OPMT 5701: Term Project Fall 2014 Instructions: This project is due in Lecture on the Wednesday of the last week of classes. You are allowed to work in Teams of 2 or 3. It may be hand written BUT must have a typed cover page and is SINGLE SIDED. Note that a signi…cant amount of the …nal exam will be based on this assignment. The more e¤ort you put into understanding these questions, the better prepared you will be for the …nal exam. 1. Indi¤erence Curves and Budget Constraints. This problem requires you to maximize utility subject to di¤erent budget constraints and to graph your results. The graph MUST be on a full page with properly labeled axis. You will be graded on the quality and detail of your graph. (a) Maximize u = xy subject to B = px x + py y and calculate the values of u; x; y for the following cases: 1. B = 240:00, 2. B = 240:00, 3. B = 169:70, px = 2:00; px = 4:00; px = 2:00; py = 2:00 py = 2:00 py = 2:00 (b) On a single, full page graph, draw the Budget lines for each case. Label the equilibrium values for x and y for each case. (c) On the same graph, add the indi¤erence curves. Getting this part correct is tricky. Be Careful! 2. Willingness to Pay versus Equivalent Compensation Skippy and Myrtle are friends who consume the same goods: yoga classes (X) and Timbits (Y ). Skippy has the utility function u = xy and faces the budget constraint: M = px x + py y. where M is income, and px ; py are prices. Myrtle has the same budget constraint as Skippy but her utility function is v = xy 2 . (a) Use the Lagrange Method to show that Skippy and Myrtles’demand functions for x and y, which will be in the form: Skippy M yrtle M M m x = 3p xs = 2p x x M 2M s m y = 2py y = 3py (b) Suppose M = 120, Py = 1 and Px = 4: What is Skippy and Myrtles’ optimal x; y and utility number? If the price of x was lowered to $2 what would be their x; y and utility numbers? (c) If the price remains as $4, how much additional income would each one require to get the same utility as they did when the price was $2 (d) Draw a graph for Skippy and another for Myrtle. In each graph show the budget constraints, indi¤erence curves and equilibrium values of x and y for both px = 4 and px = 2. Be ACCURATE and NEAT. (minimum 1/2 page for each graph) (e) Suppose that the Yoga Studio o¤ers two options: a drop-fee of $4, or a membership of $30 that lets the member take Yoga classes for $2. What will be the Utility of Skippy and Myrtle if they each buy a membership? Given the options (drop-in or member) which will each one choose? Carefully add the new budget constraint, indi¤erence curve and equilibrium x and y to each girl’s graph. (f) What is the membership fee that would make Skippy indi¤erent between drop at $4 and membership that charges $2 per yoga class? Add this to Skippy’s graph. (g) What is the membership fee that would make Myrtle indi¤erent between drop at $4 and membership that charges $2 per yoga class? Add this to Myrtle’s graph. 1 3. Skippy lives on an island where she produces two goods, x and y, according the the production possibility frontier 400 x2 + y 2 , and she consumes all the goods herself. Skippy also faces and environmental constraint on her total output of both goods. The environmental constraint is given by x + y 28: Her utility function is u = x1=2 y 1=2 (a) Write down the Kuhn Tucker …rst order conditions. (b) Find Skippy’s optimal x and y. Identify which constraints are binding. (c) Graph your results. (d) On the next island lives Sparky who has all the same constraints as Skippy but Sparky’s utility function is u = ln x + 3 ln y. Redo a, b, and c for Sparky 4. The concept of Duality 1=2 (a) Given the utility function u = (q1 q2 ) and a budget constraint Y = p1 q1 + p2 q2 use the lagrange method to …nd the optimal q1 and q2 as functions of the variables p1 ; p2 and Y: i.e. solve 1=2 L = (q1 q2 ) + (Y p1 q1 p2 q2 ) 1. Substitute your solutions for q1 and q2 into the utility function. This gives you the Indirect Utility function1 . Show that the indirect utility function is: u= Y 1=2 1=2 2p1 p2 2. The indirect utility function can be used to derive the Expenditure Function2 , which is Y = f (p1 ; p2 ; u) Re-arrange the indirect utility function to isolate Y and you will have the expenditure function. Take the partial derivatives of the expenditure function with respect to p1 and then p2 . Speci…cally, show that the following is true: 1=2 1=2 @Y p u = 11=2 @p2 p2 @Y p u = 21=2 @p1 p1 These partial derivatives are also known as the Compensated Demand functions for q1 and q2 : This result is known as Shephard’s Lemma, which you will verify in the next part of the problem (b) The previous problem can be re-cast as a minimization problem. Speci…cally, minimize spending subject to a …xed level of utility. To do this, solve the Lagrange problem L = p1 q1 + p2 q2 + u 1=2 (q1 q2 ) for q1 and q2 as functions of constants p1 ; p2 and u: 1. Show that your solutions to the mimization problem are equal to the partial derivatives of the expenditure function above. (i.e. that Shephard’s Lemma is true) 1 Utility depends on goods consumed which, in turn, are determined by the prices and budget of the consumer. Therefore Utility indirectly is a function of money and prices 2 The expenditure function tells us the MINIMUM budget NEEDED to achieve a target level of UTILITY given a known set of prices 2