BEEM103 Optimization Techniques for Economists Level Curves - PowerPoint PPT Presentation

Functions in two variables Functions in two variables Partial derivatives Partial derivatives Optimization Optimization Unconstrained Optimization Unconstrained Optimization Second order conditions Second order conditions Functions in two variables 1 Example: The cubic polynomial Example: production function Example: profit function. BEEM103 Optimization Techniques for Economists Level Curves Multivariate Functions Isoquants Partial derivatives 2 A Basic Example Dieter Balkenborg Notation A Second Example Department of Economics, University of Exeter The Marginal Products of Labour and Capital Optimization 3 Week 2 Unconstrained Optimization 4 The first order conditions Example 1 Example 2: Maximizing profits Balkenborg Multivariate Functions Balkenborg Multivariate Functions Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Example 3: Price Discrimination Functions in two variables A function z = f ( x , y ) or simply z ( x , y ) in two independent variables with one dependent variable assigns to each pair ( x , y ) of (decimal) numbers from a certain domain D in the two-dimensional plane a number z = f ( x , y ) . x and y are hereby the independent variables z is the dependent variable . Second order conditions 5 Balkenborg Multivariate Functions Balkenborg Multivariate Functions

Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Outline Example: The cubic polynomial The graph of f is the surface in 3-dimensional space consisting of 1 Functions in two variables all points ( x , y , f ( x , y )) with ( x , y ) in D . Example: The cubic polynomial Example: production function z = f ( x , y ) = x 3 − 3 x 2 − y 2 Example: profit function. Level Curves Isoquants 2 Partial derivatives A Basic Example 5 Notation A Second Example The Marginal Products of Labour and Capital 2 0 1 3 z 2 0 1 0 3 -1 Optimization -1 -2 x y -5 4 Unconstrained Optimization The first order conditions Example 1 Example 2: Maximizing profits Example 3: Price Discrimination 5 Second order conditions Exercise: Evaluate z = f ( 2 , 1 ) , z = f ( 3 , 0 ) , z = f ( 4 , − 4 ) , z = f ( 4 , 4 ) Balkenborg Multivariate Functions Balkenborg Multivariate Functions Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Outline Example: production function √ √ 1 Functions in two variables 1 1 6 6 L Q = L = K K Example: The cubic polynomial 2 Example: production function Example: profit function. capital K ≥ 0, labour L ≥ 0, output Q ≥ 0 Level Curves Isoquants 2 Partial derivatives A Basic Example 6 Notation A Second Example 4 The Marginal Products of Labour and Capital z 2 3 Optimization 0 20 4 10 Unconstrained Optimization 20 15 10 The first order conditions 5 0 0 Example 1 y x Example 2: Maximizing profits Example 3: Price Discrimination 5 Second order conditions Balkenborg Multivariate Functions Balkenborg Multivariate Functions

Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Outline Example: profit function Assume that the firm is a price taker in the product market and in 1 Functions in two variables Example: The cubic polynomial both factor markets. Example: production function Example: profit function. Level Curves P is the price of output Isoquants 2 Partial derivatives A Basic Example Notation A Second Example The Marginal Products of Labour and Capital 3 Optimization 4 Unconstrained Optimization The first order conditions Example 1 Example 2: Maximizing profits Example 3: Price Discrimination 5 Second order conditions Balkenborg Multivariate Functions Balkenborg Multivariate Functions Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Example: profit function Example: profit function Assume that the firm is a price taker in the product market and in Assume that the firm is a price taker in the product market and in both factor markets. both factor markets. P is the price of output P is the price of output r the interest rate (= the price of capital) r the interest rate (= the price of capital) w the wage rate (= the price of labour) Balkenborg Multivariate Functions Balkenborg Multivariate Functions

Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Example: profit function P = 12, r = 1, w = 3: 1 1 2 − rK − wL 6 L Π ( K , L ) = PK Assume that the firm is a price taker in the product market and in 1 1 2 − K − 3 L 6 L = 12 K both factor markets. P is the price of output 20 r the interest rate (= the price of capital) 10 w the wage rate (= the price of labour) z total profit of this firm: 0 -10 Π ( K , L ) = TR − TC 0 = PQ − rK − wL 10 1 1 20 6 L 2 − rK − wL = PK 20 10 x 0 y Balkenborg Multivariate Functions Balkenborg Multivariate Functions Functions in two variables Functions in two variables Example: The cubic polynomial Example: The cubic polynomial Partial derivatives Partial derivatives Example: production function Example: production function Optimization Optimization Example: profit function. Example: profit function. Unconstrained Optimization Unconstrained Optimization Level Curves Level Curves Second order conditions Second order conditions Outline P = 12, r = 1, w = 3: 1 1 2 − rK − wL 6 L Π ( K , L ) = PK 1 Functions in two variables 1 1 2 − K − 3 L 6 L = Example: The cubic polynomial 12 K Example: production function Example: profit function. Level Curves Isoquants 20 2 Partial derivatives 10 A Basic Example Notation z A Second Example 0 The Marginal Products of Labour and Capital 3 Optimization -10 0 4 Unconstrained Optimization 10 The first order conditions Example 1 20 Example 2: Maximizing profits 20 10 x Example 3: Price Discrimination 0 y 5 Second order conditions Profits is maximized at K = L = 8. Balkenborg Multivariate Functions Balkenborg Multivariate Functions