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W12 MIC 17 SR and LR Costs(1)

# W12 MIC 17 SR and LR Costs(1) - SHORT LONG-RUN COSTS(B&B...

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1 Northwestern University ECON 310-1: Microeconomic Theory Professors Jim Hornsten & Ron Braeutigam Winter 2012 LR Cost Minimization Comparative Statics Factor Demands & Factor Substitution SR Cost Minimization 202 Cost Curves, Revisited SHORT- & LONG-RUN COSTS (B&B Chps. 7-8) Notes for Lecture #17 A Rainbow of SR Cost Curves An Envelope LR Cost Curve Output Dollars

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2 Northwestern University ECON 310-1: Microeconomic Theory Professors Jim Hornsten & Ron Braeutigam Winter 2012 SEGUE: Chp. 7 Highlights min L , K TC = wL + rK s.t. Q [ L , K ] = Q 0 (production target) given input prices w (wage) and r (rental rate) C 0 = wL + rK K = C 0 r # w r L with intercept C 0 r and slope # w r (1) Q [ L , K ] = Q 0 Required Production (On Isoquant) (2) MP L MP K = w r Tangency (of Isoquant and Isocost) IsoCosts are parallel lines; sets of (L,K) combos that generate a constant C 0 -Slope[IsoQ] vs.- Slope[IsoC] MP L MP K vs. w r MP L w vs. MP K r The LR Cost Minimization Problem Any (K,L) combo lies on one IsoCost line. When MRTS L,K diminishing (so IsoQuant bowed-in), expect an interior solution , where two conditions hold: When we suspect a possible corner solution , we still have the same production target, but now we use the bang for the buck condition and strive to equalize the marginal product per dollar spent across inputs: An Interior Solution (L*>0,K*>0) K L A C 1 C 2 C 3 Q 0 Rising Costs A Corner Solution (L*>0,K*=0) K L B C 1 C 2 C 3 Q 0 Rising Costs
3 Northwestern University ECON 310-1: Microeconomic Theory Professors Jim Hornsten & Ron Braeutigam Winter 2012 COMPARATIVE STATICS: An Input Price Change What happens when the price of labor rises? When the wage rises , the IsoCost curve pivots inward at the K-axis and becomes a steeper IsoCost . The IsoQuant (output target) does not change! If it is relatively easy to hit Q 0 with less L and more K , then we probably will do this to contain costs. If it is difficult or impossible to substitute K for L , or L gives us a higher bang for the buck even at the higher wage, then we may end up hiring lots of expensive L . min L , K TC = wL + rK s.t. Q [ L , K ] = Q 0 , given input prices w and r IsoCost : C 0 = wL + rK K = C 0 r # w r L with intercept C 0 r # Slope[IsoCost C 0 ] = w r # Slope[IsoQuant Q 0 ] = MRTS L , K = MP L MP K Bang for the Buck : MP L w vs. MP K r C 2 /w hi K L A C 2 /w lo Q 0 C 2 /r To get the same Q 0 , we hire less L (pricey!) and more K (cheap!), and spend more. C 3 B C 2 /w hi K L D Q 0 C 2 /w lo , C 2 /r C 3 C 3 /w hi C 4 Stuck in a corner at D!

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4 Find the input demands for production function Q = 50 LK = 50 L 1 2 K 1 2 . This Cobb- Douglas function shd hv a nice interior solution. Tangency Condition : MRTS = MP L MP K = 50 K 1 2 1 2 # % & L ( 1 2 50 L 1 2 1 2 # % & K ( 1 2 = K L = w r ) rK = wL . Thus, we have L [ w , r , K ] = rK w . However, we want a reduced form equation that is only in terms of exogenous variables, so we need to get Q 0 in and K out!
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