Problem 5: Consider an economy with production. The production functions for goods X and Y are given by: X Y = Lz where Lx+Ly = Ī. The utility function of the representative consumer is given by: U (x, y) = min {2,} where all the parameters are positive. Determine the Walrasian equilibrium quantities of X and Y produced and sold in this economy.
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- Suppose the household’s preferences are given the following forms: u(c)=log(c), v(l)=log(l), that ß=2/3, and that the production functions are given by: F1(L)=L1/2 and F2(K)=K1/2. a) What are the equilibrium conditions? Derive them. b) What are the quilibrium quantities?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,…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 = X + Y, derive the Marshallian demand function for good Y and evaluate the type of good. Assume you spend your entire income on two goods X & Ywith 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. Suppose that initially Px = 2, P = 8, I = 96 and the Marshallian demand function for good Y is given by . Calculate the own price & income elasticities of demand for good Y. Interpret your computed values and say something about the type of good.
- 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, x2) = In ¤1 + In x2. 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, y½) = (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 (y7, vž) = (38, 12(1 – B)). The hunter-gatherers now have the possibility of opening up their economy to free trade. In world markets, 1 unit of meat can be exchanged for 2 units of berries, and the country would be a price-taker. Which of the following statements is correct? O a. Only agents of type g benefit from free trade. O…Elsa lives alone on an island with two goods, bananas and fresh water. Her utility function is U = BW where B is the amount of bananas she consumes and W the amount of water. Her production function for bananas is B = 6LB where LB is the amount of labour time she devotes to bananas. Her production function for water is W = 2LW where LW is the amount of labour time she devotes to water. If the total time she has available is 10, what should she do to maximize her utility?Pat is a representative consumer in the neighbourhood market for Jr Chickens. Their utility function for JCs and all other goods is given by: U(JC,Y) = 10JC – (JC^2)/2 + Y MUc = 10- JC MUy= 1 Every McDonald's franchise has the following production function for Jr Chickens: JC = 4K^(1/2) + 2L^(1/2) MPx = 2/K^(1/2) MPL= 1/L^(1/2) a. Derive Pat's Marshallian demand for Jr. Chickens. You can assume Pat has a trust fund and can always afford it. b. There are five other people in the neighbourhood who like Jr Chickens. What is the neighbourhood's market demand for JCs? c. What is the price elasticity of demand when P= 1, 5 and 9? Note: dQ/dP = -5. d. Derive the conditional factor demands for K and L for a McDonald's franchise.
- Consider an economy with a single consumer whose preferences are given by U = log(x) - , where x is consumption and labor supply. Assume that the consumption good is produced using labor alone with a constant-returns-to-scale technology. Units of measurement are chosen so that the producer prices of both the consumption good and the wage rate are equal to 1. a. Let the consumer’s budget constraint be qx = , where the consumer price is q = 1 + t, and t is the commodity tax. By maximizing utility, find the demand function and the labor supply function. b. Assume the revenue requirement of the government is 1 10 of a unit of labor. Draw the production possibilities for the economy and the consumer’s offer curve. c. By using the offer curve and the production possibilities, show that the optimal allocation with commodity taxation has x = 9/10 and = 1. d. Calculate the optimal commodity tax. e. By deriving the first-best allocation, show that the commodity…Given U = (x+2) (y+1) and Px = 4, Py + 6, and B =130: (a) write the langrangian function (b) find the optimal levels of purchase x* and y*. (c) is the second - order sufficient condition for maximum satisfaction (d) the answer ' b' give any comparative - static information? ** Can you please help me with (d)? I know the optimal level of purchase for x* and y* is 16 and 11 respectively. Thank you.Question 6: Transitivity There is only one consumption good in the world (cities). Karen is moving – suppose that her preferencesover cities are described as follows: she strictly prefers city A to city B if either (1) the average rent in cityA is cheaper than in city B, or (2) the average rent is the same in both cities, and there are more goodrestaurants in city A than in city B. Are these preferences transitive? Justify your answer.
- Consider a society consisting of just a farmer and a tailor. The farmer has 30 units of food but no clothing. The tailor has 60 units of clothing but no food. Suppose each has the utility function U=Fc, If the price of clothing is always $1, and the food price is currently $1, then we can conclude O the market is at a competitive equilibrium. the price of food will drop towards a contitive equilibrium. the price of food will increase towards a competitive equilibrium. O None of the above..Say that Laura (L), Maureen (M), and Carrie (C) are three individuals who are contemplating the purchase of some amount of good X. Units of X can be produced at a constant marginal cost of 24. The following equations show how the marginal benefit (M B) that each individual places on X varies with the quantity she consumes: L: MB(L) = 30 - X M: MB(M) = 24 - X C: (MB(C) = 20 - X (a) How much of this good should they purchase if X is a private good? (b) How much of this good should they purchase if X is a public good? (c) Construct a diagram to illustrate your answer to part b.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 + Y?, derive the Marshallian demand function for good Y and evaluate the type of good.