Economics 1011a: Intermediate Microeconomics Lecture 3: Firms with Multiple Variables Tuesday, September 23, 2008 1011a – Lecture 3 1More Choices • Last class we looked at general comparative statics in one variable. • We set up a basic model of the firm. • We found how the firm changes its labor input when the price of labor changes. • We then showed that you can do the same calculation using the general formula. 1011a – Lecture 3 2Sections • You should have received an email with your section assignment • Contact Jonathan with any problems • Friday review sections: 1-2:30 in Jefferson 250 1011a – Lecture 3 31011a – Lecture 3 4 Optimization with Multiple Variables • Today: we will extend what we did last time by using multiple endogenous variables. • We will only work out 2 variable examples. • Some “natural” applications: – How much capital and labor to use in the production process. – How many shirts and pants to own. – The quantity and quality of your children.1011a – Lecture 3 5 Two Endogenous Variables: Notation • Now we will write the optimization problem as • X and Y are endogenous. • Z is exogenous. • There could be many exogenous variables: • This would not alter our analysis. Why?1011a – Lecture 3 6 Multiple First Order Conditions • Now we are solving: • Again we use first order conditions to find the maximum of this function. • But there are two FOCs now. • These are 2 equations with 2 unknowns. – What are the unknowns?1011a – Lecture 3 7 Implicit Functions Revisited • These 2 equations with 2 unknowns define X* and Y* implicitly as functions of Z. • We will now proceed as before, only with this system of equations.1011a – Lecture 3 8 Comparative Statics (I) • We want to do comparative statics. • I.e., we want to find • So, we have to differentiate both of these equations with respect to Z.1011a – Lecture 3 9 Setting Up The Problem • The key thing is to remember that we are just taking partial derivatives of partial derivatives. • This is just the chain rule in multiple dimensions. • We will often suppress the fact that X* and Y* are functions of Z for notational simplicity. • But do not ever forget this!1011a – Lecture 3 10 Taking Derivatives • Differentiating:1011a – Lecture 3 11 A Simpler Notation • Often we will suppress the function’s argument completely, as we do here: • Again we have 2 equations and 2 unknowns. • What are the unknowns now?1011a – Lecture 3 12 Solving the Equations (I) • We will solve this system of equations using the substitution method. • You could also use Cramer’s rule (actually easier, but more technical). • Remember that we have equality of mixed partials (Young’s theorem) so FXY = FYX • Rewriting the second equation:1011a – Lecture 3 13 Solving the Equations (I) • Substituting into the first equation: • Substituting back for (or by symmetry):1011a – Lecture 3 14 Second Order Conditions • For all this to make sense, we need the FOCs to produce a maximum. So we need to check second order conditions. • In multiple variables, this means we need to check that the Hessian matrix is negative semi-definite. – Simply put, at the point (X*,Y*), F(X,Y) does not slope upwards in any directions. – M is negative semi-definite iff for all vectors v, v’Mv ≤ 0. – This is equivalent to saying the eigenvalues of the Hessian are all ≤ 0. • But for our purposes this is easy.1011a – Lecture 3 15 All You Need to Know About SOCs • Fortunately, all you need to know is that the following 3 conditions are necessary and sufficient in 2 dimensions: • Some intuition here? – For the first 2? – When would the 3rd condition fail to hold? • If F is concave, then these automatically hold. Usually we will be working with concave functions.1011a – Lecture 3 16 Back to Comparative Statics • Now what can we say about the signs of these expressions? • If Z only has a cross-effect with X, what matters? What about with Y? • What if Z impacts both variables?1011a – Lecture 3 17 A Caveat • This was all very technical and abstract. • Easy to get lost doing this. • Often you will find it more useful to just solve a problem directly, without general formulas. • We will do so now.1011a – Lecture 3 18 Application: Firms with Multiple Inputs • Recall our discussion last class. • Firms have a production function f(K,L). • The cost of a unit of capital is r, and of a unit of labor is w. • The selling price of its good is P. • Firms maximize profit:1011a – Lecture 3 19 FOCs for Firms • So firms are solving: • This is just like last time, but with two endogenous variables. • So there are two first order conditions:1011a – Lecture 3 20 K* and L* • These two FOCs implicitly define K* and L*: • Note that here we have included all the exogenous variables in the implicit functions. • Often we will only include the variable we are interested in doing a comparative static on.1011a – Lecture 3 21 Comparative Statics for Firms (I) • We have now defined K*(P,r,w) and L*(P,r,w) • Let us do a comparative static. In particular, let us find • Now we will just write K*(w) and L*(w). • Just as in the general case, we differentiate both our first order conditions.1011a – Lecture 3 22 Comparative Statics for Firms (II) • We pull out P and suppress the arguments of the partial derivatives: € ∂∂wPfKK * w( ),L * w( )( )[ ]=∂∂wr[ ]∂∂wPfLK * w( ),L * w( )( )[ ]=∂∂ww[ ]1011a – Lecture 3 23 Comparative Statics for Firms (III) • Then solve this system of equations. We will just use substitution (or you can try Cramer’s rule):1011a – Lecture 3 24 Comparative Statics for Firms (III) • What are the signs of these two terms? • What determines the sign of ? • If fKL > 0, then capital and labor are complements. • If fKL < 0, then capital and labor are substitutes. • What is the intuition here?1011a – Lecture 3 25 Putting It All Together (I) • We could have done all this just by using the general formula. • We need to match up the endogenous and exogenous