Microeconomic Theory
12th Edition
ISBN: 9781337517942
Author: NICHOLSON
Publisher: Cengage
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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…
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- A stadium can seat 60,000 people. Assume 35% of seats are not occupied (not for sale) because of Covid-19 and social distancing requirements. You are told that the current price of tickets is $70 and that the demand is linear and that the demand function (Qd) = 60,000 - 250P. examine the market for tickets for games at the stadium during these Covid times. You should Illustrate themarket for tickets using a demand and supply diagram. Show all your calculations and properly label the diagram. At $70 a ticket the cap on the number of seats available (because of Covid restrictions) is creating a shortage. How many people miss out on tickets when they are priced at $70 per ticket?arrow_forwardIs the cobb-douglas function concave if a1 +a2 = 1 whenarrow_forwardAnswer d and e. 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…arrow_forward
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