Suppose there are only two individuals A and B in the economy with utility function of first individual given as U(A) = In(x1) +y12, and utility function of second person given as, U(B) = x2^0.5 + In(y2). Which of the following provides for the equation of contract curve. In(x1) +y12 = x2^0.5 +In(y2) x1/x2 = yl/y2 yl/y2 = x^20.5/x1 x2^0.5/2 x1 = y2/y1 Next
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- Consider a small closed economy with two consumption goods: good 1 (meat) and good 2 (berries). There are two types of agents, h and g, and they have the same preferences over consumption, represented by the utility function: u(x1, 22) = In r1 + In #2. However, there are twice as many type-h agents as type-g agents. The only factors of production are their labour. When a type-h agent chooses to spend a fraction a of his day producing meat and the rest producing berries then his output is (yf, y ) = (2a, 2(1 – a)). A type-g agent is more productive. When she chooses to spend a fraction B of her day producing meat and the rest producing berries then her output is (v7, y2) = (38, 12(1 – B)). Which of the following statements is correct? a. Given equilibrium price p, each agent of type h demands one unit of good 1 (meat) and p units of good 2 (berries). Each agent of type g demands 6/p units of good 1 (meat) and 6 units of good 2 (berries). O b. Given equilibrium price p, each agent of…Suppose there are only two individuals A and B in the economy with utility function of first individual given as U(A) = In(x1) + y12, and utility function of second person given as, U(B) = x2^0.5 + In(y2). Which of the following provides for the equation of contract curve. In(x1) + y12 = x2^0.5 + In(y2) yl/y2 = x^20.5/x1 x2^0.5/2 x1 = y2/y1 x1/x2 = y1/y2The utility possibility frontier is defined as the set of maximum utility levels for all consumers that is feasible in the economy, given resource constraints and preferences. For a two-consumer economy, it is defined formally as the solution to: u (ūB) = max u A (xA) subject to ug(xB) > ūg and (xA, XB)feasible. For each of the economies above, find an equation for the utility possibilities frontier, and graph it.
- Remy and Emile consume only blueberries (x,) and raspberries (x,). Remy has utility function UR = x (x5)² and Emile has utility function UE = (xf)²x. Remy is endowed with 5 blueberries and 5 raspberries and Emile is endowed with 10 blueberries and 10 raspberries. (a) Derive the equation of the contract curve, i.e., find x%(xf). (b) Set x2 as the numeraire, i.e., assume p, = p and p2 = 1. Find the competitive equilibrium – the equilibrium price ratio, P1/P2, and the equilibrium allocation, ((xf, x5), (xf,x£ )).Consider an economy composed of 16 consumers. Of these, 5 consumers each own one right shoe and 11 consumers each own one left shoe. Shoes are indivisible. Everyone has the same utility function, which is Min(2R, L}, where R and L are, respectively, the quantities of right and left shoes con sumed. A) (10%) Is the status quo (where each individual has his own shoe) Pareto efficient? If so, briefly explain why. If not, provide a Pareto improvement b) (10%) Characterize all Pareto efficient allocationsCarol and Bob both consume the same goods in an economy of pure exchange. Carol is initially endowed with 9 units of good 1 and 6 units of good 2. Bob is initially endowed with 18 units of good 1 and 3 units of good 2. They both have the utility function U(x₁, x₂) = 1/3 2/3 x1³x2³. If we set good 1 as the numeraire (so that p. = $1), what will the equilibrium price of good 2 be?
- Reese thinks peanut butter and chocolate are great when separate, but when they combine they are even more epic. In other words, Reese likes to eat either peanut butter or chocolate, but when he eats them together, he gets additional satisfaction from the combination. His preference over peanut butter (x) and chocolate (y) is represented by the utility function: u(x, y) = xy + x + y Suppose that now Reese loses almost his entire income, so that he is left with only one dollar, i.e. his new income is I0 = 1. If prices are still px = 2, py = 4, what is his new optimal consumption of x and y (Hint: Remember that consumption of both goods must be weakly positive, i.e. x∗ ≥ 0 and y∗ ≥ 0) (a) x∗ = 0.5, y∗ = 0(b) x∗ = 0.25, y∗ = 0(c) x∗ = 0.75, y∗ = 0.25(d) x∗ = 0.75, y∗ = 0(e) x∗ = 0.5, y∗ = 1Carol and Bob both consume the same goods in an economy of pure exchange. Carol is initially endowed with 9 units of good 1 and 6 units of good 2. Bob is initially endowed with 18 units of good 1 and 3 units of good 2. They both have the utility function U(x₁, x₂) = = $1), what will the equilibrium price of x1/³x/3. If we set good 1 as the numeraire (so that p₁ good 2 be?Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. (a)How many hours are each of the spouses allocating to home production and market work? and Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer.
- a) Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. How many hours are each of the spouses allocating to home production and market work? b) Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer. [Hint there is no need to calculate the full solution to this case]For Questions 5-7 below, consider the following description of a two-goods example: Assume two consumers with utility functions of the form 1/2 1/2 UX (x₁, x₂) = x/²x² and UX (y₁, 92) = y1Y2. Further, suppose that consumer X is endowed with wx = (6,2) and consumer Y with w = (2,6). Question 5. Assume the consumers face prices p = (1, 1). Determine the net demands for each consumer. Question 6. Assume the situation of an exchange economy. (a) Determine the contract curve as an expression in which good 1 (₁) is a function of good 2 (T₂). (b) Provide a diagram that illustrates the contract curve and the core of the economy. (c) Determine the Walrasian equilibrium prices. Question 7. Provide general formulations of the First and Second Welfare Theorems for an ex- change economy and, using the specific exchange economy of this section, illustrate the theorems in a diagram.For the next three questions, assume that there are two consumers in an economy that have utility functions UA(2,3)=2¹/42/4 U" (x,y)=1¹/2¹/2 The two consumers begin with equal endowments of the two goods === e = 50 29. If the price of z and y were both set to 1, there would be (a) An excess demand for r so the equilibrium price ratio must be less than 1 (b) An excess supply of r so the equilibrium price ratio must be less than 1 (c) An excess demand for y so the equilibrium price ratio must be greater than 1 (d) An excess supply of y so the equilibrium price ratio must be greater than 1 (e) No excess supply or demand for either good, so the equilibrium price ratio is 1 30. What is the equilibrium price ratio? (a) 2/3 (b) 5/2 (c) 3/5 (d) 1 (e) None of these 31. Consumer A increases his endowment of both goods to 100 (e = e = 100). This will cause (a) No change in the equilibrium price ratio (b) The equilibrium price ratio to increase, causing consumer B to decrease their consumption of…