Econ 3030 Practice Homework Questions

Suppose a consumer is considering 3 goods. The price of good 1 is -1, the price of good 2 is +1, and the price of good 3 is +2. It is physically possible to consume any commodity bundle with nonnegative amounts of each good. Suppose the consumer has an income of 10. Could she a§ord to consume some commodity bundles that include 5 units of good 1 and 6 units of good 2: if yes, give an example; if no, show why.

Homework 0.

While abroad for a conference, Karel spent all of the local currency he had in order to buy 5 plates of spaghetti and 6 negronis. Let's not worry, for now, about what that suggests about his preferences. Spaghetti costs 8 units of the local currency per plate and he had 82 units of currency. Let s denote the number of plates of spaghetti and n denote the number of negronis. Write down an equation describing the set of all bundles of these two commodities that he could have just afforded with the local currency he had.

Homework 0.

An assistant vice president in charge of production for a small computer networking firm is asked to calculate the cost of fulflling an order of 170 customized network routers. The production function is q = min{x,y} where x and y are the amounts of two input factors used. The price of x is $18 and the price of y is $10 per unit. She thinks it will cost $1,700. Show whether she is correct in her cost estimate.

Homework 8.

The cost function c (w, r, y) of a firm gives the minimum cost of producing y units of output using labor input, L, and capital input, K, when w is the wage (i.e. the price of labor input) and r is the interest rate (i.e. the price of capital input). Derive the cost functions for the technologies modelled by the following production functions: (a) f (L, K) = min{2L, 3K} (b) f(L, K)=2L+3K (c) f (L, K) = max{2L, 3K}

Homework 8.

Consider two goods with positive prices. The price of one good is reduced, while income and other prices remain constant. Show what must happen to the size of the budget set as a result and why.

Homework 0.

Marco has well-behaved preferences and currently has a bundle with positive amounts of two goods. Let x1 and x2 denote goods 1 and 2, respectively. Thinking in terms of x1 on the horizontal axis, the absolute value of Marco's marginal rate of substitution between the two goods (MRS) at his current consumption bundle is greater than 3. Using this information and the theory we developed so far, can you predict Marco's reaction to the following propositions? (a) "I give you some of x1 and you give me 3 units of x2 for each unit of x1." (b) "You give me some of x1 and I give you 3 units of x2 for each unit of x1."

Homework 1.

Panle is an excellent statistician and is very precise. In fact, she knows that one of her indifference curves is precisely described by the following equation x2 = 20 - 4*(square root) of x1. If Panle is choosing the bundle (x1, x2) = (4, 12), what is her marginal rate of substitution?

Homework 1.

Professor Goodheart gives 3 prelim exams. He drops the lowest score and gives each student her average score on the other two exams. Polly Sigh is taking his course and has a 60 on her first exam. Let x2 denote her score on the second exam and x3 denote her score on the third exam. Draw her indifference curve for scores on the second and third exams with x2 represented by the horizontal axis and x3 represented by the vertical axis. The indifference curve goes through (x2, x3) = (50, 70).

Homework 1.

Tim consumes only apples and bananas. He always prefers more apples to fewer, but he gets tired of bananas. If he consumes fewer than 29 bananas per week, he thinks that 1 banana is a perfect substitute for 1 apple. But you would have to pay him 1 apple for each banana beyond 29 that he consumes. His indifference curve that contains the consumption bundle with 30 apples and 39 bananas also contains the bundle with 21 bananas and how many apples?

Homework 1.

A consumer has a utility function of the form U(x,y) = x^a + y^b, where both a and b are nonnegative. What additional restrictions on the values of the parameters a and b are imposed by each of the following assumptions? (note: some of the following notions we mentioned in passing during lectures, e.g. homotheticity, normal goods, but the textbook provides necessary additional information) (a) Preferences are homothetic. (b) Preferences are homothetic and convex. (c) Goods x and y are perfect substitutes. (d) Preferences are quasi-linear and convex, and x is a normal good.

Homework 2.

A consumer has preferences represented by the utility function U(x1,x2)=10(x2(1) +2*x1*x2 +x2(2))50 For this consumer, are goods 1 and 2 perfect substitutes or perfect complements or neither. Show why.

Homework 2.

Alice strictly prefers consumption bundle A to consumption bundle B and weakly prefers bundle B to bundle A. Can these preferences be represented by a utility function? Why or why not?

Homework 2.

Bob's preferences are represented by the utility function U (x, y) = x/7 if y > 0 and U (x, y) = 0 if y = 0. Draw a couple of his indi§erence curves, describing precisely your procedure, showing whether or not he 1. prefers more of each good to less 2. has quasi-linear preferences 3. has a bliss point.

Homework 2.

Max consumes two goods, x and y. His utility function is U (x, y) = max{x; y}. Show whether for Max x and y are perfect substitutes or perfect complements or neither.

Homework 2.

Suppose the utility function U(x,y) = y + x^2 represent a person's preferences. Are her preferences "well behaved"? Show precisely why.

Homework 2.

(This question requires a bit of ingenuity as it imagines something beyond what we usually discuss) Suppose the demand for a good is estimated by econometricians to be precisely p = 60 - 2q (it's called "inverse" demand since itís written as if prices are functions of quantity demanded). Now suppose that the number of consumers doubles for each consumer in the market, another consumer with an identical demand function appears. Try to model this with a graph and answer the following: does the demand curve shift to the right in a parallel way, doubling demand at every price, to model this?

Homework 3.

Joseph's utility function is given by U(J) = xA + 2*xB , where xA denotes his consumption of apples and xB his consumption of bananas. Clara's utility function is given by U(C) = 3*xA + 2*xB. Joseph and Clara shop at the same grocery store. (a) When the store observes that Joseph leaves with some apples, can they deduce that Clara also buys some apples? (b) When the store observes that Joseph leaves with some bananas, can they deduce that Clara also buys some bananas? (c) The store is interested in setting prices of apples and bananas such that both consumers buy strictly positive amounts of both goods. Can you advise the store on what those prices might be?

Homework 3.

Prudence was maximizing her utility subject to her budget constraint. Then prices changed. After the price change she ended up better off (remember though that she still remained the same Prudence, maximizing her utility subject to her budget constraint). Based on this, can we deduce anything about how the cost of her new bundle at old prices compares to the cost of the old bundle (at old prices)? Present a precise argument.

Homework 3.

In a crowded city far away, the authorities decided that rents were too high. The supply function of rental apartments was given by q = 15 + 3p and the demand function was given by q = 237 - 3p, where p is the rent (in $100s to make it realistic). The authorities made it illegal to rent an apartment at more than p = 30. To avoid a housing shortage, the authorities agreed to pay landlords enough of a subsidy to make supply equal to demand. How much would the subsidy per apartment have to be to eliminate excess demand at the ceiling price?

Homework 4.

Suppose after carefully constructing demand and supply functions, we derived that the quan- tity g of grapefruits demanded at price p is given by g = 30 - 3p and the quantity supplied by g = 6p. State government has been imposing a quantity tax at rate t, which it collects from buyers, and this rate t changes from year to year without any obvious logic behind the particular rate chosen in a particular year. A local grapefruit enthusiast is concerned that in some year the government may choose a tax rate that will actually completely shut down the grapefruit market. Is this possible? That is, what is the smallest tax rate that will result in no grapefruits being bought or sold?

Homework 4.

Suppose the following model of a small local market for burritos works perfectly in predicting the equilibrium: a demand curve, which is a downward-sloping straight line, crosses at one point a supply curve, which is an upward-sloping straight line; the absolute value of the slope of the demand curve is greater than the absolute value of the slope of the supply curve. Concerned about rising costs of public health, local legislature introduces a tax where burrito vendors must pay $2 per burrito sold. A local pro-burrito activist, however, claims that as a result the price paid by consumers will rise by at least $1 and possibly more. Show whether this claim is consistent with what the model predicts.

Homework 4.

Amaranda and Bartolo consume only two goods, X and Y . They can trade only with each other and there is no production. The total endowment of good X equals the total endowment of good Y. Amaranda's utility function is UA(x^A,y^A) = min{x^A,y^A} and Bartolo's utility function is UB(x^B,y^B) = max{xB,yB}. In an Edgeworth box for Amaranda and Bartolo, what would be the set of all Pareto optimal allocations?

Homework 5.

Stating that a contract results in a Pareto efficient allocation implies which of the following (explain each one): 1. aggregate profits are maximized. 2. there is no way to make anyone better off. 3. there is some way to make everyone better off. 4. there is no way to make someone better off without 5. making someone else worse off. there is no way to 6. make everyone worse o§ without making someone better off.

Homework 5.

Suppose the initial endowment is on the contract curve. Does there exist a competitive equilibrium in which no trade takes place? If yes, show what it is, if no, explain why.

Homework 5.

Suppose two people have identical homothetic preferences and their indi§erence curves have a diminishing marginal rate of substitution. In an Edgeworth box, what would the contact curve of allocations between them look like? Explain why.

Homework 5.

A firm has access to a technology that uses two variable factors of production. Show that if the technology is described by a production function f(x1, x2) = (2*x1 + 4*x2)^(1/2), then the technical rate of substitution between x1 and x2 that the firm has to work with is independent of the amount or either x1 or x2 that the firm chooses to employ.

Homework 6.

For each of the following production functions show whether it exhibits constant returns to scale. In each case y is output and K and L are inputs. (a) y = K^(1/2)L^(1/3) (b) y = 3(K^(1/2)L^(1/2) (c) y = K^(1/2) + L^(1/2) (d) y=2K+3L

Homework 6.

On separate axes, sketch two typical production isoquants for each of the following production functions. (a) f(x,y)=min{2x,x+y} (b) f(x,y)=xy (c) f(x,y)=x+min{x,y} (d) f(x,y)=x+y^(1/2)

Homework 6.

Suppose a firm "moves" from one point on a production isoquant to another point on the same isoquant. For each of the following, very brieáy explain why it has to happen with certainty or why it does not have to happen: 1. A change in the level of output 2. A change in the ratio in which the inputs are combined 3. A change in the marginal products of the inputs 4. A change in the rate of technical substitution 5. A change in profitability

Homework 6.

Suppose that the production function is f(x1;x2) = (xa1 + xa2)^b, where a and b are positive constants. For what values of a and b is there a diminishing technical rate of substitution?

Homework 6.

A competitive firm produces output using three fixed input factors and one variable factor. The firms short-run production function is q = 305x - 2x^2, where x is the amount of variable factor used. The price of the output is $2 per unit and the price of the variable factor is $10 per unit. In the short run, how many units of x should the firm use to maximize its profit?

Homework 7.

Jiffy-Pol Consultants is paid $1,000,000 for each percentage of the vote that Senator Sleaze receives in the upcoming election. Sleazeís share of the vote is determined by the number of slanderous campaign ads run by Jiffy-Pol according to the function S = 100N/(N+1) , where N is the number of ads. If each ad costs $10,000, how many ads should Jiffy-Pol buy in order to maximize its profits?

Homework 7.

Suppose a small competitive firm operates a technology that the firms owner knows from experience to work as follows: "Weekly output is the square root of the minimum of the number of units of capital and the number of units of labor employed that week." Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely. (a) Write down a formula that describes the marginal product of labor in the short run as a function of the amount of labor used (be careful at the boundaries, i.e. think about the "corner" cases). (b) If the wage is w=$1 and the price of output is p=$4, how much labor will the firm hire in the short run? c) What if w=$1 and p=$10? (d) Derive an equation for the firms short-run demand for labor as a function of w and p.

Homework 7.

Suppose you run a firm in a competitive industry and would like to learn about what your competition is doing. You know that one of your competitors produces a single output using several inputs. The price of output rises by $4 per unit. The price of one of the inputs increases by $2 and the quantity of this input that the competitor uses increases by 8 units. The prices of all other inputs stay unchanged. What can you deduce about the output of your competitor? Will it stay the same/increase/decrease/by how much? (Hint: use the weak axiom of proÖt maximization)

Homework 7.

A competitive firm has a long-run total cost function c(y) = 3y^2 + 675 for y > 0 and c(0) = 0. What equation(s) would describe its long-run supply function?

Homework 8.

A competitive firm has the short-run cost function c(y) = y^3 + 2y^2 + 5y + 6. (a) Write down equations for: The firms average variable cost function The firms marginal cost function (b) At what level of output is average variable cost minimized? (c) Graph the AVC and MC curves, being careful to label the key points on the graph with the numbers specifying the exact quantities and/or dollar amounts at these points.

Homework 8.

A firm with the production function f(x1; x2; x3; x4) = min{x1; x2; x3; x4} faces input prices w1 =$1,w2 =$5, w3 =$5, and w4 =$3. The firm must use at least 10 units of factor 2. What is the lowest cost at which it can produce 100 units of output?

Homework 8.

On a tropical island there are 100 potential boat builders, numbered 1 through 100. Each can build up to 20 boats a year, but anyone who goes into the boatbuilding business has to pay a fixed cost of 19. Marginal costs differ from person to person. Where y denotes the number of boats built by a boat builder per year, boat builder 1 has a total cost function c(y) = 19 + y. Boat builder 2 has a total cost function c(y) = 19+2y, and more generally, for each i from 1 to 100, boat builder i has a cost function c(y) = 19 + iy. If the market price of a boat stays at 25, how many boats will be built per year?

Homework 9.

Prove or disprove the following assertion: Average cost can never rise while marginal costs are declining.

Homework 9.

Suppose that all firms in a given industry have the same supply curve given by Si (p) = 2p when p ≥ 2 and Si(p) = 0 when p < 2. Suppose also that market demand is given by D(p) = 12 p. If firms continue to enter the industry as long as they can do so profitably, what will be the equilibrium price?

Homework 9.

Two firms have the same technology and must pay the same wages for labor. They have identical factories, but firm 1 paid a higher price for its factory than firm 2 did. They both maximize their profits and have upward-sloping marginal cost curves. Explain whether you would expect firm 1 to have a higher output than firm 2.

Homework 9.