4. Suppose the economy has 100 units each of goods X and Y and the utility functions of the (only) 2 individuals arc: Ua (Xa,YN pareto-improvement is possible if, initially, goods are divided equally between the two individuals. X0.25 y0.75, UB (XB,YB) = Xº.75 y0.25. Show that %3D
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- Darwin consumes cigarettes (x1) and other goods (x2). His preferences are given by UD(q1,q2)=4lnq1 +q2The quantity of good 2 is measured in units of income and so you can assume that throughout this problem, the price of good 2 is normalized p2 = 1 (a) Using the Lagrangian, derive Darwin’s uncompensated demand for cigarettes: D1D(p1,Y). Note that we are holding p2 =1 and so that is not a variable in the demand function. Using the Lagrangian, derive the equation λ∗(p1, Y ). Provide an economic interpretation of the Lagrange multiplier. Does it vary with income, Y ?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,…D2) Economics John is a consumer of housing and food with a Cobb-Douglas utility function of U(H, F) = HaFB, with α and β being positive constants. 1) With prices remaining constant, will John raise his housing and food consumption in proportion to his increased income? Mathematically justify your answer. 2) Will John spend the same amount of his income on housing and food regardless of how much they cost or how much he earns? Mathematically justify your answer. of their prices or her income? Justify your answer mathematically.
- The utility function of a consumer is u = √x + 2y. (a) Show mathematically that the rise of Px will decrease demand for x goods. (b) Demonstrate mathematically that an increase in Px will increase demand for goods y. (c) If currently the price of goods is four times more expensive than the price of goods x and the government wants to set a tax that results in the increase in Px′ = 1.1Px, what are the units of decrease in the number of x items requested?Assume the demand function for good X can be written as: QX = 30 - 3PX + 2PY + 0.2I Where PX is the price of good X PY is the price of good Y I is the consumer income. a) Based on the demand curve above, is X a normal or inferior good? b) Based on the demand curve above, what is the relationship between good X and good Y? c) What is the equation of the demand curve if consumer incomes are $40,000 (use $40, income in thousands) and the price of good Y is $35?Denote the consumption of food by x and the consumption of all other goods by y. The demand for food as a function of prices and income is given by: Qx(px,py,W)=5W/8px. Suppose that W=100, px=3, and py=5. The change in consumption of food that is caused by a 2% increase in W is approximately: An increase of 2% in demand of y. There is no change. A decrease of 2% in demand of y. A decrease of 2% in demand of x. An increase of 2% in demand of x.
- 1.Let x and y denote the amount of goods X and Y. Find the demand functions of X (do not need to find that of Y) when your preferences are represented by the utility function U = x^2y. Is X normal good? Can you confirm law of demand for X? What is the relationship of X with Y? Answer all of them by using the demand curve you derived. 2.Let x and y denote the amount of goods X and Y. Find the demand functions of X (do not need to find that of Y) when your preferences are represented by the utility function U = x + xy + y. Is X normal good? Can you confirm law of demand for X? What is the relationship of X with Y? Answer all of them by using the demand curve you derived 3.There are two goods F and C. Let MU and P denote marginal utility and price of each good. SupposeMUF = 3, MUC = 4, PF = 2, PC = 2.Are you maximizing your satisfaction? If not, what would you do to increase your satisfaction? Explain.The utility function of a certain consumer is U =(x1,x2)= x11/3 x22/3 , x 1and x 2 is the consumption of two kinds of goods, and the consumer's income is 100. The current prices of the two kinds of goods are P 1 =1 and P 2=2 respectively, ask: 1. If the price of the first commodity increases from 1 to 2, and other factors remain unchanged, what is the total effect of the price increase on the consumption of the first commodity? According to the Slutsky decomposition principle, what are the income effect and substitution effect? 2. Calculate the amount of income compensation that changes the price of the first commodity from 1 to 2, keeping the original effect unchangeda. Determine the demand functions of x and y in the case of a Cobb-Douglas type utility function, in the following cases: α=0.40;β=0.60 Graph the demand functions of the two goods (price as a function of quantity) assuming the individual's income is $500 - Determine what is the quantity demanded of x and y, if the price of good x is USD 1, the price of good y is USD 4, and income is USD 500 - Now, explain what happens to the quantity demanded if the prices of the goods are doubles holding income constant.
- The demand for good X is given by Qd = 6,000 – 1/2 Px – Py + 9Pz + 1/10 M Research shows that the prices of related goods are given by Py = $6,500 and Pz = $100 while the average income of individuals consuming this product is M = $70,000. A. Indicate whether goods Y and Z are substitutes or complements for good X. B. Is X an inferior or a normal good? C. How many units of goods X will be purchased when Px = $5,230? D. Determine the demand function and the inverse demand function for good X. Graph the demand curve for good X.use the demand function below to answer the questions that follow where Py and Pz are prices of related goods and M is income. using the demand function: Qdx = 3/4 - 1/3Px - 5Py + 2Pz - 2/10M where Py and Pz are prices of related goods and M is income. Using the demand function; (a) identify which type of goods is Pz and explain it (b) identify which type of good is M and expalin it (c ) identify which type of good is Py and explain it and interpret Px1) a consumer of two goods, either 1 or 2, has a utility function u (X1, X2) = X1 0.6. X2 0.4, where X1 denotes units of good 1 and X2 units of good 2. The price of good 1 is 5 euros per unit, the price of good 2 is 3 euros per unit and the consumer's income is 150 euros . Assume that only the price of good 1 changes and that the new price is P1 '= 2. Then the income that allows the consumer to consume exactly the initial basket after the decrease in the price of good 1 is Choose one: A) 86 B) 66 C) 76 D) 96 E) Not defined