Faculty of Economics Optimization Lecture 3 Marco Haan February 28, 2005

2 Last week Constrained problems. The Lagrange Method. Interpretation of the Lagrange multiplier. Second-order conditions. Existence, uniqueness, and characterization of solutions. This week: Comparative Statics (Ch. 14) Introduction to comparative statics General comparative-statics analysis The envelope theorem You’ll also learn some useful micro-economics along the way.

3 Comparative Statics How do the endogenous variables in some economic model change as some exogenous variable changes? Examples: –The effect of the tax rate on the output of a firm. –The effect of a change in national income on the interest rate. –The effect of a change in price on consumption. I won’t have time to discuss the macro examples in the book. Please do study them!

4 Example: the easiest case Suppose that demand for a good is given by D = 30 – 2p – y, with p price and y income. Supply is given by S = p Equilibrium: p* = 10 – y/3 q* = 10 – y/3 What happens with price and output as y changes? dp*/dy = – 1/3 dq*/dy = – 1/3

5 Another example A monopoly firm faces the linear demand function p = a – bq, and has total cost function C = cq 2 Government imposes a tax of t per unit of output. How does a change in tax affect the monopolist’s output? Profits: π = aq – bq 2 – cq 2 – tq = (a – t)q – (b + c)q 2 So: q* = (a – t)/2(b + c) So: dq*/dt = – 1/2(b + c)

6 So, in every example... First, save the model in terms of the variable that we are interested in. Then, take the derivative of that variable with respect to the variable for which we want to know the effect. Yet, it is not always necessary to explicitly solve the model.

7 More formally Suppose that we have a model with one endogenous variable, x, and one exogenous variable, α. The equilibrium solution is implicitly given by some function f that connects the equilibrium value x* and α: f(x*,α) = 0. We assume that f is differentiable. We want to know something about dx*/dα. Suppose we can write x*(α). We then obtain f(x*(α),α) = 0. Differentiating yields:

8 Some notes This is just an application of the implicit function theorem! This analysis only holds if f x does not equal zero. The partial derivatives are evaluated at the equilibrium point and so can be regarded as given numbers. Often, we are especially interested in the sign of the partial derivative, not so much in the exact value.

9 Example Suppose we have We then have

10 Example (from last year’s problem set!) In a given economy, a fixed amount of umbrellas is being traded among the citizens. Demand for umbrellas D is decreasing in price P. As the amount of expected rainfall R increases, people have a higher demand for umbrellas. D = D(P,R)D P 0 Supply of umbrellas S is increasing in P. As the amount of rainfall increase, people are less willing to sell their umbrella, hence supply is decreasing in R. S = S(P,R)S P > 0; S R < 0 How does an increase in rainfall affect the equilibrium price of umbrellas?

11 D = D(P,R)D P 0 S = S(P,R)S P > 0; S R < 0

12 What if we have several endogenous and exogenous variables? Suppose we have two of each. The solution will then be given by the conditions: We are now interested in signing four partial cross-derivatives. Rewrite the conditions as

13 What if we have several endogenous and exogenous variables? Differentiate through with respect to α 1 : or Applying Cramer’s rule: (section 9.4!)

14 What if we have constraints? The approach on the previous slide easily generalizes to more than two endogenous variables. Hence, we can also use it for constrained problems. The Lagrange multipliers are just additional endogenous variables.

15 Example Suppose we have a consumer problem: Applying the Lagrange method yields: Solving goes along the same lines as in the previous example. Take the derivative of all equalities with respect to an exogenous variable. Solve the system using Cramer’s rule.

16 For example, we can derive: But that implies

17 This is the Slutsky Equation First term: substitution effect. Negative. Second term: income effect. With a normal good, the total effect is negative. With a sufficiently inferior good, we may have a Giffen good, when the total effect becomes positive.

18 Definition 14.1: The general method of comparative statics. Given that the n equations can be solved to give equilibrium values of the x’s, we have that F ij is formed by replacing the jth column by

19 The Envelope Theorem First of all, consider the following maximization problem: Lagrange function: We now define the value function: the value of the objective function evaluated in the optimum: We can also write: Note:

20 But by construction, the optimal point has: So: Thus, when we want to look at the marginal effect of a change in the exogenous variable on the value of the Lagrangian, the total effect is equal to the partial effect. We only have to check how the Langrangean is directly affected by the exogenous variable. We do not have to take the effect via the equilibrium values of the endogenous variables into account.

21 Intuition At the optimum, if x changes marginally, then this will hardly have an effect on the objective function. In fact, there is only a second order effect.

22 Now consider the value function Substituting the FOCs: Note that we can write Hence: Thus:

23 This is the Envelope Theorem It says that in order to evaluate the effect of a small change of some exogenous variable on the objective function, we only have to look at the direct effect of that change on the Lagrangean.

24 Why is this called the Envelope Theorem!? Consider the firm’s long-run cost minimization problem: Exogenous variables: r, w, y. Endogenous variables: L, K. Value function: Consider the firm’s short-run cost minimization problem: Exogenous variables: r, w, y, K a. Endogenous variables: L. Value function: Thus

25 Thus, the long-run cost function is the lower envelope of short-run cost functions. At the margin when there is a small change in output, it matters little for total costs whether or not one is able to adjust the capital stock. Of course, this holds only at the margin.

26 Back to shadow prices Consider the problem What is the effect on the value function of a change in α? Just apply the envelope theorem: The Lagrangean is Hence, indeed, the Lagrange multiplier exactly gives the effect of loosening the constraint at the margin!

27 Example Consider the profit maximization problem of a firm: Exogenous variables: p, w, r. Endogenous variables: L, K, y. Value function: What is the effect of a change in exogenous variables on profit? This result is known as Hotelling’s Lemma.

28 Example Consider the standard consumer problem. We can write the value function as In the literature, this is known as the indirect utility function. It gives the amount of utility that a consumer ultimately obtains, as a function of prices and income. Thus, λ* reflects the marginal utility of income.

29 Also: Thus, a change in the price of a good has a negative effect on utility which is proportional to the amount of the good consumed. This is known as Roy’s identity.

30 We can also consider the dual of the consumer problem. Minimize expenditure to achieve some fixed level of utility: Value function: This is called the expenditure function. We can thus interpret the Lagrange multiplier as the marginal cost of utility.

31 Also: Thus, when the price of a good increases by one cent, then to maintain the same standard of living, the consumer requires approximately x i cents more income This is known as Shephard’s lemma.

32 This week’s exercises pg. 659: 1, 7. pg. 673: 1, 5. pg. 674: 3.