EECM 3714
Lecture 8: Unit 8
Multivariate Optimisation I
Renshaw, Ch. 14, 15
04 April 2022
,OUTLINE
• Multivariate functions
• Partial derivatives
• First partial derivatives
• Second partial derivatives
• Production functions
• Utility functions
• The total differential and implicit differentiation
• Examples - Production and Utility functions
• First-order and second-order conditions
• Profit maximizing levels of K and L
• Profit maximization by price discriminating monopolist
• Profit maximization by a 2-product firm
• Cost minimisation, multi-plant firm
,MULTIVARIATE FUNCTIONS
• Multivariate functions have more than one independent variable
• Most functions in economics are multivariate functions
• Function with two independent variables: 𝑧 = 𝑓(𝑥, 𝑦)
• Function with 3 independent variables: 𝑧 = 𝑓(𝑥1 , 𝑥2 , 𝑥3 )
• Extension to n independent variables: 𝑧 = 𝑓(𝑥1 , 𝑥2 , … , 𝑥𝑛 )
• Again consider the function 𝑧 = 𝑓(𝑥, 𝑦)
• This function needs to be graphed on 3 axes in 3D space and the resulting graph is a surface
Fig 14.2)
• If all three variables are raised to the power 1, and if there are no cross-products, then
resulting graph is called a plane (e.g. Fig 14.3)
• If one of the variables is kept constant, one can take ”slices” through the surface. These slice
called sections (e.g. Fig 14.1, Fig 14.4)
• These sections are known as iso-sections (if z is constant: iso-𝑧 section, if 𝑥 constant, then
section, if y constant, then iso-y section)
• Can represent these functions with lines/curves in 2D space
, FIRST ORDER PARTIAL DERIVATIVES
• Suppose 𝑧 = 𝑓(𝑥, 𝑦), then:
𝜕𝑧
• = 𝑓𝑥 , slope of surface in direction of x (y is treated as a constant)
𝜕𝑥
𝜕𝑧
• = 𝑓𝑦 , slope of surface in direction of y (x is treated as a constant)
𝜕𝑦
𝜕𝑧
• Suppose that 𝑧 = 𝑔 𝑥1 , 𝑥2 , … , 𝑥𝑛 , then = 𝑓𝑘 , slope of surface in direction of 𝑥𝑘 (other 𝑥
𝜕𝑥𝑘
independent variables treated as constants)
• Note the change in notation:
𝑑𝑦
• For univariate functions, the derivative is
𝑑𝑥
𝜕𝑦
• For multivariate functions, the partial derivative is
𝜕𝑥
• When partially differentiating, all variables except the one that you are differentiating with
respect to are treated as constants (or kept constant)
Lecture 8: Unit 8
Multivariate Optimisation I
Renshaw, Ch. 14, 15
04 April 2022
,OUTLINE
• Multivariate functions
• Partial derivatives
• First partial derivatives
• Second partial derivatives
• Production functions
• Utility functions
• The total differential and implicit differentiation
• Examples - Production and Utility functions
• First-order and second-order conditions
• Profit maximizing levels of K and L
• Profit maximization by price discriminating monopolist
• Profit maximization by a 2-product firm
• Cost minimisation, multi-plant firm
,MULTIVARIATE FUNCTIONS
• Multivariate functions have more than one independent variable
• Most functions in economics are multivariate functions
• Function with two independent variables: 𝑧 = 𝑓(𝑥, 𝑦)
• Function with 3 independent variables: 𝑧 = 𝑓(𝑥1 , 𝑥2 , 𝑥3 )
• Extension to n independent variables: 𝑧 = 𝑓(𝑥1 , 𝑥2 , … , 𝑥𝑛 )
• Again consider the function 𝑧 = 𝑓(𝑥, 𝑦)
• This function needs to be graphed on 3 axes in 3D space and the resulting graph is a surface
Fig 14.2)
• If all three variables are raised to the power 1, and if there are no cross-products, then
resulting graph is called a plane (e.g. Fig 14.3)
• If one of the variables is kept constant, one can take ”slices” through the surface. These slice
called sections (e.g. Fig 14.1, Fig 14.4)
• These sections are known as iso-sections (if z is constant: iso-𝑧 section, if 𝑥 constant, then
section, if y constant, then iso-y section)
• Can represent these functions with lines/curves in 2D space
, FIRST ORDER PARTIAL DERIVATIVES
• Suppose 𝑧 = 𝑓(𝑥, 𝑦), then:
𝜕𝑧
• = 𝑓𝑥 , slope of surface in direction of x (y is treated as a constant)
𝜕𝑥
𝜕𝑧
• = 𝑓𝑦 , slope of surface in direction of y (x is treated as a constant)
𝜕𝑦
𝜕𝑧
• Suppose that 𝑧 = 𝑔 𝑥1 , 𝑥2 , … , 𝑥𝑛 , then = 𝑓𝑘 , slope of surface in direction of 𝑥𝑘 (other 𝑥
𝜕𝑥𝑘
independent variables treated as constants)
• Note the change in notation:
𝑑𝑦
• For univariate functions, the derivative is
𝑑𝑥
𝜕𝑦
• For multivariate functions, the partial derivative is
𝜕𝑥
• When partially differentiating, all variables except the one that you are differentiating with
respect to are treated as constants (or kept constant)
