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HW-4-soln

# HW-4-soln - Economics 101 problem set 4 solutions 1 Total...

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Unformatted text preview: Economics 101: problem set 4 solutions 1. Total capital is K = 4 and total labor is L = 25. Production functions are x = K 1 2 x L 1 2 x y = K 1 2 y L 1 2 y Find and sketch the production possibility frontier. Solution: in this setup, if K x is the level of capital allocated to the production of good x , K- K x = K y = 4- K x will be the level of capital allocated to the production of good y . Similarly, if L x is the level of labor allocated to the production of good x , L- L x = L y = 25- L x will be the level of labor allocated to the production of good y . Recall that along the production possibility frontier, RTS x = RTS y . We may then compute RTS x =- MPL x MPK x =- ∂x ∂L x ∂x ∂K x = 1 2 K 1 2 x L- 1 2 x 1 2 K- 1 2 x L 1 2 x = K x L x Since the calculations for RTS y will be identical, we know as well that RTS y = K y L y . Then the production possibility frontier is defined by K x L x = K y L y . Equating the rates of technological substitution we obtain RTS x = RTS y ⇐⇒ K x L x = K y L y ⇐⇒ K x L x = 4- K x 25- L x ⇐⇒ K x (25- L x ) = L x (4- K x ) ⇐⇒ L x = 25 4 K x Then we have x = 5 2 K x = 2 5 L x This is the same as K x = 2 5 x L x = 5 2 x Then for y , we find y = 4- 2 5 x 1 2 25- 5 2 x 1 2 = 10 1- 1 10 x 1 2 1- 1 10 x 1 2 = 10- x March 1, 2011 1 Economics 101: problem set 4 solutions With this, we have a neat closed form of our production possibility frontier, y = 10- x . Graphically, this is seen in Figure 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 2 4 6 8 10 Production possibility frontier x y Figure 1: production possibility frontier for question 1 2. Total capital is K = 4 and total labor is L = 25. Production functions are x = K 2 3 x L 1 3 x y = K 1 3 y L 2 3 y Find and sketch the production possibility frontier....
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HW-4-soln - Economics 101 problem set 4 solutions 1 Total...

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