A consumer has the utility function u(x,y) = xy and the price of both x and y are $4 a unit. If the consumer has an income of $224, what is the optimal amount of good x for the consumer to purchase? O28 O 48 O 110
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- 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.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.It is estimated that a grocer’s daily profit from the sale of two brands of cherry juice is given by thefunction:P(x,y) = (x - 30)(70 - 5x + 4y) + (y – 40)(80 + 6x – 7y) centswhere x is the price per can of Brand V Juice and y is the price per can of Brand Triple V Juice. CurrentlyBrand V Juice sells for 50 cents per can and Brand Triple V Juice sells for 52 cents per can.i. Use marginal analysis to estimate the change in the daily profit that will result if the grocer raises theprice of Brand Triple V by one cent per can while keeping the price of Brand V unchanged.ii. Find a stationary point of the daily profit function. Classify this stationary point.
- Brit-Brick is a company that produces bricks and cement in the UK. Their largest consumer is ConstrUK, a UK construction company. The manager of Brit-Brick has asked the research department to find out how sensitive ConstrUK's demand for bricks is. The research department has estimated that ConstrUK's preferences over bricks (x) and cement (y) can be described by the utility function wwwww U(x,y) xb/10 y1-b/10 , where b=8, and where x and y are measured in bags. Example: if b=4, then U(x,y) x*/10 yl-4/10 = x²/5 y3/5. %3D The price for one bag of cement is equal to £1. It is estimated that ConstrUK's budget is £10,000. Find the price-consumption curve for bricks and the corresponding demand curve.Electrical Components Profit A company’s dailyprofit from the production and sale of electrical components can be described by the equation P(x) =6.45x - 2000 dollars, where x is the number of unitsproduced and sold. What level of production andsales will give a daily profit of more than $10,900?A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $5, and the society sells an average of 24 per week at a price of $9 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 3 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (Let x represent the price per feeder and P represent the profit.) P(x) = (b) What price should the society charge for each feeder to maximize profits? $ What is the maximum weekly profit? $
- Your 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.The utility function and the prices are the following: U = 51 x1 + 23x2 P1-5, P2-20 and 1=3157 What is the optimal amount of x2?A large company in the communication and publishing industry has quantified the relationship between the price of one of its products and the demand for this product as Price = 160 -0.02 × Demand for an annual printing of this particular product. The fixed costs per year (i.e., per printing) = $51,000 and the variable cost per unit = $35. What is the maximum profit that can be achieved? What is the unit price at this point of optimal demand? Demand is not expected to be more than 4,000 units per year. The maximum profit that can be achieved is $144,313. (Round to the nearest dollar.) The unit price at the point of optimal demand is $ per unit. (Round to the nearest cent.)
- Maximize the following equation: w (x, y) = -3x² + 30x – 6y² + 48y Subject to x + y = 5 O x=2.33; y=2.67 O x=3.33; y=2.67 O x=2.33; y=3.67 O x=12.33; y=12.67It is known that a certain company sells each kg of the product it manufactures at $80, it is also known that the total manufacturing cost "CT" is given by the function CT=(1/1000)x2 +100 , where "x" are the kg of product produced. a) How many units of "x" must the company sell to break even? Value = 100 points.b) How many units of "x" is the optimal quantity that should be sold to optimize the producer's profit? Value = 100 points.c) How many monetary units does that optimal profit for producers amount to? (remember, you will have to prove it mathematically either by the method of the first or by the method of the second derivative). Value = 100 points.For a certain Utility function, which is a simple sifted parabolic equation, the maximum utility is 10 for n=5. Determine the mathematical expression of this Utility function. Hint: look up (h,k) values for a parabola. Multiple Choice 10n + 5 10n + 5n - 2n2 +10n + 5 - (n-5)2+10