HW2-Solutions - Homework Assignment 2 Solutions Econ 382...

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Homework Assignment 2 Solutions Econ 382, Professor Platt, BYU Do not distribute — only for current students 1 Warmup Optional questions. Points earned in this section can offset points lost elsewhere (though the maximum score for the assignment is still 100 points). Credit is only given if you show all steps taken to obtain your answer. 1. Suppose that Alex must produce his own cereal, using raw inputs of raw materials (obtained from his parent’s food storage) and labor (provided by Alex’s cooking skills). (a) (4 pts) Suppose Alex has a utility function U A ( c, n ) = c 0 . 5 0 . 5 . He has no en- dowment of cereal, but is endowed with 120 hours of time (which can be sold as labor or kept as leisure, ) and 40 food storage cans ( s ). Set up his household problem. Solution: max c,‘ c 0 . 5 0 . 5 s.t. p c · c + p · 120 p + 40 p s + Π Any desired notation may be used for the prices, but it should be clear. It is appropriate to include p s · s in the expense side, even though no s will be demanded by Alex in equilibrium. (b) (4 pts) Next, consider production in this economy. Alex sets up a firm to produce cereal. Production of cereal is given by F C ( L, S ) = L 0 . 5 S 0 . 5 . Set up the firm’s problem. Solution: max C,L,S Π = p c · C - p · L - p s · S s.t. C L 0 . 5 S 0 . 5 Price notation should match those used in the household problem. The allocation must be distinguishable from that used in the household problem ( e.g. use capital letters). While it is not essential to indicate that the profit function is Π, it is useful. 1
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(c) (4 pts) Define a competitive equilibrium for this economy. (You may refer to the previous two parts as appropriate, rather than repeating them, but be precise.) Be sure to specify the market clearing condition for each market. Solution: A competitive equilibrium is an allocation ( c * , ‘ * ) and ( C * , L * , S * ), and prices ( p * c , p * , p * s ) such that: ( c * , ‘ * ) is the solution to the household problem in (a), given ( p * c , p * , p * s ). ( C * , L * , S * ) is the solution to the firm’s problem in (b), given ( p * c , p * , p * s ). Markets clear: c * = C * , * + L * = 120, and S * = 40. In the market clearing conditions, notation must be consistent with the previous parts. If s was included in the consumer’s budget set, it must be included in the market clearing condition for S . (d) (4 pts) Solve for the aggregate demand for leisure and for cereal (in terms of input and output prices – there should be no other variables in the resulting equations). Solution: £ = c 0 . 5 0 . 5 + λ (120 p + 40 p s + Π - p c · c - p · ) £ ∂c = 1 2 c 0 . 5 - λp c £ ∂‘ = 1 2 c 0 . 5 - λp £ ∂λ = 120 p + 40 p s + Π - p c · c - p · = p c c = p = c = 120 p + 40 p s + Π 2 p c = = 120 p + 40 p s + Π 2 p If they included s in the budget equation, they should also note here that s * = 0. (e) (5 pts) Solve the firm’s problem.
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