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- Only typed answer and please don't use chatgpt otherwise I downvote the answer Assume that the relationship between test scores and the student-teacher ratio can be modeled as a linear function with an intercept of 698.9 and a slope of (-2.28). A decrease in the student teacher ratio by 2 will: A) reduce test scores by 2.28 on average B) result in a test score of 698.9 C) reduce test scores by 2.56 on average D) reduce test scores by 4.56 for every school district PLEASE EXPLAIN WHY C IS CORRECT.Give U=XY-2Y and Px=4 ,Py=2 want to minimize budget that subject to is PX+PY=288 1).write out the Lagrangian function 2.Use First order condition to find X and Y 3.Test for the second order condition for minimum satisfied.What is a controllable source of budget variance
- (b) It is thought that a consumer measures the utility, u(x, y), of possessing a quantity, z, of apples and a quantity, y, of oranges by the formula u(x, y) = rªyla, where a is a constant. Suppose that apples cost p dollars each and oranges q dollars each. If the consumer has M dollars to spend on apples and oranges, use the method of Lagrange multipliers to find the quantity of each that he will buy in terms of a, p, q and M. [You are not required to justify the use of the method of Lagrange multipliers here.] For equally priced apples and oranges when he has one dollar to spend, he is actually observed buying one apple and two oranges. What value of a does this imply?Population of Fish in a Lake A lake is stocked with 100 fish. Let f(1) be the number of fish after i months, and suppose that y = f(1) satisfies the differential equation y' = .0004y(1000 – y). Figure 7 shows the graph of the solu- tion to this differential equation. The graph is asymptotic to the line y = 1000, the maximum number of fish that the lake can support. How fast is the fish population growing when it reaches one-half of its maximum population? 1000 800 600 400 200 10 15 Figure 7 Growth of a fish population.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.)
- The vending machine in Katherine's office building offers cans of pop and candies. Katherine's utility function is U = 3PC, where P is the amount of pop consumed per %3D week and C is the amount of candy consumed per week. Pop costs $1 and candy costs $0.5 per bag. If Katherine has $10 to spend, she will consume A bags of candy.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 )correctlyA consumer is maximising her utility function: U(x, y) = (x¹/³+y¹/³)³, subject to the budget constraint x + 3y = 100. (a) Set up the Lagrangian function of this utility maximisation problem and derive the first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the Lagrange multiplier. (c) What are the utility maximising amounts of x and y if the budget constraint changes to x + 3y = 50? Also, calculate the Lagrange multiplier.