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- Find the values of x1 and x2 where the following two constraints intersect. (Round your answers to 3 decimal places.) (1) 10x1 + 5x2 ≥ 50 (2) 1x1 + 2x2 ≥ 12The manager of a hockey arena is pricing tickets for an upcoming game. She knows that if she increases the ticket price she will sell fewer tickets. The situation is modelled by the relation, R = -88.9p^2 + 2667p, where R is the total revenue and p is the ticket price, both in dollars. The graph is given.U.S. food markets consumers viewed beef as a normal good from 1960-1976, but viewed it as an inferior good after that point. This type of change is not abnormal, in that as average household incomes rise, preferences might change. For instance, as households move from poor to middle-class, their consumption of beef might increase. However, as households move from middle- class to upper-middle-class, they might choose to purchase more exotic foods products. Assuming you are a beef producer in 1983, what will happen if incomes continue to increase? a. The marginal cost of beef will increase. b. The marginal cost of beef will decrease. c. The demand for beef will increase. d. The demand for beef will decrease.
- 12) Melquiades is worried about his grades from the microeconomics course, according to the rules for grading, the final grade is obtained as the minimum between the grade for the midterm test (x) and the final test (y). The utility function of Melquiades will depend on the grades obtained from those two tests Now, Melquiades has researched that the average student needs to study 24 minutes an additional point in each test (tests are graded from 0 to 100) obtain a) If Melquiades has 20 hours to study for both tests (x and y), how much time would he employ for each test? Figure out the grade that Melquiades will obtain in each test and his final grade in microeconomics. Elaborate a graph. b) How much additional study time would Melquiades require to increase his final grade by 10 points. Elaborate a graph. c) Suppose the difficulty of the final test increases in such a way that now each additional point would require 26 minutes of study. What would be the total cost (in minutes)…Consider a consumer with the utility function U (x1, x2 ) = 10x12/3x21/3 −50. Suppose the prices of x1 and x2 are 10 and 2 respectively and the consumer has an income of 150. (a) Write out the consumer’s constrained optimization problem. Specifically, write out the objective function and constraint for the problem (e.g. max _?_ subject to _?__). (b) Write the Lagrangian equation corresponding to the constrained optimization problem. Derive the Necessary First Order Conditions. (c) Use the NFOCs to solve for the consumer’s optimal bundle. (d) Show that at the solution you found in (c), the tangency condition is satisfied: MRS = p1 / p2. (e) How did the ‘50’ in the utility function influence the optimal con- sumption bundle? How did the ‘10’ in the utility function influence the optimal consumption bundle? (i.e., how would the optimal bun- dle change if these coefficients were to change?). How would the optimal bundle change if the utility function was x12x2? Lastly, how would…PROBLEM 2.25% PARTIAL DIFFERENTIAL LAGRANGE OPTIMIZATION In the era of the Covid 19 pandemic, we are required to consume lots of fresh fruits. The satisfaction of a consumer in consuming two kinds of fruit, namely Apples (X) and Oranges (Y) is: U = X3 Y2 income or budget to buy the two fruits is Rp. 250,000, the price of apples IDR 75,000/fruit and the price of oranges is IDR 25,000/fruit. a. How many apples and oranges must be obtained in order to achieve optimum satisfactionb. Calculate the optimum satisfaction value that can be achieved
- Exercise 1 A consumer's utility function is u(x, y) = x²y³ and the budget constraint is prx +Pyy ≤ w, where the parameters pr, Py and we are all strictly positive. (a) Solve the consumer's utility maximization problem. (b) Use the envelope theorem to estimate the change in the indirect utility function (i.e., the problem's value function) when the price py is changed to py + e, with e > 0.Suppose that the satifaction received from a number of rides taken on a roller coaster is given by the function S(x) = 200 (2x+1/3x+4)0.5 ,where S is the satisfaction received and x is the number of rides taken. Find the rate at which the satisfaction changes with respect to the number of rides taken.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 your
- 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.)A consumer buys (x1, X2 , X3 ) at price vector (P,, P2 , P3). We make three observations of his behavior: At (P,P,,P;) = (1,1,2), the consumer chooses (x,,x,,x3) = (5,19,9) At (P, P,,P,) = (1,1,1), the consumer chooses (x,,x,, x3 ) = (12,12,12) At (P, P,,P,) = (1, 2,1) , the consumer chooses (x,,x,,x; ) = (27,11,1) %3D %3D Does the consumer's behavior satisfy WARP? What about SARP?Suppose a consumer's utility is given by: u (x1, x2) = a/xT + b/x2 Where a= 8, and b = 4 %3D What is the value of MU1 (x1, X2)when x1=9 and x2=8. (Note: The answer may not be a whole number, so round to the nearest hundredth) (Note: The numbers may change between questions, so read carefully)