Final solution Q1 to Q4

# 1 wv 10 the inequality can also be shown by

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(1 + w/v ) . (10) The inequality can also be shown by considering that: C ( w, v, q ) = SC ( w, v, k ( q ) , q ) , (11) where k ( q ) = q (1 + v/w ) is the conditional input demand for capital found in c). So: ∂C ∂q = ∂SC ∂k d k d q + ∂SR ∂q . (12) From part a), one verifies that ∂SC ∂k < 0 for k < q (1 + v/w ). The desired inequality hence obtains by noting from part b) that d k d q > 0, and that SR is increasing in q . For part a), 2 pts were given for showing reasonable but unfruitful effort; 0 pts, if one at- tempted a minimization problem (unless the exact answer was provided, in which case full credit was given); writing down the exact expression gave 5 pts by itself. For part b), 8 pts were given for a reasonable derivation that led to the incorrect answer; 5 pts, for starting a maximization problem that ultimately led to nowhere after a few lines; 10 pts were automat- ically given for writing down the exact expression. In part c), showing the condition gave 4 pts; otherwise, partial credit was given according to how close the derivation got to the actual answer (in most cases of partial credit, 2 or 3 pts were given). For part d), 1 pt was given for writing down each marginal cost function, and partial credit (1-2 pts) was also given based on how close the derivation got to the actual answer. Writing down the condition gave 5 pts. Question 3: (a)(3 points) Given prices p x , p y and w = 1, firm 1’s problem is max l x p x l x - l x by first order condition, we have l x = 1 4 p 2 x Similarly, we have l y = 1 4 p 2 y (2 points)So, the excess demand for labor is l x + l y - 2000 = 1 4 ( p 2 x + p 2 y ) - 2000 2

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(2 points)The profits from the two firms are respectively π x = p x l x - l x = 1 4 p 2 x π y = p y l y - l y = 1 4 p 2 y (6 points)Now consider the consumer’s problem max { x,y } x 4 y s.t. p x x + p y y = π x + π y + 2000 Define the Lagrangian function L = x 4 y + λ ( π x + π y + 2000 - p x x - p y y ) ∂y : x 4 = λp y ∂x : 4 x 3 y = λp x x = 4 y p y p x (Note: with right first order condition but wrong budget constraint, you will get 2 points.) Plug π x , π y and x into the budget constraint, we have y * = 2000 + 1 4 ( p 2 x + p 2 y ) 5 p y x * = 4 5 p x [ 1 4 ( p 2 x + p 2 y ) + 2000] (2 points)So, excess demand for y and x are respectively: y * - 1 2 p y = 2000 + 1 4 ( p 2 x - 9 p 2 y ) 5 p y x * - 1 2 p x = p 2 y 5 p x + 1600 p x - 3 10 p x (b)(2 points)Walras’ law: the total value of excess demand for all markets in one economy equals zero for any prices. That is, n
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