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(1). derive and sketch the optimization condition for profit maximization.
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- Assume that F(X1, X2) = X11/3 X21/3 w1 = 1, w2 = 2, p = 3 Please solve the above profit maximization problem (unconstrained optimization) for the optimal quantities X1* and X2* showing step-by-step solution.Solve the following problems related to optimization problemsA restaurant finds that, when priced at $8, it sells 60 sandwiches a month. For every dollar that the restaurant discounts their sandwiches, they sell 5 more per month. If the cost to the restaurant to make a sandwich is $4, find the price that the restaurant should charge for a sandwich to maximize its profit. must show all stepsA local newspaper currently has 84,000 subscribers at a quarterly charge of $30.Market research has suggested that if the owners raise the price to $32, they wouldlose 5,000 subscribers. Assuming that subscriptions are linearly related to theprice, what price should the newspaper charge for a quarterly subscription tomaximize their revenue?a) Find the cost function (Hint: find slope and use point-slope form to find thecost function) b) Find the revenue function c) Find the maximum revenue d) Find the profit function
- The variable x represents the number of HDTVs manufactured and sold in a given month. The variable p is the price charged when you sell a TV. A demand curve is given by p = 660-3x. Assume you must pay salaries totalling 8000$ in a month. Also, it costs $110 to manufacture one TV. Graph the profit function, M(x)=550x-3x²-8000, and using the Maximum program, find the maximum possible profit and fill in the details (round off x to the nearest whole number): number of sales:The demand function for a particular brand of LCD TV is given by p = 1480 − 20x where p is the price per unit in dollars when x television sets are sold. (a) Find the revenue function. R(x) = (b) Determine the number of sets that must be sold in order to maximize the revenue.sets(c) What is the maximum revenue? $ (d) What is the price per unit when the revenue is maximized?$ per unitExplain the Envelope Theorem for unconstrained and constrained optimization. Give an economic example of these concepts.
- A firm manufactures two goods labeled 1 and 2. It sells Qi items of good i for a fixed price per unit of pi. The total cost of producing good i is ciQi2.Explain briefly why the profit function is given byπ(Q1, Q2) = p1Q1 + p2Q2 − c1Q12 − c2Q22Find the values of Q1 and Q2 which maximize π and verify that the second-order conditions for a maximum are satisfied. Find an expression for the maximum profit.James mainly sells confectionery items, newspapers, magazines and cigarettes in his convenience store. Noting his small business is not thriving, he thought of selling hot pies and rolls too. Suppose the total cost function for rolls and pies is, TC = 900 + 50Q, Q = Q1 + Q2 Where Q1 and Q2denote the quantities of rolls and pies respectfully. If P1 and P2 denote the corresponding prices, the inverse demand equations are. Q1 = 70 - P1 and 0.5Q2 = 100 - P2 a)If James decides to make a total of 48 rolls and pies per day and charges different prices as above (that is, P1 ≠ P2 ), how many of rolls and pies each should he make in order to maximize the profit of a particular day? Estimate and interpret the Lagrange Multiplier λ [note: assume second-order conditions are satisfied]. b)Using your knowledge of input-output tables, explain which components of the economy will be affected if all convenience stores, including James’, closed down for three months due to the COVID-19. What would…Objective: Find optimal L, Q, and profits for every change in market conditions. •Case 1: Economists calculated that Amazon’s productivity function in 2000 was: Q=40K^0.55 L^0.45 . Also, average price per unit (mostly books) = $22, wage per unit = $8, and at that time, Amazon had one distribution center (K) at $10,000 (including renting rate, inventory costs, etc.) Case 2: In 2005, Amazon expanded its distribution centers to 5. Case 3: In 2012, Amazon bought robot maker Kiva System, changing Amazon’s productivity function to: Q=80K^0.55 L^0.45 , keeping the rest of the information as in Case 2. Case 4: In 2019, Amazon announced hourly wage of $15 per unit. Compare optimal number of workers, Q, and profits under these corporate and market changes (Table). Case1: L1 = Q1 = Profits1 = Case2: L2 = Q2 = Profits2 = Case3: L3 = Q3 = Profits3 = Case4: L4 = Q4 = Profits4 =
- 1. A firm is said to be earning normal profit whenever:A. Accounting profit is zero.B. Economic profit is positive.C. Accounting profit is positive.D. Total revenue equals explicit and implicit costs.2. If a 10% increase in inputs results in a 20% increase in output, then the firm is said to exhibit:A. Diseconomies of scaleB. Diminishing marginal returns to scaleC. Constant returns to scaleD. Economies of scale3. Economists have traditionally attempted to avoid the trap of:A. Thinking in terms of total utility.B. Computing marginal utilityC. Making interpersonal utility comparisons.D. None of the above.4. According to the water-diamond paradoxA. Water has a lower MU than Diamond.B. Water has a higher TU than Diamond.C. Prices reflect MU and not TUD. All of the above.The next three problems are from Unit 10. You are provided with a pair of demand and cost functions. P=20,000-15.60 TC = 400,000 + 4640 Q + 100 Use the above functions and derive the maximum profits for the firm: Optimal Q = Optimal P = Optimal TR = Optimal profits - =Identify the point of diminishing returns for the input-output function. R is the revenue and x is the amount spent on advertising. Use a graphing utility to verify your results. R = − 4/9(x3 − 4x − 3), 0 ≤ x ≤ 4 (X,Y)=