹/² (a) Calculate Cynthia's marginal rate of substitution. (b) Calculate Cynthia's Marshallian Demand functions x(p, 1) and x(p,1). (c) Does Cynthia consider either of these goods to be Inferior Goods? (d) Show that Cynthia's demands generate an indirect utility function V(p,1)=√√IP₁P2 (e) Solve for Cynthia's expenditure function.Suppose your utility for goods x1 and x2 is represented by the following utility function: U(x1,x2)= x11/5 x24/5 a) What is your marginal rate of substitution, MRS12? b) If the price for good x1 is p1 = 2, the price for good x2 is p2 = 4, and your available income is m = 20, write down your budget constraint. c) Using the prices and income given at b) above, find your optimal consumption choice bundle (Marshallian demand) and its corresponding utility level. d) Illustrate your optimal consumption choice on a graph. e) For the prices given in b), what income would you need to achieve a utility level of 25?