9. Perfect competition Econ 494 Spring 2013 Most of these notes are taken directly from Silb §4.4

Agenda Quick recap from last class Perfect competition 2

Recap… 3

Where are we going with this? Why do we care? Production function defines the transformation of inputs into outputs Postulates of firm behavior Profit maximization Cost minimization Results: shape of production fctn is key to FONC and SOSC Especially in evaluating comparative statics. 4

Application: Profit maximization and perfect competition 5

Strict Concavity 6

Perfect competition 7

Interpret FONC 8

Interpret SOSC 9

Solve FONC implicitly 10

Comparative statics: How do prices affect factor demand? 11

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Express in matrix form Matrix form: 13

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We already know where this will wind up: Apply Cramer’s rule… 17

18 No refutable implications emerge. PS6#2, show both cannot be negative.

The supply function 19

Homogeneity 20 Chiang §12.6 SH, §12.6 Hoy, §11.5

Production functions: What makes sense? 21

Euler’s theorem 22 Silb p. 56 Chiang, p

Corollary to Euler’s Theorem 23

Paying for the inputs 24

Paying for the inputs 25

Paying for inputs All this together implies: Or…total costs identically equal total revenues, and the product of the firm is exactly “exhausted” in making payments to all factors. 26

Homogeneity of factor demands 27 See Silb §4.5

Homogeneity of factor demands 28

Homogeneity of factor demands 29

Compare results 30

Compare results 31

Le Châtelier Principle Short-run vs long-run 32 See Silb §4.6

Le Châtelier Principle Consider the following constrained maximization problem: 33 What are the choice variables? Parameters?

Optimality conditions 34

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Interpret result 37

Result is local 38 w1w1 x1x1 x 1 * (p,w 1,w 2 ) x 1 s (p,w 1, x 2 º ) Note that the comparative static is dx/dw, but the graph is in the opposite order (w-x, not x-w).

Le Châtelier Principle Alternative derivation Consider the identity: 39

Example 40

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Find sign 42 Recall the reciprocity relation: Reciprocity

Find sign (cont) 43

Long- and short-run output supply Show that the long-run output supply function is more elastic than the short-run supply. Fundamental identity: 44 What does this mean?

45 Reciprocity relation from PS6: Need to find this

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