2. Suppose that two products, i= 1, 2, are located at the extreme ends of the [0,1] interval. Let li denote the location of the product and x the consumer locations. Consumers buy one unit and are uniformly distributed on the unit interval; their indirect utility is given by r - pit |li - xl, where r is the willingness to pay, pi the price set by firm i, and t a parameter that measures the disutility from not consuming their ideal product. Marginal cost of production is constant and equal to c. Solve for the Stackelberg equilibrium in prices where the firm located at zero is the leader firm. Is there an advantage for being a price leader, that is, is there a first-mover advantage? Why? Explain. Compare the results with simultaneous choice of prices.
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- Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain.Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain. (b) Calculate consumers’ income elasticity as a function of the parameters. (c) Anaya is targeting a market whose income is expected to double over the next five years.Explain whether she should expect the demand for chargers to increase or decrease, andby how much, using the language of normal, inferior, luxury, and necessity goods. (d) Calculate the cross-price elasticity ofCwith respect toPF. (e)…Consider the market for a single network good and suppose thatconsumers differ in their valuation of both the stand-alone and thenetwork benefits (it can indeed be argued that it is more plausiblethat a user who has a higher value for the stand-alone component of atechnology also assigns more importance to the size of its network.) Tocapture this idea, write the consumer’s utility function for joining thenetwork as U(θ) = θ(a + νne), where a is the stand-alone benefit, ν > 0measures the network effect, neis the expected number of users joiningthe network, and is uniformly distributed on the unit interval.a. What happens if p > a + νne? (Hint: Identify all consumer who buythe good for at price p and a given expected network size ne).b. What is the consumer’s willingness to pay for the nth unit of thegood when ne units are expected to be sold? Is the demand downwardsloping? Does the demand increase with the expected network size?c. Express the fulfilled-expectations demand curve…
- 1 Assume there are two firms in a Hoteling Linear city. One firm is located at zero and the other at one. Assume the following utility equations: U0=v−p−tx+μ, U1=v−pn−t(1−x). Where U0 is utility from buying the good from the firm located at zero, p is the price for the firm at point zero, t is the travel cost and μ is bonus utility because the firm at point zero makes people a little bit happier. U1 is utility from buying the good from the firm at point one and pn is the price. Part a) Find the indifferent consumerA consumer’s preferences over two goods x and y are given bythe utility function U(x, y) = xαyβ with α, β > 0. The prices of the goods are px = 2 and py = 4.The consumer has an income of I > 0.(a) For what values of α and β are these utility functions strictly monotone?(b) For what values of α and β will the consumer demand (i.e., Walrasian demand) be more x than y?(c) For what values of α and β are these goods gross substitutes? For what values of α and β are these goods gross complements? Provide a justification for your answer.14.3 (0) Quasimodo consumes earplugs and other things. His utility function for earplugs x and money to spend on other goods y is given by u(x, y) = 100x 2:² 2 +y. (a) What kind of utility function does Quasimodo have? (b) What is his inverse demand curve for earplugs?. (c) If the price of earplugs is $50, how many earplugs will he consume? (d) If the price of earplugs is $80, how many earplugs will he consume? (e) Suppose that Quasimodo has $4,000 in total to spend a month. What is his total utility for earplugs and money to spend on other things if the price of earplugs is $50?
- Two friends, Karol and Manuel, like to drink kombucha (x₁) and matcha (x₂). Both X₁ and X2 are expressed in ounces. The following utility function represents Karol's preferences: 1 1 2 2 u (x₁, x₂) = x ₁ x ² The following utility function represents Manuel's preferences: u (x₁, x₂) = √√x1 + x2 Karol's income in dollars is denoted by mk, and Manuel's income is denoted by mm. Both face the same prices in the market, denoted by p₁ and p2, for kombucha and matcha, respectively. Both prices are expressed in dollars per ounce. Assume p2=1 throughout the whole question.Consider the following utility function и(х, у) %3 3х + 4y a. Is the utility function homothetic? Explain. b. Under what condition will the consumer purchase good x alone?This problem set tells the story of four friends Alice, Bob, C'arol, who are consumers, and Fanny, who owns a firm. Alice, Bob and Carol have preferences over two goods, Dumplings and Eggs. Fanny's firm produces eggs. In what follows you should use m; for i = A, B, C to denote the income of each of the three consumers and p; for j = D, E to denote the prices of each of the two goods. Similarly, d; and e; denote the consumption of dumplings and eggs respectively by consumer i = A, B, C. Fanny uses two (non-fixed) inputs, called 1 and 2, to produce eggs; the price of these inputs is denoted by w; for j = 1,2.
- Suppose we have a market with 200 individuals with preferences over two goods, and y. 3. Suppose now that we know m = 135, P = 3, 4. What is the market demand for x? and Py This should be a number. 4. Consider now that 100 individuals (type A) have the utility function U = x²/³y¹/³ and the other 100 (type B) have the utility function U = x¹/3y2/3. Each individual still has the same 'y income and is subject to the same prices. Calculate the market demand for x as a function of income and prices. 5. Suppose now that we know m = 250, P = 5, and Py 3. What is the market demand for x? This should be a number. F =Fi Consider the utility function: u(x1,x2) = Axf x}-« (-a where 0 0. (a) Compute the Marshallian demand functions. (b) Derive the indirect utility function. O Type here to :Guerdon always puts half a sliced banana, q, , on his bowl of cereal, q, The two 14 goods are perfect complements. What is his utility function? Derive his demand curve for bananas graphically and mathematically. 12- 10- e, Guerdon's utility function (as a function of q, and q2) is 6- VA U= min/2q1.92} 4- O B. U=q, +92- 2- 0+ OC. U=29, + q92- 10 12 14 16 18 20 6 22 Bananas O D. U=0.5q1 + q2- OE. U=q, xq2- 5.00- 4.50- Guerdon's demand function for bananas (as a function of the price of bananas, P1. the price of a bowl of cereal, p2, and income, Y) is 4,00- 3.50- 91 = (p1 + 2p2) (Properly format your expression using the tools in the E 3.00 palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ character.) 2.50- * 2.00- P. 1.50- 1.00- 0.50- 0.00+ 10 12 14 Bananas tv 80 DI DD F1 F3 F4 F5 F6 F7 F8 F10 @ %23 $ & 1 3 4 7 Q W E Y U S F H. J く C V M option command command optic ーの * CO Cereal (bowls) p. S per banana B * * N A