Comparative statics and first-order cond

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Published on November 5, 2008

Author: dstallings

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Mathematics for Economists : Mathematics for Economists Lecture 8: Implicit functions, comparative statics, introduction to optimization A Cobb-Douglas Function and its isoquants : 2 A Cobb-Douglas Function and its isoquants Use the implicit function theorem to define an equation for the slope of this isoquant. First, take the partial derivatives with respect to L and K. We can now find the derivative of K with respect to L according to the implicit function theorem: A Cobb-Douglas Function and its isoquantsIn-class problem 1, part a : 3 A Cobb-Douglas Function and its isoquantsIn-class problem 1, part a When K = 6, and L = 2, what is the slope of the isoquant? When K = 3, and L = 14, what is the slope of the isoquant? A Cobb-Douglas Function and its isoquantsIn-class problem 1, part b, marginal rate of technical substitution : 4 A Cobb-Douglas Function and its isoquantsIn-class problem 1, part b, marginal rate of technical substitution The marginal rate of technical substitution (MRTS) is the negative of the slope: When K = 6, and L = 2, what is the MRTS? When K = 3, and L = 14, what is the MRTS? An isoquant and the MRTS : 5 An isoquant and the MRTS K L A general production function and its isoquants : 6 A general production function and its isoquants For a production function written: The level curve given by Is defined formally by the set: Which represents the input combinations which will generate the same level of output The total differential of y is: Since then and Is the slope of the isoquant, the negative of which is the marginal rate of technical substitution. A general utility function and its indifference curves : 7 A general utility function and its indifference curves For a utility function written: The level curve given by Is defined formally by the set: Which represents the consumption combinations which will generate the same level of utility The total differential of u is: Since then and Is the slope of the indifference curve, the negative of which is the marginal rate of substitution. A feature of utility functions : 8 A feature of utility functions Consider the indifference curves below: A B1 B2 C What does this formally mean? : 9 What does this formally mean? For a utility function that describes a preference ordering, such as: Then so does any positive monotonic transformation of that function: If accurately reflects a preference ordering, then so does: An example of a monotonic transformation : 10 An example of a monotonic transformation Suppose we have the utility function Then The slope of the indifference curve is Take the monotonic transformation: The slope of the indifference curve is The same! General Functions : General Functions Comparative static analysis A linear market model : 12 A linear market model Suppose we have linear supply and demand functions which appear as: Where y is income. Equilibrium implies: How does a change in income affect the equilibrium price? The higher is c, the larger the effect on price; the higher are b and , the lower the effect. A linear market model, implicit case : 13 A linear market model, implicit case We can also note that: In equilibrium: As a total differential A general market model : 14 A general market model Suppose we have supply and demand functions which appear as: Where y is income. In equilibrium: As a total differential The change in price as a result of a change in income is therefore: What does this tell us about normal goods, inferior goods and Giffen goods? Two endogenous and two exogenous variables : 15 Two endogenous and two exogenous variables Suppose we have two functions with endogenous variables xi and exogenous variables yi: Assume that we can solve for xi as differentiable functions of yi: We are now in a position to see the effect of a change of, say, y1 on the endogenous variables. The total differential : 16 The total differential With respect to y1, the total differentials appear as: We can then write this as a linear system: The determinant to solve the system. Back to Cramer : 17 Back to Cramer Applying Cramer’s rule: What do we need to know to assign the proper signs? The IS curve in the IS-LM model : 18 The IS curve in the IS-LM model The aggregate expenditure function, a function of aggregate income Y and the interest rate R appears as: In equilibrium: Differentiating: The LM curve in the IS-LM model : 19 The LM curve in the IS-LM model The demand for money is a function of aggregate income Y and the interest rate R appears as: In equilibrium: Differentiating: General equilibrium in the IS-LM model : 20 General equilibrium in the IS-LM model The IS and LM curve together determine overall equilibrium values of Y and R: The comparative statics model asks us to determine how changes in the exogenous variables affect the equilibrium values of Y and R. Take the total differential of the IS-LM model The total differential of the IS-LM model : 21 The comparative statics model asks us to determine how changes in the exogenous variables affect the equilibrium values of Y and R. Make a 1-unit change in , assuming that The total differential of the IS-LM model Put this into matrix form: The changing equilibrium in the IS-LM model : 22 The changing equilibrium in the IS-LM model The determinant of the coefficient matrix: And its sign is ________________ Using Cramer’s rule And its sign is ________________ And its sign is ________________ negative positive positive An import-export model, from before : 23 23 An import-export model, from before Consider the import-export model Where X represents exports, M represents imports, E is the exchange rate, YD is domestic income, and YF is foreign income. The endogenous variables are X, M, and E. The exogenous variables are YD and YF. Put the model in implicit function form: An Import-Export ModelMatrix form and determinant of the coefficient matrix : 24 An Import-Export ModelMatrix form and determinant of the coefficient matrix Write as a matrix system: The determinant of the coefficient matrix: An Import-Export ModelSolve for changes using Cramer’s rule : 25 An Import-Export ModelSolve for changes using Cramer’s rule Does it work? Changes in the solution to an import-export model, the easy way : 26 26 Changes in the solution to an import-export model, the easy way The model in implicit function form: The total differential: Changes in the solution to an import-export model, the easy way, in matrix form : 27 27 Changes in the solution to an import-export model, the easy way, in matrix form The total differential in matrix form: Looking at a change in foreign income: Changes in the solution to an import-export model, the easy way, the solution using Cramer’s rule : 28 28 Changes in the solution to an import-export model, the easy way, the solution using Cramer’s rule A Linear IS-LM modelIn-class problem 2 (a) : 29 A Linear IS-LM modelIn-class problem 2 (a) The following is a linear IS-LM model A Linear IS-LM modelIn-class problem 2 (b and the first part of c) : 30 A Linear IS-LM modelIn-class problem 2 (b and the first part of c) Take the total differential: Place in matrix form: Hold M constant and restate the matrix form: A Linear IS-LM modelIn-class problem 2 (continuation of c) : 31 A Linear IS-LM modelIn-class problem 2 (continuation of c) Take the determinant of the coefficient matrix: Use Cramer’s rule to give us the comparative static results for a change in E: Optimization : Optimization Equilibrium by extremes Extreme values : 33 Extreme values What is the basic goal of the business firm? Consumers? Policymakers? The extreme value of a univariate function can occur within the interval over which it is defined, or at its endpoints. Monotonic functions do not have extreme values, except at their endpoints. We will therefore be looking at non-monotonic, differentiable functions. The tax revenue function : 34 The tax revenue function Remember the tax revenue function that we had from earlier: Where 0 < t < 1. Finding the maximum : 35 Finding the maximum Take the derivative of R with respect to t: The tax revenue function reaches a stationary point where We can then find maximum revenue by going back to the original function: First-order condition : 36 First-order condition The location of a stationary point by the use of a first derivative is called the first-order condition. A stationary point can be a maximum, a minimum, or an inflection point. In addition, we can define global extremes, or local extremes. Many examples follow. Global maximum : 37 Global maximum Take the derivative: Consider the function: Set the derivative equal to zero and solve: How do we know we have a global extreme? How do we know whether we have a maximum or minimum? : 38 How do we know whether we have a maximum or minimum? What does the value of h’(x) represent at each point, x0 and h(x0)? What does a positive value of h’(x) represent? What does a negative value of h’(x) represent? Thus, if x0 is less than its stationary value, and h’(x0) is greater than 0, then h(x) is increasing. If x0 is greater than its stationary value, and h’(x0) is less than 0, then h(x) is decreasing. If the two statements are true, the stationary value is a maximum. Global minimum: in-class problem 3 : 39 Global minimum: in-class problem 3 Take the derivative: Consider the function: Set the derivative equal to zero and solve: How do we know whether we have a maximum or minimum? More than one extreme : 40 More than one extreme Take the derivative: Consider the function: Set the derivative equal to zero and solve: Local maximum and minimum : 41 Local maximum and minimum It is easy to see that, for the function on the previous slide, that the extreme points, where are not global maxima and minima. At x = -1, the value of j(x) is at a maximum from - until x reaches 3.5, so this point represents a local maximum. Similarly, at x = 2, the value of j(x) is at a minimum from -2.5 to , so this point represents a local minimum. Neither maximum or minimum : 42 Neither maximum or minimum Take the derivative: Set the derivative equal to zero and solve: Test for maximum or minimum. So we are in the running for a maximum, until: Another general function exampleExercise 9.2 problem 4 : 43 Another general function exampleExercise 9.2 problem 4 Finding the marginal function M and the average function A: Show that, when A is at an extremum, M=A. What does this imply about average and marginal diagrams? What is the elasticity of T when A is at an extremum? The limits of the first-order condition : 44 The limits of the first-order condition Identifies only a stationary point. The stationary point may not be an extreme point. If the stationary point is an extreme point, the first-order condition: Does not, by itself, distinguish between maxima and minima. Does not distinguish between global and local maxima and minima. Requires us to evaluate the first derivative several times to determine maxima or minima. Next week : 45 Next week Last lecture More optimization problems. This is the time to ask those burning questions. Final exam given out

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