3133_note_2.pdf - 2 Cost Functions Cost minimization problem min wl vk s.t f(k l = q k,l Equal-cost lines c1 = wl vk vk = c1 \u2212 wl c1 w k= \u2212 l v v

3133_note_2.pdf - 2 Cost Functions Cost minimization...

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2. Cost Functions Cost minimization problem min k , l wl + vk , s.t. f ( k , l ) = q Equal-cost lines, c 1 = wl + vk vk = c 1 - wl k = c 1 v - w v l The intercept is c 1 v . The slopt of equal-cost lines is - w v , which is a input price ratio. When technology advances, physical capital becomes cheaper, the equal-cost lines become steeper. 1
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Example, Cost minimization for f ( k , l ) = Ak α l β min k , l wl + vk , s.t. Ak α l β = q If the isoquant is “well-behaved” (decreasing and convex), we can use the tangency condition RTS = f l f k = A β k α l β - 1 A α k α - 1 l β = β k α l = w v Together with the constraint Ak α l β = q , we can solve two unknowns k and l from two equations. k = α w β v l A ( α w β v l ) α l β = q l α + β A α α w α β α v α = q l α + β = β α v α A α α w α q l * = l c ( v , w , q ) = β α v α A α α w α q 1 α + β = β α α + β v α α + β A 1 α + β α α α + β w α α + β q 1 α + β k * = k c ( v , w , q ) = α w β v l * = α w β v β α α + β v α α + β A 1 α + β α α α + β w α α + β q 1 α + β = β α α + β - 1 v α α + β - 1 A 1 α + β α α α + β - 1 w α α + β - 1 q 1 α + β = α β α + β w β α + β A 1 α + β β β α + β v β α + β q 1 α + β The solution to the cost minimization problem are conditional/contingent input demand func- tion l c ( v , w , q ) = β α α + β v α α + β A 1 α + β α α α + β w α α + β q 1 α + β k c ( v , w , q ) = α β α + β w β α + β A 1 α + β β β α + β v β α + β q 1 α + β Here q is fixed. Later, we have l * ( v , w , p ) and k * ( v , w , p ) as solution to profit-max problem when q is chosen (not fixed). They are called input demand functions. 3
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Cost function Cost function is the value function of cost minmization problem. (The objective function, wl + vk , has not been optimized). If we evaluated the objective function at the solution, this is the cost function C ( v , w , q ) = wl * + vk * = wl c ( v , w , q )+ vk c ( v , w , q ) For the Cobb-douglas example, the cost function is C ( v , w , q ) = w β α α + β v α α + β A 1 α + β α α α + β w α α + β q 1 α + β + v α β α + β w β α + β A 1 α + β β β α + β v β α + β q 1 α + β = w β α α + β v α α + β A 1 α + β α α α + β w α α + β + v α β α + β w β α + β A 1 α + β β β α + β v β α + β q 1 α + β C ( q ) = Bq 1 α + β Average cost AC ( q ) = C ( q ) q = Bq 1 α + β - 1 = Bq 1 - ( α + β ) α + β Shape of cost function and average cost depend on parameter α and β .
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