Comparative statics Single variable unconstrained optimization Multiple parameters Econ 494 Spring 2013

Agenda Quick review of profit max. Comparative statics One variable, one parameter One variable, multiple parameters Problem set 3 due Mon, Feb 11 2

Aside: Identity vs. equality Identity (always holds for all values of x) Equality (may hold for some values of x) Function 1 st derivative 2 nd derivative 3 Operations, such as differentiation, when applied to identities result in identities, but when applied to equations produce nonsensical results in general. Also see Chiang p. 6-7

Identities & equalities: Some other examples 4 IdentitiesEqualities

Identities, equalities and the FONC 5

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8 Note the identity

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Quick review 11 What are the endogenous variables? exogenous parameters? Also see BBT section Keep this slide handy

Where are we? 12

Comparative statics Definition: Mathematical technique by which an economic model is investigated to determine if refutable hypotheses are forthcoming. (see Silb., p. 15) We usually want to know if we can say anything about the sign of the comparative static: 13

Step 5. Comparative statics (one variable, one parameter) 14

Step 5. Comparative statics 15

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18 Because this is an identity, we must also differentiate the right- hand side (RHS). The derivative of any constant is zero.

Comparative statics Step 5c. Use algebra to solve 5b: 19 Step 5d. What is the sign? Hint: Look at denominator.

Step 6. Interpret Recall the SOSC: Use SOSC to sign the comparative static: Interpret result: A profit-maximizing firm’s output will decrease (increase) as the tax rate increases (decreases). 20

What have we accomplished? 21

Review 22 Set up a generic profit max problem. Make no assertions about the market or the cost function. Firm faces a per-unit tax (t) on output. See Handout #4 for example with ad valorem tax. Keep this slide handy

More than one parameter 23

Example 24 What are the choice variables? Parameters?

1. Set up optimization problem 25

2. Find FONC 26

3. Find SOSC FONC alone do not guarantee a maximum. The SOSC tell us 2 things. What are they? 1. As long as the SOSC hold, we know profits are maximized. 2. We also know that we can use the IFT to solve the FONC for the explicit choice function. 27

4. Find explicit choice function (specific case) 28

4. Find explicit choice function (general case) 29

5. Comparative statics 30

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Use algebra to solve: 33 What sign?? Recall that the SOSC for a max are negative Compare with denominator.

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Step 5c. Solve 5b (general case) 37 Use a little algebra… What sign?? Recall that the SOSC for a max are negative Compare with denominator.

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6. Interpret result 39

40 Typo corrected

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42 Specific: General: Again, the SOSC < 0 is the denominator, so we can sign the comparative static. Interpret:For the firm, advertising fees are another cost of producing output. As these fees increase, the firm will reduce output.

Notice a pattern??? 43

Notice a pattern??? 44 The numerator is from the explicit appearance in the FONC. The denominator is the SOSC. Here is the comparative static result:

Calculus review: Optimization with 2 variables MaximumMinimum FONC (both must hold) SOSC (all 3 must hold) 45 These are the optimality conditions when there are only 2 variables. Later, you will see a more general version of this using matrices.

Calculus review: Young’s theorem The order of differentiation does not matter Result extends to any 2nd partial derivative of a function of many variables This symmetry result will come in handy… 46