Question one.

Chepa's utility function is given by U (x, y) = ln x + 4 ln y. Assume that Chepa has endowments (10, 10) and that Py = 10 throughout the problem.

I need help with questions G) and H) only please.

a) (Note: this part of the question is intended to reduce your workload for parts (b) to (e). If you prefer not to work with a general demand function but calculate the demand separately for each of the cases, you can do part after part (f).) Given Py = 10, solve for the optimal bundle for Chepa as a function of Px and money income M.

(b) Suppose Px = 10. Calculate Chepa's endowment income. How much of each good will Chepa consume? Call this bundle A.

(c) Now suppose Px drops to 5. Calculate Chepa's new endowment income. Find Chepa's new consumption bundle. Call this bundle B.

(d) What is the income required such that Chepa can just afford bundle A under the new prices? Find Chepa's consumption bundle if she has this income and is facing the new prices. Call this bundle H1.

(e) Find Chepa's consumption bundle if she has her original endowment income and is facing the new prices. Call this bundle H2.

(f) Using your answers from (b) to (e), calculate the substitution effect, ordinary income effect, endowment income effect and the price effect associated with the change in Px.

(g) Express Chepa's endowment income as a function of Px. Using this expression and your answer to (a), find the range of values of Px such that Chepa will be a net seller of good x.

(h) This part of the question is to investigate Chepa's welfare under different prices. We will do it step by step.

(i) By substituting out the M with the expression of Chepa's endowment income (see part (g)), obtain Chepa's gross demands as functions of Px.

(ii) Plug your answer to (i) into Chepa's utility function (that is, replacing the general x and y in her utility function by the optimal x and y given Px) to obtain an expression of the maximal utility achieved by Chepa as a function of Px.

(iii) Find the value of Px that gives Chepa the lowest utility. (Hint: Take the answer to (ii), differentiate it with respect to Px, set the derivative to zero and solve for Px in that equation. It is a good practice to check the second order condition to make sure you are getting a minimum ? but if you feel uninterested or that this is too hard, you can trust that I am giving you a "nicely behaved" minimisation problem and skip checking the SOC.)

(iv) Explain the economic meaning of your result in (iii).

Question 2.

1. Budget sets. Say we have 2 goods, and that the absolute price of good 1 is 10, and of good 2 is 20 (say P = (P1;P2) = (10;20), and income m is 100. a. Dene the consumption set, and then plot the budget set at this P: b. Show the budget set gets "smaller" under set inclusion assuming either component of P increases, or m decreases. c. Show that the imposition of positive sale tax of good 1 (not good 2) has the same impact as a rise in the P1: d. Say the price of good 1 increases from 10 to 20 whenever more than 1 unit of good 1 is purchases. Draw the new budget set, and show its convex. Is it strictly convex? Explain.

2. Preferences. Let denote the consumers preference relation on C =Rn +: Answer the following: a. Sayis reexive, complete, but not transitive. Show that the consumers preferences could "cycle" (i.e., if for j = 1;2;3;:::;n; and consumption bundles xn we could have xj xj1 and x0 xn: b. Say is reexive, complete, and transitive. (i) Can indi?erence curves "cross"? (ii) what additional assumption rules this out. Show also that this assumption indeed does rule out crossing indi?erence curves. (iii) Show the consumer cannot "cycle" (i.a., part (a) cannot happen). (iv) Show that under "strictly monotonic" preferences, indi?erence curves cannot be "thick".

3. Convex Preferences and optimal solutions. Let denote the consumers preference relation on C =Rn +: We say a preference relation is convex (re-spectively, strictly convex) if for any two bundles x and y such that x~y (i.e., x and y indi?erent), then for any 2 [0;1] (respectively, 2 (0;1)), and z = x+(1)y; z x~y (respectively, z x~y): We say a preference relation is continuous if the two sets: weakly less preferred: WLP(x) = fy 2 Cjx y;x 2 Cg and weakly preferred: WP(x) = fy 2 Cjy x; x 2 Cg are "closed" (i.e., contain their boundaries. See discussion in class. Answer the following questions. Let the consumption set be C =Rn +: (a) Show if is convex, WP(x) convex. (b) Show if is strictly convex, WP(x) is strictly convex. (c) Is WLP(x) convex?

Consider a consumer facing a budget set B(p;m) = fx 2 Cjp x mgfor p >> 0: Dene the best choice set X(p:m) = fx 2 Cjx x for allx 2 B(p;m)g (d) Show if is convex, X(p;m) might have many elements (i.e., many optimal choices. (e) Show if is strictly convex, X(p;m) is a unique element.

4. Say we have a utility function u(x) = x 1 x1 2 for 2 (0;1):(a) Construct the Marginal rate of substitution. (b) Discuss how the Marginal rate of substitute in related to the slope of an indi?erence curve at a point x >> 0 (i.e., each component of x is strictly positive).

5. Answer the following: (i) Why is a utility function considered to be an "ordinal" concept? (ii) Show that if u(x) represents a consumers preference relation, any ^ u(x) = 10u(x) represents that same utility function. (iii) In question 4, of show that if u(x1;x2)=x 1 x(1) 2 for 2 (0;1); theMRS between the two goods for ^ u(x) is not impacted by this strictly increasing transformation. (iv) Actually, show if h(y) : R ! R is a strictly increasing continuously di?erentiable transformation, ^ u(x) = h(u(x)) represents that same preferences as u(x). (v) Show in part (iv) that the MRS at the same for both ^ u(x) and u(x): (vi) Show in part (v) that the MRS is the same if (vii) let h(y) = lny: Show the MRS is the same for both ^ u(x) and u(x) if u(x1;x2)=x 1 x(1) 2 for 2 (0;1):

Section B.

In reporting on real GDP growth in the second quarter of 2016, an article on Reuters news noted that: "U.S. economic growth unexpectedly remained tepid in the second quarter as inventories fell" and also that the "inventory drawdown was almost across the board." Source: Lucia Matikani, "Inventory Reduction Curbs U.S. Economic Growth; Rebound Expected," reuters.com, July 20, 2016. If companies are drawing down inventories is aggregate expenditure likely to have been larger or smaller than GDP? O A. Smaller than GDP, because more is being bought than produced. O B. Larger than GDP, because less is being bought than produced. O C. Larger than GDP, because more is being bought than produced. O D. Smaller than GDP, because less is being bought than produced. The chief economist at UniCredit Research is quoted in the article as stating that: "The U.S. economy just went through a meaningful inventory correction cycle." What would an "inventory correction cycle" be? O A. A situation where companies deplete low levels of inventory to earn economic profits. O B. The process of companies increasing production levels to create more inventory. O C. A situation where inventory levels are held constant for stability. O D. The process of companies decreasing or increasing actual inventories to match planned inventories. The article states that: "Though the inventory drawdown weighed on GDP growth, that is likely to provide a boost to output in the coming quarters." Why would an inventory drawdown boost output in the coming quarters? O A. Once the firms are able to sell off their excess inventories, consumption will be lower. O B. Once inventories reach desired levels, firms will increase production to match expected planned aggregate expenditures. O C. This would indicate that the quantity of output demanded is greater than the quantity of output supplied, and production levels will rise. O D. An inventory drawdown would reduce storage costs, thus giving firms an incentive to produce more output.Question 1: Louis the retired Canadian lives on a fixed budget and consumes only two goods: toques (T) and maple syrup (M). Suppose Louis' monthly budget is 100 and the price of the two goods are (PT, PM) = (4,2). (a) Make a properly labeled diagram illustrating Louis' budget constraint with T on the hori- zontal axis and M on the vertical axis. Indicate the area corresponding to the set of bundles (M, T) that Louis can afford (b) What is the maximum T that Louis can afford? What about the maximum M? (c) What is the slope of the budget line, and what is the economic interpretation of it? (d) Suppose the Canadian government decides to ration maple syrup, limiting each person to a maximum of 40 units of maple syrup per month. Draw a new diagram showing Louis' new budget constraint and indicate the area corresponding to the set of bundles that Louis can afford now. Question 2: Annie and Marty are both single parents with full-time jobs. Consequently, both value child care services (5) and children's clothing (C) but may have different preferences for the two goods. The government decides it wants to make life easier for single parents and is considering providing single parents with monthly child care vouchers or, simply, providing single parents with a cash payment every month. Suppose Annie and Marty's incomes are both 400, the price per unit of child care is 10 per unit and the price per unit of clothing is 20. (a) Draw Annie's budget constraint assuming the government decides to offer a $100 voucher for child care. (b) Draw Marty's budget constraint assuming the government decides to offer a $100 cash payment to single parents. (c) Draw an indifference curve representing Annie's preferences that is consistent with Annie strictly preferring the cash payment to the voucher. (d) Draw an indifference curve representing Marty's preferences that is consistent with Marty being indifferent between the cash payment and the voucher. Question 3: After winning the lottery, Australian Artie decides to retire, stay home, and watch sports on television all day. Artie likes both Rugby, R, and Aussie Rules Football, F and views them as imperfect substitutes. In particular, Artie's preferences can be represented by the follow- ing utility function: U = R04Foe. Since Artie is rich, the only cost of watching F is the opportunity cost of not watching R: one hour of R necessarily means one hour less of F. In other words, even though Artie is rich, there is still a price on his time. 1-2 of 2(a) What is Artie's marginal rate of substitution between R and F, MRSay? Interpret it. (b) What is the price ratio, PR/PF? Interpret it. (c) Suppose Artie chooses some bundle, (R, F). If this bundle maximizes Artie's utility, what relationship must hold between R and F? (d) Define ME. What is the interpretation of this? If Artie is maximizing his utility, what must FR MUs be equal to? (e) Suppose there are 24 hours in a day. What bundle (R, F) will Artie consume? [Now you should find actual numbers.] Question 4: Picky Pete drags Indifferent lan out for coffee at an upscale Italian cafe which sells only espresso (E) and biscotti (B). Picky Pete simply must have one biscotti for every espresso consumed, whereas Indifferent lan views espresso and biscotti as perfect substitutes. One of their utility functions is given by U = min [E, B] and the other's by U = E + B. (a) Given the description, which utility function corresponds to which person? (b) Suppose the prices of the goods are (PEPs) = (2,1) and Pete has allocated $12 for the outing while lan brought only $4. Solve for each person's utility-maximizing bundle of espresso and biscotti. (c) Draw a fully labeled diagram illustrating Pete's budget constraint, his optimal bundle, and an indifference curve representative of his utility at his optimal bundle. Indicate how much utility Pete obtains. (d) Do the same for Ian. Question 5: Energetic Erin consumes a worrisome amount of caffeinated beverages, particularly soda (S) and coffee (C). A utility function representing her preferences is given by U = 450503. Erin's Income is 120 and the prices of the two drinks are given by (ps. pc) = (4,2). (a) Solve for Erin's utility maximizing bundle of beverages and calculate how much utility she obtains from it. (b) Suppose Erin moves to Berkeley where there is a soda tax and, consequently, the price of soda is 6. Solve for Erin's new utility maximizing bundle of beverages and calculate how much utility she obtains now. What is the "total effect" of the price change on Erin's consumption of soda? (c) Draw a diagram showing Erin's budget constraint before and after the move, and repre- sentative indifference curves of her choices before and after the move. (d) The total effect you found in part "b" can be broken down into a "substitution effect" and an "income effect". What is the sign of: (i) the total effect, (ii) the substitution effect, and (iii) the income effect? Explain why for (ii) and (uil). 1-2 of 2