Suppose that consumer has the following utility function: U(X,Y) = In(2X) +Y. Suppose also that Px 1, Py = 12 and I = 120. What would be the optimal consumption of X and Y at the cquilibrium, respectively? !3! 48, 6 O 12, 9 24, 8 72, 4
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- Michael does not like to mix peanut butter and jelly in the same sandwich. However, he will consume them separately; for him, a sandwich with 1 spoon of peanut butter is exactly the same as a sandwich with 2 spoons of jelly. Michael has an income of m = 50, and the prices per spoonful of peanut butter and jelly are pPB=5 and pJ=11. Please write down Michael’s utility function over peanut butter (PB) and jelly (J). 2. Please determine Michael’s Marshallian demands PB*m and J*m3. Please determine Michael’s new Marshallian demands PB*m and J*m, when the price of peanut butter falls to pPB = 1.4. What are the (Hicks) SE and IE? Draw a diagram to show your analysis, with peanut butter on thehorizontal axis, and jelly on the vertical axis. 5. Recall that there are two different types of substitution effects. For example, in Q2 we have used theHicks SE. Does your answer to the last part change if we use the Slutsky SE? Justify. 6. Nam likes his peanut butter and jelly sandwiches with exactly…Mats, who has reference-dependent preferences over beer and money, goes to the local pub with a friend, but is not planning on drinking any beer or spending any of his 50 Euro in cash. Let his end-of-evening outcomes in pints of beer consumed and cash be c1 and c2, respectively, and let his reference point in pints of beer and cash be r1 and r2, respectively. Then, Mats’ utility is given by v(6c1 − 6r1) + v(c2 − r2), where v(x) = x for x ≥ 0, and v(x) = 1.5x for x < 0. (a) Suppose that the price of beer is pB. Calculate Mats’ utility from drinking one pint of beer at this price. What is Mats’ utility from drinking no beer? And, comparing these two utility values, what is the maximum price pB that Mats would pay for one beer? (b) Suppose that Mats unexpectedly gets a pint of beer as part of a promotion at the pub, and incorporates its consumption into his reference point in beer. [Hint: this means that (r1, r2) = (1, 50).] Suppose that Mats could sell the beer at a price pS.…Assume an individual spends all of the their income on a bundle comprised of good #1 and good #2. In particular, their utility function is given by: U(q1,q2) = q12/3q21/3 Assume the price of good #1 is $1 (p1=1) and the price of good #2 is $3 (p2=3). What must the individual's income be if they maximize their utility by purchasing 10 units of good #1?
- When utility function U(q1, q2)=min{ 34.6 q1, 17 q2} and q1= 33 and q2= 60 are given, find utility level consumer gains?Bob has a utility function U(x, y) = √x1 + 0.8√x2 + 0.64√x3 over his incomes x1, x2, x3 in the next three years. This function is an example of (A) expected utility; (B) quasi-hyperbolic utility function; (C) discounted utility; (D) none of the above. . Which of the following preferences agree with Bob’s utility? (A) (9, 10, 11) ≻ (9, 10, 12); (B) (9, 10, 11) ≻ (11, 10, 9);(C) (9, 10, 11) ≻ (9, 11, 10); (D) none of the above. Bob’s utility function implies (A) time stationarity; (B) transitivity; (C) impatience; (D) all of the above.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.
- Q12. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F.Suppose as Case X, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $15, respectively. a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e (F*,C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space.Suppose that you have two opportunities to invest $1M. The first will increase the amount invested by 50% with a probability of 0.6 or decrease it with a probability of 0.4. The second will increase it by 5% for certain. You wish to split the $1M between the two opportunities. Let x be the amount invested in the first opportunity with (1-x) invested in the second. Find the optimal value of x. Using expected value as the criterion (linear utility) Using the flowing utility function: u(x)=2.3 ln〖(1+4.5x)Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…
- Q)Suppose we have two goods, x1 and x2, with price p1 and p2, respectively. Consider the budget constraint x1p1+x2p2≤m. If both p1 and m double (i.e. p’1=2p1 and m’=2m), the area of the feasible consumption set: a) doubles B)stays the same D) quadruples C)none of the above E) falls by halfSolve; a consumer utility function is given as 64q10.5q20.25q30.4 2. derive the second-order partial derivatives of the utility function with respect to the three pairs of commodities1. Use budget constraints to express consumption levels, ct and ct+1. (Hint: Use income conditions given above in the budget constraint. Notice that there are two possible states in the second period.)2. Rewrite the utility maximization problem as choosing the optimal at alone. (Hint: Replace ct and ct+1 in the utility function with your answers from point 1. Use probabilities to derive the expected value in the utility function. Remember that a random variable that takes values x1 in state one with probability p and x2 in state two with probability 1 − p has the expected value E [x] = p.x1 + (1 − p).x2)3. Derive the first order condition and find the optimal value of savings, at. (Hint: The only control (choice) variable is at)4. Does household accumulate precautionary savings to self-insure against the scenario of low income in the second period? Why or why not?