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Assume a consumer has a utility function given by u(x,y)=x+y , with an income of $100. Consider a $1 per-unit tax on good 1. Suppose a lump sum tax raises the same amount of income as the per-unit tax. How much income must it raise?
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- Consider your preferences for Gasoline (X) and a composite good (Y). Your utility function is U(X,Y) = X^(2/5 )+ Y^(2/5). You have an annual income of $40,000. Suppose the price of the composite good is $1. a) Due to shortages, the government has introduced a rationing system such that you can only consume a maximum of 5,000 liters a year at $1 a liter. What would be your optimal consumption bundle? b) The government removes the rationing system and the free market price of gasoline jumps to $2. What would be your new optimal consumption bundle? Are you better off with or without the rationing? Show by finding the utility levels. c) Illustrate your solution in a clearly labeled graph.Suppose that the price of good X is $6 and the price of good Y is $2. You have $144 to spend and your preferences over X and Y are defined as: U(x,y) = x2/3y1/3 Calculate the marginal utility of X Calculate the Marginal Utility of Y What is the optimal Choice of X and Y given the PX = $6, PY = $2 and I = $144 If Income is increased to $150 calculate how the optimal choice of X and Y changeExercise 2 Suppose you have the following hypothetical demand or sales function. Qx= -4Px+2Py+0.201+0.04A and PX = $200, (price of good X) PY =$230, (price of good Y) 1 = $1,500 (disposable per capita income) A=$12,000 (advertizing expenditures) a. Write out a reduced demand function b. Solve for and write out the demand curve equation c. Give a narrative interpretation for the coefficient of I d. How could we classify good X and good Y? Explain why.
- Suppose that an individual has a Utility function represented by a CES function. The utility function of the individual is given as: U(x,y) = x1/2 + y1/2 b. Calculate the own price elasticity using the "share elasticity" of any good. Let us assume that the prices of both goods are equal.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.An individual consumes two goods, X and Y, and receives utility according to a Cobb-Douglas production function of the form: U(X,Y)= X1/3Y2/3. Give that MUX= 13(MUYMUX)2/3 and MUY= 23(MUXMUY)1/3 find the consumers optimal demand functions for X and Y in terms of PX, PY, and I. Explain why I could have solved the problem above using the utility function U(X,Y)= XY2 and would have obtained the same answer. What is it about utility that makes this possible? Economists define the ‘certainty equivalent’ of a risky stream of income as the amount of guaranteed money an individual would accept instead of taking a risk. The certainty equivalent varies between individuals based on their risk preference.
- A 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.Suppose you have the following indirect utility function: V(Pa, Py, I) = In PxPy What are marshallian demands for x and y? I (a) (9x9y) = (22) (b) (9,9y) = (In, In 2) (c) (9, 9y) = (exp(2p/py), exp(2ppy)) I (d) (9x, gy) = (2pr+py' px+2py) What is the expenditure function for the associated expenditure minimization problem? (a) E(pa, Py, U) = (P + Py) ln(U) (b) E(pa, Py, U) = √exp(U)Papy (c) E(pa, Py, U)= (p²+p²) In(U) (d) E(pa, Py, U) = exp(U)²papy What are the individual's Hicksian demands for goods x and y? (a) (h₂, hy) = ((BU)¹/², (PU) ¹/²) (b) (ha, hy) = (RU, DU) (c) (ha, hy) = ((2 exp(U))¹/², (exp(U))¹/²) -1/2 (d) (hx, hy) = ((P₂PzU)−¹/², (P₂PzU)-¹/2) Are x and y complements or substitutes?Marvin has a Cobb-Douglas utility function, 0.5 0.5 U= 91 92 his income is Y = $900, and initially he faces prices of p₁ = $1 and p2 = $4. If p₁ increases from $1 to $2, what are his compensating variation (CV), change in consumer surplus (ACS), and equivalent variation (EV)? Marvin's change in consumer surplus (ACS) is $ minus sign if necessary.) Marvin's compensating variation (CV) is $. (Enter your response rounded to two decimal places and include a minus sign if necessary.) (Enter your response rounded to two decimal places and include a Marvin's equivalent variation (EV) is $. (Enter your response rounded to two decimal places and include a minus sign if necessary.)
- Utility maximization with a budget constraint. A hypothetical consumer spends all tgheir income on ramen noodles (N) and wild rice (W). N is the quantity of noodles; W is the quantity of wild rice. Their income is $1,600 per month. the price of noodles is $2 per package and the price of wild rice is $20 per pound. The utility function is U=sqrt(N*W). the MRS = -N/W. The budget constraint is: 1,600 = 2*N + 20*W Graph Qty of noodles (N) on vertical axis and Qty of wild rice (W) on horizontal axis. SOLVE: a. Graph the budget constraint. label all points. What is the slope of the budget constraint? b. Find the optimal quantities of noodles(# of packages) and the wild rice (# of pounds) given the budget constraint. graph these optimal quantities. draw your indifference curve on the same graph. c. Show on your graph what happens when the price of wild rice increases to $40 per pound. Find your new optimal quantities of noodles and wild rice. label all points on graph. label the…Suppose that an individual has a Utility function represented by a CES function. The utility function of the individual is given as: U(x,y) = x1/2 + y1/2 c. Is the demand more elastic or inelastic than a Cobb-Douglas Utility function? Use the Slutsky matrix to illustrate.Suppose a consumer had a utility function given by: U=X + 4Y. If the price of Good X (Px) is $1 and the price of Good Y is $8 then what is the utility maximizing quantity of Good X the consumer will purchase with a budget of $4? (Round to the nearest two decimal places if necessary.)