4. Suppose you had to eat a cake over 3 days. The share of cake consumed in period t is Ct, so that Σ1Ct ≤ 1. Instantaneous utility is derived as u(ct) = log(Ct). a) = Find the optimal consumption path for an exponential discounter with a daily discount rate of p 0.1. b) Find the consumption path for a naïve hyperbolic discounter with a discount factor of 1 in the current day and B (1) for utility delayed by 7 days, where ß = 0.7 and p = 0.1. c) Discuss what a sophisticated hyperbolic discounter should do on the first day.

Let u(x) be a utility function that represents % and let f(.) be a continuousmonotonic function. f(x) is monotonic when x > y ⇐⇒ f(x) > f(y) a) Show that any monotonic transformation of the utility function (f ◦u) can also represent the same preferences.

Gina is at a toy store and is buying toy T-Rex's (x), which cost på, and toy yaks (y), which cost py. Her income is I. She likes 3 T-Rex's as much as she likes 4 yaks. These toys are perfect substitutes for her.

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Which of the following utility functions represents preferences over consumption bundles (X, Y) that violate the assumption of diminishing MRS? (a) U(X,Y) = 3 ln(X) + 5 ln(Y) (b) U(X,Y) = X² + y² (c) U(X, Y) = XY (d) U(X,Y) = X² + Y² (e) U(X, Y) = X¹Y³

Given the total utility function for a consumer, consuming two goods is U = x+2y + xy+2 the prices of x and y are 4 TL and 6 TL respectively (Px = 4 and Py = 6), and the total budget of the consumer (B) is 130 TL; a) Write the constraint ( budget) function. b) Write the lagrangian function. c) What are the first-order conditions for utility maximization? d) Find the critical values of x, y and \lambda (x*,y*and \lambda *) according to the first order conditions. e) is the second order suficent condition for maximum utility satisfied 2 4 10 [ [:] 8 Write the appropriate commands to create A and d matrixes in R software. A-1. din R language? Given A = 1 3 And d = How can you formulate the equation X =

Emma has a utility functionU(x1, x2, x3) = logx1+ 0.8 logx2+ 0.72 logx3over her incomes x1, x2, x3 in the next three years. This is an example of(A) expected value;(B) quasi-hyperbolic utility function;(C) standard discounted utility;(D) none of the above.  Emma’s preferences can exhibit which of the following behavioral patterns?(A) preference for flexibility;(B) context effects;(C) time inconsistency;(D) intransitivity.

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Anthony seeks to maximize the following utility function u(x, y) = x'/3y2/3 subject to the budget constraint Pæa + PyY = I 1 where pr, Py, x, y, I > 0. a) Find Anthony's utility-maximizing bundle (x*, y*) as a function of pæ, Py, and I. b) Show that y* is decreasing in py and increasing in I (hint: use partial derivatives). c) What share of Anthony's income is spent on x? What share is spent on y? In other words, calculate Pa and Pu. Are these shares a function of prices? Pyy* Note: The above utility function is Cobb-Douglas, and all Cobb-Douglas functions have these share formulas for any values for the exponents. d) What is the impact of a change in pr on Anthony's utility?

Merge the two budget constraints into a single lifetime budget constraint by eliminating savings from the system of two constraints.