Let preferences of both individuals be given by log(ci)+ log(c). Suppose that the endowment vectors are wA = (5, 10) and wB price and the equilibrium consumption bundles of each individual. (10,5). Solve for the market clearing
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- Individual that consumes two goods (X and Y) and has a CES Utility Function of the form: U = 100(X^(0.75) + Y^(0.75)). Income of 1000, the price of Good X is 10 and the Price of Good Y is 20 a) Find the Marginal Rate of Substitution as a function of the quantities consumed of Good X and Good Y. b) Write out the Lagrangian for this problem. c) Solve to find the demand for Good X, the demand for Good Y, and the highest level of utility for this individual. d) Now consider an increase in the price of Good X to 20. What is the demand for Good X and Good Y? What is the Utility of the consumer following the price change? e) Considering the change in demand for each good between parts c) and d), how much is due to the substitution effect and how much is due to the income effect? f) Show your answers on a graph.Utility function U(X,Y)=(x^(3/4))(Y^(1/4)) 1)Suppose Px=2, Py=1, and Income is 100. What are the new levels of demand for X and Y? 2) Digrammatically show the income and substitution effect from 3 to 2 on the consumption of x. You must explicitly specify the optimal consumption bundles for the different sets of prices (appropriatly graph and label actual quantities) but you may qualitatively specify anything else deemed relevant. 3)Which effect (substitution or income) will move you along a given indifference curve? And for this question, do you move to the left or right along the given indifference curve and does this increase of decrese MRS. Also, explicitly specify what the MRS is equal to before and after you move along the indiferrence curve.A consumer has set a budget of $400 for the consumption of good X and Y. The price of Good X is S"Px", and the price of good Y is S5. The consumer has a Utility function given by U(X,Y) = ху Find the optimal consumption choice of the individual and the utility obtained in terms of Px. Make a graph that illustrates the solution to the problem. Why do we use parameters?
- For each of the following utility functions get the marginal utility of consumption of each of the goods. 1/2 1/2 (a) u(a,b) = a b (b) u(x, y) = x³/4y1/4 (c) u(a, b) = n(a) + In(x) (d) u(x, y) = ln(x) + ln(y) (e) u(a, b) = 2xa +XbEach week, an individual consumes x and y of two goods, and works for 1 hours. These three quantities are chosen to maximize the utility function U(x, y, l) = a ln(x) + ß ln(y) + (1 − a − ß) ln(L − 1), which is defined for 0 ≤ 1 0, and where a, ß > 0 and a + ß < 1. The individual faces the budget constraint px + qy = wl, where w is the wage per hour. • State the optimization problem • Solve completely for the utility-maximizing levels of x, y, and IIf the utility function of an individual takes the form: U = U ( x 1, x2) = (x1 + 2) 2 (x2 + 3) 3 Where U is total utility, and x1 and x2 are the quantities of two commoditiies consumed: (a) Find the marginal-utility function of each of the two commodities (b) Find the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.
- 1 Consumption utility functions Consider the following utility functions defined over the outcome space X = R². 1. U(1, ₂) = 1X2 2. U(1, ₂) = 2x₁ + x₂ 3. U(1, 2) = min{2x1,4x2} 4. U(₁, ₂) = 100 - (₁ - 10)²(x₂ - 10)² 5. U(x₁, x₂) = 1 + ln(x₂ + 1) 6. U(x₁, x₂) = x² + x² For each function: (a) Draw two indifference curves and an arrow indicating the direction of increasing prefer- ence. (b) Shade the upper contour set above the more-preferred indifference curve you drew in the previous part. (c) Indicate which of the following properties are satisfied by the underlying preference relation: convexity, monotonicity, local nonsatiation. (A given preference may satisfy more than one!) Explain why each property does or does not hold.Suppose we are able to model the total utility function for the consumption of two goods, good x and good z. The utility function is structured as U(x, z) = 3x2 + z2 - 2xz. The consumer is faced with the prices of goods x and z. The price for each unit of good x and z is $1 each. The consumer has an income $1 (in thousands). How many units of each good should the consumer consume so as to maximize his/her utility?Using the homogeneity of indirect utility, which of the following cannot be an indirect utility function? (a) V: = (b) V: = (c) V: = [2 PzPy I Pz+Py p²+p² (d) V = 1 Pz
- Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3Calculate the optimal consumption bundles for the following scenarios: (a) U(x,y) = x+47y-3y2.Income is $107. The price of x is $1 and the price of y is $23. (b) U(x,y) = 3x2 + 6y2. Thepriceofxis$5, thepriceofyis$11, and the income is $49. solve by maximizing and using derivatives for utility functionSuppose the economy has 100 units each of goods X and Y and the utility functions of the (only) 2 individuals are: UA (XA,YA) = X0.25Y 0.75, UB (XB,YB) = X0.75Y 0.25 . Show that pareto-improvement is possible if, initially, goods are divided equally between the two individuals.