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Unformatted text preview: P a w a r P a w a r F LF KF C LC KC = + = × + × = = + = × + × = ( ) ( ) ( ) ( ) 1 2 1 2 4 4 2 1 2 10 (1) P a w a r w r P a w a r w r F LF KF C LC KC = = + = × + × = = + = × + × 4 1 1 15 4 1 ( ) ( ) ( ) ( ) (3) $ $ $ $ . w p p r C F = = 50% (2) Homework #6: Answers Text questions, Chapter 7, problems 12. 1. Suppose there is only one technique that can be used in clothing production. To produce one unit of clothing requires four laborhours and one unit of capital; in food production each unit requires a single laborhour and one unit of capital. At an initial equilibrium suppose the wage rate and the capital rental are each valued at $2. a. If both goods are produced, what must be their prices? Note that these are Leontief (i.e. fixed coefficient) production functions. The zero profit conditions for these industries are: The second equality follows from substitution. The, if both goods are produced, P F = 4 and P C = 10. b. Now keep the price of food constant and raise the price of clothing to $15. Trace through the effects on the distribution of income. Rank the relative changes in the wage rate, the price of clothing, the price of food (unchanged by assumption), and the rent on capital. Relate your results to the StolperSamuelson theorem. This problem assume a 50% increase in P C , i.e. It is P P P C C C " ' ' . . = = 15 10 10 5 assumed that clothing is the Lintensive good, i.e. Thus, the a a a a LC KC LF KF = = 4 1 1 1 . StolperSamuelson theorem asserts that: Thus, labor gains unambiguously and capital loses unambiguously from this price change. With Leontief production functions, equations (1) are linear equations in w and r , so they can be solved simultaneously for any values of P F and P C . That is: One way to do this is is to transform equations (3) into an explicit relationship between w P P a a a a w r w r F C LF KF LC KC = ⇒ = 4 15 1 1 4 1 . L a y a y K a y a y LF F LC C KF F KC C = + = + (6) $ . $ . $ $ . . w p p r C F = = = =  83 5 83 (5) and r , plot both lines, and find the wage and rental that satisfy these two equations. Alternatively, one can solve the equations directly by substitution or using Cramer’s rule. To do the latter, note that (3) can be rewritten as: The determinant of the matrix A = [ a ij ], * A * = 3, * A w * = 11, * A r * = 1. Thus: w = * A w * / * A * = 11/3; r = * A x * / * A * = 1/3. These can be compared to w = 2 and r = 2 to give the ranking, given in (2), derived from the StolperSamuelson theorem: 2. Retain the assumptions about technology in problem 1: a LC = 4, a KC = 1, a LF = 1, a KF = 1....
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 Fall '09
 Economics, Econometrics, Slope, Relative price, KF, yc

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