If an individual has the following utility function, U(x, y) = ln(x) +ln(y), then 2ex,px + ex,py ex, I is equal to: (a) -1 ANSWER (b) -1/2 (c) 0 (d) 1/2 (e) 1 PC
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- Suppose a consumer with income I has preferences over consumption bundles that can be represented by the utility function U(X,Y) = XY. Suppose the price of X, px, is 1 and the price of Y, py, is 2. At the consumer's optimal consumption bundle, (X*,Y*), what must the value of the ratio be? (a) 10 (b) √5 (c) 4 (d) 2 (e) 8If the utility function (U) between food (F) and clothing (C) can be represented as U = F*C, then which combination of food and clothing will a rational consumer choose? a. (F, C) = (9, 25) O b. (F, C) = (25, 4) O c. (F, C) = (4, 16) d. (F, C) = (4, 25)The utility function of a consumer is u = √x + 2y. (a) Look for the demand function of goods x and goods y. (b) Demonstrate mathematically that the demand function of goods x does not depend on income m. (C) Demonstrate mathematically that the demand function of goods y depends on m, Px, and Py.
- A consumer consumes pizza and soda always in the fixed proportion of 3 slices of pizza (x) with 2 cups of soda (y). Any pizza or soda not consumed together with the proportional amount of the other is useless to her. (a) Write down a utility function u(x,y) that represents her preferences. (b) If she has I = $60 budget and px = 4, py = 4, what is her optimal bundle? (c) In (b), If the price of soda increases to $6, how much money should she be compensated so that she achieves the same level of utility before the price increase? (d) Returning to (b), If the price of soda increases to $6, how much should the price of pizza be reduced so that she achieves the same level of utility before the price hike? (e) In (a), what values of a and b she is indifferent between the bundle "30 slices of pizza and a cups of soda" and the bundle "b slices of pizza and 40 cups of soda". Find at least two (a,b) pairs to get full points.Philip sells homegrown heirloom tomatoes at a local farmer's market. Suppose for the past several weeks, Philip has been selling his tomatoes for $2.25 per pound and he has been selling 90 pounds of tomatoes each week. Suppose this week he drops the price of the tomatoes to $1.75 per pound and as a result of the price drop, he is stilling 110 pounds of tomatoes. (a) Using the midpoint rule, what is the percentage change in the quantity of tomatoes sold from the drop in price? % (Enter your answer as a percentage with no decimals rounded to the nearest whole percent (i.e. 33% or -10%)) 4 (b) Using he midpoint rule, what is the percentage change in the price of Philip's tomatoes? % (Enter your answer as a percentage with no decimals rounded to the nearest whole percent (i.e. 33% or -10%)) (c) Using the midpoint rule, what is the price elasticity of demand for Philips, tomatoes? (Enter yourA consuMER hAS A UTILITY FUNCTION v(x, y) = MAX (2x+y) (x+3y) he consumes(1,0) in equilibrium THEN WHICH Of the fOLLOWING MUST BE TRUE? (a) Px ≤PV Px < (3/₂) Py PX ≤ (2/3) PY Px = Py (b) (C) (d)
- Ice cream and cakes are perfect substitutes for a child, and 2 units of ice cream is always worth 3 units of cakes (however many ice creams or cakes she might have, she would be willing to give up 2 ice creams to get 3 more cakes to keep the same utility level) . (a) Write down a utility function u(x,y) that represents the child's preferences, where x is the number of ice creams and y is the number of cakes she has. (b) If the prices are px= 8 and py =5, and she has $140 to spend on the two goods this summer, what is her optimal bundle? (c) If the price of ice creams decreases slightly, down to px = 7, what happens to her optimal bundle in this case? Did it change "slightly" compared to (b)?Wrong answer will be dislikedYour own a chocolate producing company which can advertise on both television (T) and internet(I). The effect of TV and online commercials on sales is again given byS(T,I) = 500 + 48T−6T2+ 112I−6I2+ 4TI. You have a budget of $25 that you can spend on T and I. The price of aTV commercial is $12per unit and the price of an online commercial is also $12 per unit. 1. Determine the optimal level of TV commercials T and online commercials I if you have to spend all of your budget. You should provide two methods to solve this, by direct substitution and by setting up the Lagrangian. Is the Lagrange multiplier positive or negative? Give an intuitive interpretation of why this is the case? 2. Now determine the optimal level of TV commercials T and online commercials I if you DO NOT have to spend all of your budget. Do you obtain the same answer as subquestion 5.1? What is the Lagrange multiplier equal to in this case? Discuss.
- A consumer is maximising her utility function: U(r. y) = (r05 +y0.5)2, subject to the budget constraint 4x + 2y = 108. (a) Set up the Lagrangian function of this utility maximisation problem and first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the multiplier. (c) What are the utility maximising amounts of x and y if the budget constrain to x+ 2y = 36? How would change? Explain your reasoning. (Hint: Yc need to calculate A, rather comment on how it would change and why.)For the following utility function find the MRS at (x, y) = (2, 3) a) U(x, y) =2x +y b) U(x, y) =x^3 y^2 c) U(x, y) =min(x, 2y) Solve subparts (c )correctlyProblem 1 Jane has the following utility function: u(w) w^.5 or the square root. function Which of the following prospects would she prefer? P1(.4, 3,000, 600) P2(.7, 1,200, 1,000) P3(.6, 2,000, 100)