Assignment #31.Suppose a firm sells two products labeled: x & y. The total revenue function for these products is: TR = 36x â 3×2 + 40y â 5y2. And the total cost function is: TC = x2 + 2xy + 3y2. Find:a.The optimal levels of x & y this form should produce to maximize profits.b.Be sure to check and show the second order conditions confirming your results.c.What are the prices of both goods x & y at the optimal levels?2.Suppose a firm has a cost function: TC = 4×2 + 10y2. The production manager want to know the quantities of each product that will minimize the total cost if the total output of x & y must equal 800.a.Solve this problem for the manger using the Lagrangian method.b.How would this manager use the resulting Lagrangian multiplier in this case?3.The supply function for a particular kind of cheese is: Qs = 100 + 3P where Qs is the quantity supplied of this cheese in millions of pound per year, and P is the price of this cheese in dollars per pound. If the demand function is: Qd = 106 million pound of cheese per year and if the government imposes a price floor of $1 per pound:a.Will there be excess supply or excess demand for this cheese? How big will it be?b.What if the government sets a price floor of $3 per pound? Will there be excess demand or supply? How big will it be?4.Given the following Demand and Supply functions for wheat in the U.S.:Qd = 3.1 â 0.25PQs = 1.3 + 0.5PFind: a.Equilibrium price and quantity.b.Price elasticity of demand at equilibrium.c.If wheat prices increased slightly, would we expect total revenues to increase or decrease? Explain using the elasticity coefficient you calculated above.5.The B-2 bomber was designed as a stealthy long range bomber. Two Northrop engineers used mathematical analysis to determine how an aircraftâs volume should be proportioned between the wing and the fuselage to maximize its flying range. Taking the derivative of range with respect to volume, they found that this derivative equaled zero when the total volume was almost all wing. Hence the result was the flying wing configuration. Thus the engineers concluded that this configuration would maximize the flight range. However, in reality, this aircraft exhibits one of the worst flying ranges in the Air Force inventory. There was a flaw in the engineersâ analysis. What was it?6. Given the following demand functions: x = 25 â 0.5 Px y = 30 â Pywith the combined cost function: c = x2 + 2xy + y2 + 20Find:a.The profit maximizing output for each product.b.Price of each product.c.Profit.
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