Problem (1) has one very important similarity to the initial problem: the utility function in the new problem is the square of the utility function in the old problem. The value of going to war will depend on whether Peru wins or loses. Therefore, one Problem Set 3: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Cobb-Douglas Utility Functions) 1.1: Optimal fraction of income spent on (berries) x 2: b a+b. Marginal analysis and consumer choice. Step 1: Simplify the problem (further) by expressing x 2 from the budget con-straint and plug it into the utility function This is the rst necessary step to solve the consumer problem. Utility maximization: equalizing marginal utility per dollar. 2. More on this in a while. The hourly wage is w, and assume that the price of each unit of consumption is $1. The value of not going to war is 0. 10. utility function does the P equilibrium exist? POLI 2500 Expected Utility Functions Expected Utility Function Practice Problems - Solutions Scenario 1: Peru must decide whether going to war with Chile is worth more than not going to war with Chile. However, if we restrict ourselves Solution No. POLI 2500 Expected Utility Functions Expected Utility Function Practice Problems - Solutions Scenario 1: Peru must decide whether going to war with Chile is worth more than not going to war with Chile. u(x) ≥u Notice that the utility function ( ) which is a standard Cobb-Douglas, is a monotonic transfor-mation of the utility function ( ) with ( )=exp( ( )) Hence, the two utility functions represent the same preferences, and thus the two utility maximization problems are identical. Max U H,P (H,P) = P + 4H s.t. Second, observe that under strict concavity of f(z) assumption, the only solution to the rms maximization problem de ned in (a), is ' = 0. Solution (a) The expenditure function is the minimal expenditure needed to attain a target utility level. (b) Pratt's formula for the relative risk premium (p. 18, eq. 3P + 6H = 300 f. Solve for the values of P and H that maximizes Petra's utility. (b) of the problem we get: Therefore, the consumer will buy 12 units of good X. Philip's utility function is: U = 2q_{1}^{0.5}+q_{2} What are his demand functions for the two goods, assuming he is not at a corner solution. Suppose the utility function is of the form U(x,y)=− e −γx y for some γ>0, and assume that the interest rate r t =r is constant. Problem 2, where you are actually required to check for corner solutions. Answer Consider . Katie likes to paint and sit in the sun. H P 50 0 3.1 Solution Method 1: Graphical Approach The agent wishes to choose a point in her budget set to maximise her utility. (The problem only asks for berries.) Thus,at . The expenditure function is found as a solution to the following problem: 2 12 1 2 1 (, ,) ..(, )o ii i ep p u Min pxstux x u = =≥∑ If we choose as the given utility level in the expenditure problem the solution to the utility maximization problem for given income m we have in general : vp p m vp p ep p u u(, ,) (, ,( , ,))12 12 12== 1. u(x) ≥u L If preferences are strictly monotonic, then the constraint will be satis ed with equality L Denote the solution to the expenditure minimization problem as: xh(p;u) =argmin x p ⋅x s.t. (b) of the problem we get: Therefore, the consumer will buy 12 units of good X. 2. Assuming that The utility function is quasilinear, which may give either an interior. Provide an example of a utility function that leads to at least one good being an inferior good. Question 4 (a) Deflne the expenditure function (either mathematically or in words). 10. View Answer Consider an individual with income I and utility function where X and Y are two products. The price of a paint brush is $1 and the price of a straw hat is $5. Figure Figure7 7 illustrates with a solid line the function which, based on the Complex Number model, relates utility (Y-axis) and QALYs (X-axis) when time is set to 1 year. (b) Suppose p1 rises. (Quiz: Why?) Consumer Theory - Indirect Utility Function Indirect Utility Function - V(P,I) ≡ Max U(x) st P⋅x ≤ I and x ≥ 0; optimized value function (i.e., solve the maximization problem, then plug solution back into U(x) to get V(P,I)); lists the solutions to the maximization problem for the various values of the parameters P and I Problem Set - Chapter 3 & 4 Solutions 1. Notice how neither fraction depends on income m or the prices of . Problem (1) has one very important similarity to the initial problem: the utility function in the new problem is the square of the utility function in the old problem. Provide a general proof that all goods cannot be inferior goods. Channel donations are much appreciated:https://www.paypal.com/cgi-bin/webscr?cmd=_do. Marginal utility and total utility. This rule, combined with the budget constraint, give us a two-step procedure for finding the solution to the utility maximization problem. 2 This utility function has indifference curves that exhibit diminishing MRS which goes to zero and infinity as they touch the x and y-axis. However, if we restrict ourselves If I keep my . Solve Katie's utility-maximization problem using a . Optimal fraction of income spent on (nuts) x 1: a a+b. Show all the probabilities and outcome values. (Quiz: Why?) Graph a typical indifference curve for the following utility functions and determine whether they obey the assumption of diminishing MRS: a. U(x, y) = 3x + y Since the indifference curves are not bowed towards the origin, they do not obey the Solutions to selected problems from homework 1. Expenditure Function L The expenditure function is the minimum amount of expenditure necessary to achieve a given utility level u at prices p: e(p;u) =min x p ⋅x s.t. For the utility-of-consequences function u(w) = w1/2 we have u0(w) = 1 2 The value of going to war will depend on whether Peru wins or loses. This violates monotonicity. † There is an interior solution to the agent's maximisation problem. Solution No. Therefore, one Show your work. Show your work. 12. If he does not consume one of these goods, there will be a corner solution. However, no argument can be provided to exclude corner solutions to this utility maximization problem. More on this in a while. Assuming that utility is the one that is the highest, where there is a corner solution and Petra only eats hamburgers. This rule, combined with the budget constraint, give us a two-step procedure for finding the solution to the utility maximization problem. First, in order to solve the problem, we need more information about the MRS. As it turns out, every utility function has its own MRS, which can easily be found using calculus. That is, the agent 1. Visualizing marginal utility MU and total utility TU functions. This video gives an example of a utility maximization problem with a corner solution. (b) Intuitively explain why the expenditure function is concave in prices. 3.1 Solution Method 1: Graphical Approach The agent wishes to choose a point in her budget set to maximise her utility. e. Petra is a utility maximizer. Problem Set - Chapter 3 & 4 Solutions 1. the same demand functions for goods x and y as we derived above by intuitive means. 1. If he does not consume one of these goods, there will be a corner solution. (1.15) in the book) is Π(˜ˆ z) = 1 2 σ2 R(w) where σ2 is the variance of the proportional risk ˜z, and R(w) the coefficient of relative risk aversion. Draw a decision tree for this simple decision problem. 1 Let U(x) denote the patient's utility function, wheredie (0.3) x is the number of months to live. x = w, x ≥ 0 It is easy to show that V is non-increasing in p, strictly increasing in w and that the set {(p,w) : V (p,w) ≤ u} Katie has $50 to spend on paint brushes and straw hats. What we do is notice that once we are thinking of choosing a value for x 1 this automatically implies the value that x 1 must have if the budget has to be exhausted. Problem 3: Taxes and the Labor Market (20 points) Suppose a worker has preferences over consumption and leisure that can be repre-sented by the following utility function: U = ln(c) + ln(l) There are 16 hours per day available for leisure (l) and work (L). As a result, any solution to the tangency conditions constitute a maximum. Solution (a) The expenditure function is the minimal expenditure needed to attain a target utility level. 3This procedure only works if there is an interior solution|that is, the individual consumes some xand some y. 4. This violates monotonicity. Optimal fraction of income spent on (nuts) x 1: a a+b. This video shows how to maximize utility subject to a budget constraint. Consumer Theory - Indirect Utility Function Indirect Utility Function - V(P,I) ≡ Max U(x) st P⋅x ≤ I and x ≥ 0; optimized value function (i.e., solve the maximization problem, then plug solution back into U(x) to get V(P,I)); lists the solutions to the maximization problem for the various values of the parameters P and I Problem 2, where you are actually required to check for corner solutions. 3This procedure only works if there is an interior solution|that is, the individual consumes some xand some y. First, in order to solve the problem, we need more information about the MRS. As it turns out, every utility function has its own MRS, which can easily be found using calculus. This video gives an example of a utility maximization problem with a corner solution. 4The budget constraint holds with equality because the utility function is strictly increasing in both arguments (Quiz: Why?). You know that there will be an interior solution if each marginal utility is a function of the quantity of the good and thus the rst order conditions will be . Expenditure Function L The expenditure function is the minimum amount of expenditure necessary to achieve a given utility level u at prices p: e(p;u) =min x p ⋅x s.t. Draw a decision tree for this simple decision problem. Of course, if g is the exponential utility function g(x)=−e −γx, then this form of the utility is equivalent to the separable form considered here. Give the description of the equilibrium in this case. Question 1 Let ube a utility function which generates demand function x(p;w) and indirect utility function v(p;w). This utility function has indifference curves that exhibit diminishing MRS which goes to zero and infinity as they touch the x and y-axis. Solution: op U(3) no op live (0.7) U(12) U(0) 2. 4The budget constraint holds with equality because the utility function is strictly increasing in both arguments (Quiz: Why?). † There is an interior solution to the agent's maximisation problem. Sorememberthedistinction: - Directutility: utilityfromconsumptionof(x. Problems and Applications 29 Kevin has a utility function U = wi, where Was his wealth in millions of dollars and is the utility he obtains from that wealth, in the final stage of a game show, the host offers Kevin a choice between (A) $9 million for sure, or (b) a gamble that pays $1 million with probability and $16 million with probability 0.6. Practice: Total Utility and Marginal Utility. Since the utility function in the old problem was always positive (for x>0 and y>0),it follows that the utility function in the new problem is an increasing function of the . (The problem only asks for berries.) If the utility function u is de ned by u(x) = F(u(x)) what are the demand functions Her utility function is U(P, S ) = 3PS + 6P, where P is the number of paint brushes and S is the number of straw hats. Notice how neither fraction depends on income m or the prices of . If I keep my . That is, the agent (b) Intuitively explain why the expenditure function is concave in prices. u(x) ≥u The function takes the form of a hyperbola in which the vertex is located at the co-ordinate (0.7071,0). (b) Suppose p1 rises. Question 4 (a) Deflne the expenditure function (either mathematically or in words). Solution. Proposition 3.3 Exponential utility. We call V()the "Indirect Utility Function." This is the value of maximized utility undergivenpricesandincome. Because (at the utility maximizing solution to this problem), x and y are alreadyoptimized,aninfinitesimalchangein Idoesnotalterthesechoices. You know that there will be an interior solution if each marginal utility is a function of the quantity of the good and thus the rst order conditions will be . Let U(x) denote the patient's utility function, wheredie (0.3) x is the number of months to live. As such, a. This is the currently selected item. View Answer The utility function is quasilinear, which may give either an interior. Show all the probabilities and outcome values. Write down the full optimization problem with the objective function and the constraint. Let F: R !R be a strictly increasing function. Solution: op U(3) no op live (0.7) U(12) U(0) 2. Problems and Applications 29 Kevin has a utility function U = wi, where Was his wealth in millions of dollars and is the utility he obtains from that wealth, in the final stage of a game show, the host offers Kevin a choice between (A) $9 million for sure, or (b) a gamble that pays $1 million with probability and $16 million with probability 0.6. Marginal utility free response example. However, no argument can be provided to exclude corner solutions to this utility maximization problem. Since the utility function in the old problem was always positive (for x>0 and y>0),it follows that the utility function in the new problem is an increasing function of the . The expenditure function is found as a solution to the following problem: 2 12 1 2 1 (, ,) ..(, )o ii i ep p u Min pxstux x u = =≥∑ If we choose as the given utility level in the expenditure problem the solution to the utility maximization problem for given income m we have in general : vp p m vp p ep p u u(, ,) (, ,( , ,))12 12 12== 2 we can see that solving the Lagrangian problem gives us the same solution, i.e. Graph a typical indifference curve for the following utility functions and determine whether they obey the assumption of diminishing MRS: a. U(x, y) = 3x + y Since the indifference curves are not bowed towards the origin, they do not obey the Consider an individual with income I and utility function where X and Y are two products. Philip's utility function is: U = 2q_{1}^{0.5}+q_{2} What are his demand functions for the two goods, assuming he is not at a corner solution. The value of not going to war is 0. As a result, any solution to the tangency conditions constitute a maximum. u(x) ≥u L If preferences are strictly monotonic, then the constraint will be satis ed with equality L Denote the solution to the expenditure minimization problem as: xh(p;u) =argmin x p ⋅x s.t. Problem Set 3: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Cobb-Douglas Utility Functions) 1.1: Optimal fraction of income spent on (berries) x 2: b a+b. First, make standard assumption f(0) = 0 and without any loss of generality set p= 1.
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