Economics 326: Duality and Supply. Ethan Kaplan

Transcription

1 Economics 326: Duality and Supply Ethan Kaplan Octobe 29, 202

2 Outline. Cost Minimization 2. Cost Minimization: Example 3. Maginal, Aveage and Aveage Vaiable Cost Cuves 4. Supply Cost Minimization The Dual appoach to po t maximization is a to step appoach called cost minimization.

3 Stage : Minimize costs fo a given amount of poduction by choosing input mixes subject to a quantity constaint. Stage 2: Choose the level of poduction by maximizing po ts as a function only of quantity poduced. Fist stage: Endogenous Vaiables: K; L Exogenous Paametes: ; ; q Solve fo: K (; ; q) L (; ; q) Second stage:

4 Endogenous Vaiable: q Exogenous paametes: ; ; p Solve fo: q (; ; p) What is the elationship beteen these to appoaches: They solve fo di eent things The po t maximization appoach solves fo input demands as a function of input and output pices The cost minimization appoach solves fo input demands as a function of quantity of output and input pices To ecove supply Po t maximization: plug in the input demand functions into the poduction function

5 Cost minimization: plug input demands into po t function and pefom second stage maximization 2 Cost Minimization Example Minimize costs to poduce q units of output fo the poduction function q = K L If e minimized costs, hat ould choose? Economically? Mathematically? min K L K;L

6 So, e minimize costs subect to the constaint that output is at least q: subject to K L = q min K L K;L Witing a Lagangian: min K;L; K L h q K L i In theoy, e should check ou second ode conditions. The Lagangian, K L h q K L i, should be convex. This is the same thing as saying that the poduction function, K L ; should be concave. What ae the endogenous vaiables? Paametes?

7 Solving = K L = = K L = = q K L = 0 (3) Solving fo in equation () and equation (2), e get: = K L = K L o K = L o K = L (4) No comining ith the st ode condition fo in

8 equation (3) ; e get: q = K L = L L L q = L = q L = q No using equation (4) ; e can solve fo K as ell: K = L = = q q

9 So, e no kno ho much capital and labo the m uses as a function of output. Theefoe, e can constuct cost functions hich ae functions of output: C (; ; q) = K (; ; q) L (; ; q) e can no eite this as: = q q q This is called the cost cuve (o total cost cuve). It gives the cost of poduction as a function of output and facto pices.

10 3 Maginal, Aveage and Aveage Vaiable Costs Cuves Fom the total cost cuve, e can deive maginal, aveage and aveage vaiable cost cuves. What is maginal cost? The maginal cost incued fom an exta unit q Computing ith ou computed cost cuve fo Cobb- Douglas: C (q) = q

11 Thus maginal cost is equal to dc dq : q Notice that maginal cost is positive. When is maginal cost inceasing? When the deivative of maginal cost (o the second deivative of total cost is 2 q 2 = q 2 2 Thee ae 3 cases: > 0 : Inceasing Maginal Costs = 0 : Constant Maginal Costs < : Deceasing Maginal Costs

12 This is the same as etuns to scale Inceasing Maginal Costs = Deceasing Retuns to Scale Constant Maginal Costs = Constant Retuns to Scale Deceasing Maginal Costs = Inceasing Retuns to Scale Aveage cost is the aveage cost pe unit of poduction and is given by: C q In the Cobb-Douglas case, it is equal to: C q = = " q # q q

13 Gaphs of maginal and aveage cost cuves. Why does maginal cost intesect aveage cost at its bottom? What is the di eence beteen aveage cost and aveage vaiable cost? 4 Supply In the second stage of cost minimization, e can gue out ho much the m ants to poduce. Lets ty ou Cobb-Douglas example: max q pq q

14 Computing the st ode conditions - d p dq and e set this equal to zeo. This then leads to: " p ( ) and nally solving fo q : q = 0 " p ( ) 3 # = q # C A 7 5 q This is the supply cuve: ho much the m ants to poduce as a function of the pice. Notice that as long as > 0, the exponent on pice is positive and thus the function is upad sloping. In othe ods, if e have deceasing etuns to scale o

15 ineasing costs, the m ill alays ant to supply moe as the pice goes up. What is the poblem if < 0? In geneal, maximizing po ts, e maximize: max q (q) = pq c (q) What is the di eence beteen this po t maximization poblem and the oiginal one? Maximizing po ts by choosing quantity of output, e get a famous esult: d dq = p c 0 (q) = 0 =) p = c 0 (q)

16 This says that the m sets pice equal to maginal cost. Intuition? Suppose e anted to gaph supply. We could gaph quantity on pice o quantity on maginal cost. That is hy the supply cuve is sometimes called the maginal cost cuve. One note: supply is the maginal cost cuve hen pice is above aveage cost? Why?