Consider a consumer with expenditure function e(p, u) = p2u - √ Suppose that the consumer has wealth 4. Initially, prices are (3,2). (a) What value of a would make the consumer indifferent between a fall in prices to (1,1) with wealth remaining at 4 and an increase in wealth of 2 with prices remaining at (3, 2)? How is this conceptually related to the compensating or equivalent variation? (You do not have to do any additional calculations to answer the second part of the question.) (b) Find the substitution and income effects on Good 1 associated with a marginal increase in the price of Good 2 when prices are (3,2).
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- Existence of representative consumer Suppose households 1 and 2 have one-period utility functions u(c1) and w(c2), respectively, where u and w are both increasing, strictly concave, twice-differentiable functions of a scalar consumption rate. Consider the Pareto problem: Subject to the constraint c1 + c2 = c. Show that the solution of this problem has the form of a concave utility function vθ(c), which depends on the Pareto weight θ. Show that vθ(c) = θu (c1) = (1 − θ)w (c2). The function vθ(c) is the utility function of the representative consumer. Such a representative consumer always lurks within a complete markets competitive equilibrium even with heterogeneous preferences. At a competitive equilibrium, the marginal utilities of the representative agent and each and every agent are proportional.A consumer is faced with the followlling Utility Function, U( x 1 x2) = ( xp +xp ) 1/ρ, where 0<ρ<1. The consumer also faces the prices and and has income level m. 1. Set up the Lagrangian 0ptimisation function for the consumer and Compute the optimal consumption bundle for the consumer. 2. The solution in (a) represents the Marshallian demand function for and . Using the solution in (a) compute the indirect utility function. 3. Derive the corresponding expenditure function for the consumer and the Hicksian demand function.Consider a consumer with utility function u(x1, x2) = α_1x_1^( 2) + α_2x_2^( 2) where α1 > 0 and α2 > 0. Assume that p1, p2 > 0.? (a) Derive expenditure function e(p, u). Verify that it is homogeneous of degree 1 in p and increasing in u. (b) Using expenditure function and Hicksian demand, calculate Walrasian demand and indirect utility
- If total utility increases as wealth increases, the first derivative of the utility function is negative. TRUE OR FALSEConsider a three-commodity consumer setting with the expenditure function:e(p, u) = up1α p2βp3γ 1. Find the indirect utility function2. Find the Walrasian demand function3. Verify Roy's identity4. Recover consumer's direct utility functionConsider the following function describing the utility of a consumer: U(x1, x2, x3) = a1*ln(x1) + a2*ln(x2) + a3*ln(x3), where ln = natural logarithm and a1, a2, a3 constants a. Pose the primal problem (using Langrange's method), obtaining the Marshallian demands for each good and the individual's indirect utility function. b. From the results obtained from question a., find the minimum expenditure function and the Hicksian demands.
- An individual´s utility function is U = x0.5 y0.5 While the budget constraint is x + 4y = 100 Derive the expenditure function. Calculate the CV and EV when the price of the good x increases from 1 to 4.Jane has utility function u(x,y)=x4y5. The price of x is px, the price of y is py, and Jane’s income is W. What is the expression of her consumption of y as a function of prices and income (her optimal consumption is determined by the tangency rule)? Qy(px,py,W)= 9W/5py. Qy(px,py,W)= 5W/9py. Qy(px,py,W)= 4W/3py. Qy(px,py,W)= 5W/4py. Qy(px,py,W)= 4W/5py.A consumer is maximising her utility function: U(x, y) = (x¹/³+y¹/³)³, subject to the budget constraint x + 3y = 100. (a) Set up the Lagrangian function of this utility maximisation problem and derive the first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the Lagrange multiplier. (c) What are the utility maximising amounts of x and y if the budget constraint changes to x + 3y = 50? Also, calculate the Lagrange multiplier.
- An individual’s utility function is given by where is the amount of leisure measured in hours per week and is income earned measured in cedis per week. Determine the value of the marginal utilities, when = 138 and = 500. Hence estimate the change in utility if the individual works for an extra hour, which increases earned income by GH¢15 per week. Does the law of diminishing utility hold for this function?Consider an overtime rule that requires that workers get paid double for any weekly hours over 40. Draw a picture that shows how a worker decides how much to work. Label everything in your picture and explain what is happening. Consider an investment that costs $100 and pays back $10 each year as long as the person making the investment is alive. Construct an equation for the net present value of the investment. An individual has a utility function, U = AX1 X2 where X1 and X2 are consumption of goods 1 and 2. The individual also faces a budget constraint. Show mathematically how an increase in Aa§ects the individualís decisions about consumption of each good.Q 2. Xinyi has an income of I = 140, and faces prices px = 1 and py = 2. Her utility function is U(x,y) = xy + x. (a) (a) Find her optimal consumption bundle, using the Lagrangian. (Note: You are not required to do so here, but it would be good practice to also solve for Xinyi's demand function for x by leaving prices and income as variables.).