I have done the first 2 questions
UNIVERSITY OF CALGARY DEPARTMENT OF ECONOMICS ECONOMICS 301 Assignment 1 Wojciech (Victor) Fulmyk Winter 2023 This assignment is due at 11:59pm on Friday, February 17. Submit your assign- ment using the dropbox on D2L. Make sure to submit your assignment as a .pdf file (preferred) or a .jpg/.jpeg file. If you cannot submit a .pdf or .jpg file, try to submit a .png, .tiff, or other image file that is not locked. Do not submit your assignment in a proprietary/locked format such as any Apple format that must be opened on Apple devices only. If you submit your assignment in a locked format it WILL NOT be graded and you WILL get a zero for this assignment. 1. Define what it means for preferences to be: (a) Reflexive (b) Monotonic (c) Convex (d) Strictly Convex (e) Complete (f) Transitive You may use the notation of preference relations (∼,≻,⪰), diagrams, and/or examples to illustrate the concept wherever necessary. 2. In your own words, define utility functions and indifference curves. Then, in one paragraph, explain the relationship between utility functions and indifference curves. 3. Consider the following utility functions: i u(x1, x2) = x 0.3 1 x 0.7 2 ii u(x1, x2) = 2x1 + 8x2 iii u(x1, x2) = min[4x1, 20x2] iv u(x1, x2) = x2 + 4x 0.5 1 Page 1 ECON 301 Assignment 1 Winter 2023 (a) Which of these utility functions represent perfect substitutes? (b) Which of these utility functions represent perfect complements? (c) Which of these utility functions represent quasilinear preferences? (d) A good is essential if there is no way to attain utility greater than what one would attain at the origin without consuming at least some of that good. For each of the above functions, which good is essential? Which good is not essential? 4. Viktor survives on a typical undergraduate diet of pizza and beer: he does not con- sume any other items. Suppose Viktor’s preferences over pizza (x1) and beer (x2) are represented by the utility function u(x1, x2) = 4x 2 1 √ x2 − 17. (a) Apply a positive monotonic transformation to convert Viktor’s utility function into a Cobb Douglas utility function with exponents summing to 1. (b) What percentage of Viktor’s income is Viktor spending on beer? What percentage of Viktor’s income is Viktor spending on pizza? (c) Suppose the price of beer doubles, but the price of pizza and Viktor’s income stay the same. How much beer is Viktor consuming as a percentage of his previous beer consumption? How much pizza is Viktor consuming as a function of his previous pizza consumption? (d) Would Viktor be willing to switch bundle A = (4, 2) for bundle B = (2, 4)? Why or why not? 5. Consider the following utility functions: i u(x, y) = (1/5)xy, with MU1 = (1/5)y and MU2 = (1/5)x ii u(x, y) = 11xy, with MU1 = 11y and MU2 = 11x iii u(x, y) = 9(xy)2, with MU1 = 18xy 2 and MU2 = 18x 2y (a) For each of these utility functions, what is the equation of an indifference curve associated with a utility value of 10? (Hint: provide an equation for y as a function of x). (b) For each of these utility functions, compute the MRS. (c) Diminishing marginal rate of substitution occurs when the MRS decreases (in absolute value) as we increase x1. Note that only strictly convex preferences exhibit this property. For each of the utility functions above, show whether the function exhibits the diminishing marginal rate of substitution property. Page 2 ECON 301 Assignment 1 Winter 2023 (d) Do the above utility functions represent the same preference ordering? Why or why not? 6. Consider your tastes for $5 bills and $10 bills. Suppose that all you care about is how much money you have, but you don’t care whether a particular amount comes in more or fewer bills. Moreover, suppose that you could have partial $10 and $5 bills; for instance, assume that if you own 60% of a $10 bill , the value of that portion is $6. (a) With the number of $5 bills on the horizontal axis and the number of $10 bills on the vertical axis, illustrate 3 indifference curves from your indifference map. (b) What is your marginal rate of substitution of $10 bills for $5 bills? (Hint: how many $10 bills would you be willing to give up to increase your consumption of $5 bills by 1 unit?) (c) What is your marginal rate of substitution of $5 bills for $10 bills? (d) Are averages better than extremes? How does this relate to whether your tastes exhibit diminishing marginal rates of substitution? (e) Are your tastes quasilinear? (f) Are either of the goods on the axes essential? 7. Suppose Peter has $800,000 to spend on a house and “other goods” (denominated in dollars). The price of 1 square foot of housing is $300, and Peter chooses to purchase a house of 2,000 square feet in size. Assume that houses do not differ in quality: their price is solely determined by their size. Also, assume throughout that Peter spends money on housing solely for its consumption value, not as part of his investment strategy, and that Peter has well-behaved, strictly convex preferences. (a) On a graph with “square feet of housing” on the horizontal axis and “other goods” on the vertical axis, illustrate Peter’s budget constraint and Peter’s optimal bundle A. (b) After Peter has bought the house, the price of housing falls to $200 per square foot. Given that Peter can sell his house from bundle A if he wants to, is Peter better or worse off? Illustrate Peter’s new budget constraint on the graph. (c) Assuming Peter can easily buy and sell houses, will Peter now buy a different house? If so, is Peter’s new house smaller or larger than his initial house? (d) Suppose that, instead of the price of housing dropping to $200 as in (b), the price of housing increases to $350 instead. Is Peter now better or worse off? If Peter Page 3 ECON 301 Assignment 1 Winter 2023 can easily buy and sell houses, will Peter now buy a different house? If so, is Peter’s new house smaller or larger than his initial house? 8. Sam’s preferences are represented by u(x1, x2) = min[2x1, 4x 1/2 2 ]. Draw Sam’s indiffer- ence curves for utility levels u1 = 1, u2 = 4, and u3 = 16. 9. Suppose Alex’s preferences are represented by u(x1, x2) = x1x 3 2. The marginal utilities for this utility function are MU1 = x 3 2 and MU2 = 3x1x 2 2. (a) Show that Alex’s utility function belongs to a class of functions that are known to be well-behaved and strictly convex. (b) Find the MRS. [Note: find the MRS for the original utility function, not some monotonic transformation of it.] (c) Write down the tangency condition needed to find an optimal consumption bundle for well-behaved preferences. (d) Write down Alex’s budget constraint. (e) Using the equations you wrote in (c) and (d), find Alex’s ordinary demand func- tions for goods 1 and 2 (in other words, find out how much Alex will consume of good 1 as a function of p1, p2, and m, and how much Alex will consume of good 2 as a function of p1, p2, and m). (f) Suppose that m = 100, p1 = 2 and p2 = 3. How much of good 1 will Alex choose to consume? How much of good 2 will Alex choose to consume? 10. Picky eater Becky eats only waffles (x) and cereal (y) for breakfast. Her preferences are described by u(x, y) = 16x+ 24y. (a) If her income is $48 and prices are px = py = $6, how much of each food will she consume? (b) If the price of cereal increases to $8, calculate Becky’s compensating and equiva- lent variations. Comment on your results. Use a graph to show all your results. Page 4