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hw7_soln

# hw7_soln - ECON 121 Winter 2011 Professor Faingold Noam...

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ECON 121 Winter 2011 Professor Faingold Noam Tanner Homework Assignment 7 Solutions to Homework Assignment #7 Economics is common sense made di ffi cult -Anonymous Congratulations everyone! You have o ffi cially graduated from the simple supply- demand equations of intro micro and entered the Competitive Equilibrium framework used to by pros! In this assignment, you will solve for an equilibrium in a model economy with much more detail than possible with the supply and demand line approach. Now you can not only discuss price e ff ects (and compar- ative statics) on outputs, you can discuss price e ff ects (and comparative statics on inputs. This assignment can get computationally intense if you don’t proceed the right way. Because of this, it might be a good idea to set β = 1 2 in for the function f 1 . Throughout the problem set I will say the solutions for the case where β = 1 2 . 1. The first thing to do is to write what you know about this economy: We have two consumers, A and B, one firm (with production function f ), and two consumption goods (1 and 2). Technology: The firm’s production functions are: y 1 = f 1 ( k 1 ,l 1 ) = k β 1 l 1 - β 1 and y 2 = f 2 ( k 2 ,l 2 ) = k 2 l 2 where β ( 1 2 ,1] is a technology parameter and y 1 , k 1 , and l 1 denote the output and input levels in the production of good 1, respectively, and likewise for good 2. Preferences: Consumers derive utility functions as in HW #6: u A ( x A 1 ,x A 2 )= ( x A 1 ) α ( x A 2 ) 1 - α and u B ( x B 1 ,x B 2 )= x B 1 x B 2 Endowments: One important detail to pick up here is that each con- sumer gets L units of labor and K units of capital. Remember this when it comes to writing the budget constraints! Now for the feasibility constraints: x A 1 + x B 1 = y 1 x A 2 + x B 2 = y 2 k 1 + k 2 = 2 K l 1 + l 2 = 2 L 1

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Now let’s get down to business: (a) The question asks why the production bundle ( y 1 ,y 2 ) in a Pareto op- timal allocation ( x A 1 ,x A 2 , x B 1 ,x B 2 , y 1 ,y 2 ) is on the production possibili- ties frontier (PPF) . Notice that the allocation must specify the output quantities of goods 1 and 2 consumed by A and B in addition to the production bundle ). This is an easy way to lose points on homeworks and exam. MAKE SURE YOU INCLUDE ALL OF THE QUANTITIES ( x A 1 ,x A 2 , x B 1 ,x B 2 , y 1 ,y 2 ) WHEN SPECIFYING A PARETO OPTIMAL ALLO- CATION IN AN ECONOMY WITH PRODUCTION! Back to (a): Let’s say that ( y 1 ,y 2 ) is NOT on the PPF, in other word’s y 1 = Y 1 ( y 2 ) where Y 1 ( y 2 ) = max k 1 ,l 1 ,k 2 ,l 2 f 1 ( k 1 , l 1 ) subject to f 2 ( k 2 , l 2 ) = y 2 k 1 + k 2 = 2 K l 1 + l 2 = 2 L In words, this means that Y 1 ( y 2 ) is the largest amount of good 1 that can be produced given that the amount of good 2 produce id y 2 and that the fea- sibility conditions are satisfied (all the inputs are used and the economy is not using more than this amount of inputs). Thus, if y 1 = Y 1 ( y 2 ) then this means that MORE y 1 CAN BE PRODUCED WHILE KEEPING THE AMOUNT OF y 2 CONSTANT. Thus, since both utility functions are strictly increasing, if we produce Y 1 ( y 2 ) we can take the extra amount of good 1 pro- duced ( y 1 - Y 1 ( y 2 )
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