Problem 5: Consider an economy with production. The production functions for goods X and Y are given by: X Y = L ပိဋက where Lx+Ly = L. The utility function of the representative consumer is given by: xyl U (x, y) = min 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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![Problem 5: Consider an economy with production. The production functions
for goods X and Y are given by:
X
Y
||
La
Ly
= 3
where Lx+Ly =Ī. The utility function of the representative consumer is given
by:
y
U(x, y) = min { // }
where all the parameters are positive. Determine the Walrasian equilibrium
quantities of X and Y produced and sold in this economy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F82b46645-14e3-440f-8c2c-fd00e1b61743%2F1eaef796-1dc0-4728-a87c-770598f7ab29%2Fp1f17g_processed.png&w=3840&q=75)
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- 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..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.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 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?Assume that the prices of good X, Y and Z are as follows R5,R1 and R4 respectively, and the Judith has an income of R37 to spend. HOW much of each good will judith consume in order to maximise her utility? What will be her total utility and marginal utility of the last rand spent on each good? Show all the calculationsConsider an economy inhabited by George and Harriet, whose utility functions are Ug : (ac)² (bc)2 Он тан + 2bн The total quantities of ale and bread that can be produced by the economy are a and b, and they are constrained by the production function b = 2(10 – a)/2 There are infinitely many Pareto optimal allocations. In one of them, Harriet's utility is 8. a) An allocation in this economy is described by a list of four variables. What are these variables? b) What four equations describe the Pareto optimal allocation in which Harriet's utility is 8? c) Find this Pareto optimal allocation.
- Elsa lives alone on an island with two goods, bananas and fresh water. Her utility function is U = 4BW where B is the amount of bananas she consumes and Wthe amount of water. Her production function for bananas is B=Lg where Lg is the amount of labour time she devotes to bananas. Her production function for water is W = Lwwhere Lwis the amount of labour time she devotes to water. If the total time she has available is 70, what should she do to maximize her utility? (Enter your answers in whole numbers.) LB= Lw=[5. Consider a two-consumer one-firm economy with a single input good and a single output good. Consumers' utility functions are u4(x4) Suppose that the input good is divisible and that wA = wB = 1/2. Suppose that the output good is indivisible (i.e., that xg, q = 0, 1, 2, ...), and that the firm's production technology is such that it requires 1 unit of input to produce one unit of output. rA + 3r and uB(xB) 4x? + x. (a) Derive the Pareto set of this economy. (b) For all Pareto-efficient allocations from (a), derive a competitive equilibrium for this economy that generates these allocations.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.
- Throughout this problem set, we will look at exchange economies with two goods and two agents. Let X = R², let u denote agent i's utility, and let wie X denote agent i's endowment. 1. Suppose u¹(x¹) = min{ri, 2} and wi = (4,8) for both agents i. (a) Argue that every Pareto optimal allocation has r≥r for both agents i. (b) Argue that every allocation z with r≥r for both agents i is Pareto optimal. (c) Draw an Edgeworth box, with a picture depicting every Pareto-optimal allocation. In this picture, also draw the endowment allocation, and draw each agent's indifference curve through the endowment. (d) Argue that, in any competitive equilibrium, the price of good 2 must be zero. (e) Find all competitive equilibria.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.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.