📖 Functions of several variables

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References and additional materials

[Sydsæter, Hammond, Strøm, and Carvajal, 2016]

Chapters

WARNING

This section of the lecture notes is still under construction. It will be ready before the lecture.

We want to extend our discussion of differential calculus from single-real-valued univariate functions, \(f(x_1)\), to single-real-valued multivariate functions, \(f(x_1,x_2,\dots,x_n)\).

• functions of two and more variables

• level curves and sets

• partial derivatives

• rules for partial derivatives

• composition of functions and chain rule

• implicit functions and differentiation

• homogeneous functions

• homothetic functions

• examples, examples, economic applications

\[ \huge{(x_1,x_2,\dots,x_n)} \]

\[ \huge{\downarrow} \]

\[ \huge{\downarrow} \]

\[ \huge{f(x) \in \mathbb{R}} \]

Functions of two variables

Let’s have a look at some functions of two variables

• How to visualize them

• Slope, contours, etc.

Example: Cobb-Douglas production function

Consider production function

\[\begin{split} f(k, \ell) = k^{\alpha} \ell^{\beta}\\ \alpha \ge 0, \, \beta \ge 0, \, \alpha + \beta < 1 \end{split}\]

Let’s graph it in two dimensions.

Fig. 41 Production function with \(\alpha=0.4\), \(\beta=0.5\) (a)

Fig. 42 Production function with \(\alpha=0.4\), \(\beta=0.5\) (b)

Fig. 43 Production function with \(\alpha=0.4\), \(\beta=0.5\) (c)

Like many 3D plots it’s hard to get a good understanding

Let’s try again with contours plus heat map

Fig. 44 Production function with \(\alpha=0.4\), \(\beta=0.5\), contours

In this context the contour lines are called isoquants

Can you see how \(\alpha < \beta\) shows up in the slope of the contours?

We can drop the colours to see the numbers more clearly

Fig. 45 Production function with \(\alpha=0.4\), \(\beta=0.5\)

Example: log-utility

Let \(u(x_1,x_2)\) be “utility” gained from \(x_1\) units of good 1 and \(x_2\) units of good 2

We take

\[ u(x_1, x_2) = \alpha \log(x_1) + \beta \log(x_2) \]

where

• \(\alpha\) and \(\beta\) are parameters

• we assume \(\alpha>0, \, \beta > 0\)

• The log functions mean “diminishing returns” in each good

Fig. 46 Log utility with \(\alpha=0.4\), \(\beta=0.5\)

Let’s look at the contour lines

For utility functions, contour lines called indifference curves

Fig. 47 Indifference curves of log utility with \(\alpha=0.4\), \(\beta=0.5\)

Example: quasi-linear utility

\[ u(x_1, x_2) = x_1 + \log(x_2) \]

• Called quasi-linear because linear in good 1

Fig. 48 Quasi-linear utility

Fig. 49 Indifference curves of quasi-linear utility

Example: quadratic utility

\[ u(x_1, x_2) = - (x_1 - b_1)^2 - (x_2 - b_2)^2 \]

Here

• \(b_1\) is a “satiation” or “bliss” point for \(x_1\)

• \(b_2\) is a “satiation” or “bliss” point for \(x_2\)

Dissatisfaction increases with deviations from the bliss points

Fig. 50 Quadratic utility with \(b_1 = 3\) and \(b_2 = 2\)

Fig. 51 Indifference curves quadratic utility with \(b_1 = 3\) and \(b_2 = 2\)

Notation

To simplify notation we will sometimes denote vectors with bold letters, e.g. \(\mathbf{x}=(x_1,x_2,\dots,x_n)\). Hence, we can write our multivariate function as a function of a vector input, \(f(\mathbf{x})=f(x_1,x_2,\dots,x_n)\).

Single-real-valued multivariate functions

Definition

A single-real-valued multivariate function takes as input \(n\) real numbers and outputs a single real number,

\[ f: X \subseteq \mathbb{R}^{n} \rightarrow Y \subseteq \mathbb{R}. \]

Recall that:

• \(X\) is the domain of \(f\).

• \(Y\) is the target space of \(f\).

Often we simply write a single-real-valued multivariate functions as

\[ y = f(x_1,x_2,\dots,x_n), \]

where \(y\) is refered to as the dependent variable, and \(x_i\) is referred to as an independent variable. Alternatively, \(y\) and \(x_i\) can be referred to as the endogenous and exogenous variables, or the output and input variables.

Example

Consider the single-real-valued multivariate function \(f(x_1,x_2) = \sqrt{x_1 - 1} + \sqrt{x_2 - 2}\).

• the domain of \(f\) is \(\left\{ (x_1,x_2) \in \mathbb{R}^{2} | x_1 \geq 1, x_2 \geq 2 \right\}\).

• the target of \(f\) is \(\left\{ y \in \mathbb{R} | y \geq 0 \right\}\).

Fig. 52 Illustrations of domain and target of \(f\).

In economic analysis we will often use single-real-valued multivariate functions to represent

• production functions

• cost functions

• profit functions

• utility functions

• demand functions

Example

Important single-real-valued bivariate functions used in economic analysis:

• Linear function

\[f(\mathbf{x}) = a_{1} x_{1} + a_{2} x_{2}.\]

• Input-output function

\[f(\mathbf{x}) = \text{min}(a_1 x_1, a_2 x_2).\]

• Cobb-Douglas function

\[f(\mathbf{x}) = k x_{1}^{b_1} x_{2}^{b_2}.\]

• Constant elasticity of substitution (CES) function,

\[f(\mathbf{x}) = k (c_1 x_1^{a} + c_2 x_2^{a})^{b/a}.\]

These functions are straight forward to extent to more than two inputs.

Partial derivatives

Our primary goal in economic analysis is to understand how a change in one independent variable affect the dependent variable. By using partial derivatives we take the simplest approach by changing one variable at a time, keeping all other constant.

Definition

The partial derivative of the single-real-valued multivariate function \(f\) with respect to the variable \(x_{k}\) is defined as

\[ \begin{aligned} \frac{\partial f\left(\mathbf{x}\right)}{\partial x_{k}} = \lim _{h \rightarrow 0}\left\{\frac{f\left(x_{1}, \dots, x_{k} + h, x_{k+1}, \dots, x_n\right) - f\left(x_{1}, \dots, x_{k}, x_{k+1}, \dots, x_n\right)}{h}\right\} . \end{aligned} \]

Fig. 53 Illustration of the partial derivative with respect to \(x_k\).

Note that the partial derivative, \(\partial f\left(\mathbf{x}\right)/\partial x_k\), is a single-real-valued multivariate function itself, as it in general depend on all the independent variables.

As a matter of convenience partial derivatives are often denoted by \(f_{k}\left(\mathbf{x}\right)\) or \(f'_{k}\left(\mathbf{x}\right)\).

Definition

The gradient is defined as the vector of first-order partial derivatives of \(f\)

\[ \nabla f(\mathbf{x}) = \bigg(\tfrac{\partial f(\mathbf{x})}{\partial x_1}, \tfrac{\partial f(\mathbf{x})}{\partial x_1}, \dots, \tfrac{\partial f(\mathbf{x})}{\partial x_n} \bigg)^T \]

The gradient is commonly denoted by one of \(\nabla f(\mathbf{x}), D_{x} f(\mathbf{x})\), or \(\operatorname{grad} f(\mathbf{x})\).

Note that when calculating the partial derivative with respect \(x_k\) we treat the remaining independent variables as constants. Hence, we can use the same rules for partial differentiation as for differentiation of univariate functions.

Fact: Rules for partial differentiation

• \(f(\mathbf{x}) = c g(\mathbf{x}) \Rightarrow f_k\left( \mathbf{x} \right) = c g_k\left( \mathbf{x} \right)\).

• \(f(\mathbf{x}) = g(\mathbf{x}) + h(\mathbf{x}) \Rightarrow f_k\left( \mathbf{x} \right) = g_k\left( \mathbf{x} \right) + h_k\left( \mathbf{x} \right)\).

• \(f(\mathbf{x}) = g(\mathbf{x})h(\mathbf{x}) \Rightarrow f_k\left( \mathbf{x} \right) = g_k(\mathbf{x})h(\mathbf{x}) + g(\mathbf{x})h_k(\mathbf{x})\).

• \(f(\mathbf{x}) = g\left(h\left(\mathbf{x}\right)\right) \Rightarrow f_k\left( \mathbf{x} \right) = g_h\left(h\left(\mathbf{x}\right)\right)h_k\left(\mathbf{x}\right)\).

Example I

Consider the bivariate function \(y=f(x_1,x_2)=a x_1 + b x_2\). The partial derivative with respect to \(x_1\) is then

The partial derivative with respect to \(x_1\) is then

\[ f_1(x_1,x_2)=a. \]

The partial derivative with respect to \(x_2\) is then

\[ f_1(x_1,x_2)=b. \]

Example II

Consider the bivariate function \(y=f(x_1,x_2)=x_1 x_2\).

The partial derivative with respect to \(x_1\) is then

\[ f_1(x_1,x_2)=x_2. \]

The partial derivative with respect to \(x_2\) is then

\[ f_2(x_1,x_2)=x_1. \]

Definition

Some terminologi:

• If \(\partial f\left(\mathbf{x}\right)/\partial x_k\) exist for each \(k\), we say that \(f\) is differentiable at \(\mathbf{x}\).

• If these \(n\) partial derivative functions are continuous, we say that \(f\) is continously differentiable at \(\mathbf{x}\). This is denoted by \(f \in C^{1}\) on \(X\).

Implicit functions in many variables

Until now we have been working with explicit functions where the dependet variable, \(y\), is on the left side of the equation and the independet variables, \(x_i\), are on the right side

\[ y = f(x_1,x_2,\dots,x_n). \]

Frequently, we have to work with equations where the dependent variable cannot be separated from the independent variables

\[ G(x_1,x_2,\dots,x_n,y) = c. \]

We say that \(G(x_1,x_2,\dots,x_n,y) = c\) represent a relationship between \(y\) and \(x\) that defines an implicit function \(y\) of \(x\).

When dealing with implicit functions we want to know the following two answers:

• Does \(G(x_1,x_2,\dots,x_n,y) = c\) determine \(y\) as a continuous implicit function of \((x_1,x_2,\dots,x_n)\)?

• If so, how does changes in \(x\) affect \(y\)?

Example: The line of the cicle

The unit cicle has the representation

\[ x^2 + y^2 = 1. \]

Consider the point \(x=0\), \(y=1\). In the neighborhood of this point \(y\) is an explicit function of \(x\)

\[ y(x) = \sqrt{x^2 - 1}. \]

\(y(x)\) can be represented by a different explicit function around the neighborhood at the point \(x=0\), \(y=-1\)

\[ y(x) = -\sqrt{x^2 - 1}. \]

In contrast, \(y\) cannot be represented by a well defined function around the points \((-1,0)\) and \((1,0)\).

Example: Polynomial equation of order five

Consider the polynomial equation

\[ y^5 + y^3 + y + x = 0. \]

\(x\) can be expressed as an explicit function of \(y\)

\[ x(y) = - y^5 - y^3 - y. \]

However, polynomials of order \(k \geq 5\) has no explicit solution. Hence, there exist no explicit function \(y\) of \(x\). Instead, the polynomial equation defines an implicit function for \(y\) of \(x\).

Suppose we could find a function \(y=y(x)\) which solves this equation for any \(x\)

\[ y(x)^5 + y(x)^3 + y(x) + x = 0. \]

Use the chain rule to find \(y'(x)\)

\[\begin{split} 5 y(x)^4 y'(x) + 3 y(x)^2 y'(x) + y'(x) + 1 = 0 \\ \Leftrightarrow y'(x) =-\frac{1}{5 y(x)^4 + 3 y(x)^2 + 1} \end{split}\]

The point \(x=-3\), \(y=1\) solves the polynomial equation, and

\[ y'(3)=-\tfrac{1}{5 + 3 + 1} = -\tfrac{1}{8} \]

We conclude that there exists an implicit function \(y=y(x)\) in the neighborhood around (-3,1) that solves polynomial equation and it is differentiable.

Now, let’s carry this computation out more generally for the implicit function \(G(x,y)=c\) around the specific point \(x=x^*\), \(y=y^*\), and let’s suppose there exist a solution \(y=y(x)\). Use the chain rule to differentiate wrt \(x\)

\[ \frac{\partial G\left(x^*, y(x^*)\right)}{\partial x} + \frac{\partial G\left(x^*, y(x^*)\right)}{\partial y} y'(x)=0. \]

Solving for \(y'(x)\) yields

\[ y'(x) =-\frac{\tfrac{\partial G\left(x^*, y(x^*)\right)}{\partial x} }{\tfrac{\partial G\left(x^*, y(x^*)\right)}{\partial y}}. \]

We see that if the solution \(y(x)\) of \(G(x,y)=c\) exists and is differentiable it is necessary that \(\partial G\left(x_0, y(x_0)\right) /\partial y\) be nonezero. The implicit function theorem stated below implies that this necessary condition is also a sufficient condition.

Fact (Implicit function theorem I)

Let \(G\left(x, y\right)\) be a \(C^1\) function around the point \((x^*,y^*)\) in \(\mathbb{R}^2\). Suppose \(G(x^*,y^*)=c\) and consider the expression

\[ G(x,y)=c. \]

If \(\partial G\left(x^*, y(x^*)\right) / \partial y \neq 0\), then there exists a \(C^1\) function \(y=y(x)\) defined in neighborhood around the point \(x^*\) such that

• \(G(x,y(x)) = c\) for all \(x\) around \((x^*,y^*)\)

• \(y(x^*)=y^*\)

• the derivative of \(y\) wrt \(x\) at \((x^*,y^*)\) is

\[ y'(x) =-\frac{\tfrac{\partial G\left(x^*, y(x^*)\right)}{\partial x} }{\tfrac{\partial G\left(x^*, y(x^*)\right)}{\partial y}}. \]

The implicit function theorem provides conditions under which a relationship of the form \(G(x,y)=c\) implies that there exists a implicit function \(y=y(x)\) locally.

Example: The line of the cicle

When the implicit function theorem is applied to the line of the circle we get that

\[\begin{split} y'(x) &= -\frac{\tfrac{\partial y^2 + x^2}{\partial x}}{\tfrac{\partial y^2 + x^2}{\partial y}} \\ &= -\frac{2x}{2y} \\ &= -\frac{x}{y} \end{split}\]

which is not defined in the point (-1,0) and (1,0) .

The implicit function theoriem can be extended to \(G(x_1,x_2,\dots,x_n)=c\).

Fact (Implicit function theorem II)

Let \(\mathbf{x}=(x_1,x_2,\dots,x_n)\) and \(G\left(\mathbf{x}, y\right)\) be a \(C^1\) function around the point \((\mathbf{x}^*,y^*)\) in \(\mathbb{R}^{n+1}\). Suppose \(G(\mathbf{x}^*,y^*)=c\) and consider the expression

\[ G(\mathbf{x},y)=c. \]

If \(\partial G\left(\mathbf{x}^*, y(\mathbf{x}^*)\right) / \partial y \neq 0\), then there exists a \(C^1\) function \(y=y(\mathbf{x})\) defined in the neighborhood around the point \(\mathbf{x}^*\) such that

• \(G(\mathbf{x},y(\mathbf{x})) = c\) for all \(\mathbf{x}\) around \((\mathbf{x}^*,y^*)\)

• \(y(\mathbf{x}^*)=y^*\)

• for each \(i\) the derivative of \(y\) wrt \(x_i\) is

\[ \frac{\partial y(\mathbf{x}^*)}{\partial x_i} = -\frac{\tfrac{\partial G\left(\mathbf{x}^*,y^*\right)}{\partial x_i} }{\tfrac{\partial G\left(\mathbf{x}^*,y^*\right)}{\partial y}}. \]

Example: \(G(x,y,z)=c\)

Consider the relationship

\[ - 3 x^2 y + y z - 4 x z = 7. \]

We will show that near \((-1,1,2)\) we can write \(y=y(x,z)\) and we will find the partial \(\partial y / \partial x\) to that point

\[ - 3 \cdot (-1)^2 \cdot 1 + 1 \cdot 2 - 4 \cdot (-1) \cdot 2 = - 3 + 2 + 8 = 7 \]

Find the partial derivatives of \(G\) with respect to \(y\) and \(x\)

\[\begin{split} \frac{\partial G}{\partial x} &= -6xy - 4z. \\ \frac{\partial G}{\partial y} &= -3x^2 + z. \end{split}\]

The partial derivative is given by the implicit function theorem as

\[ \frac{\partial y}{\partial x} = -\frac{\tfrac{\partial G}{\partial x} }{\tfrac{\partial G}{\partial y}} = -\frac{-6xy - 4z}{-3x^2 + z} \]

We conclude that there exists a \(C^1\) function \(y=y(x,z)\) in the neighborhood around the point (-1,1,2).

Level sets

Level curves can be used to visualize single-real-valued bivariate functions, \(f: X \subset \mathbb{R}^2 \rightarrow Y \subset \mathbb{R}\).

Fact (Level curve)

A curve \(\ell\) in \(\mathbb{R}^2\) is called a level curve of \(y=f(x_1,x_2)\) if the value of \(f\) on every point of \(\ell\) is some fixed \(c\).

Example: Level curve

\(x^2 + y^2 = c\) is the level curve of \(z=x^2 + y^2\).

Similarly, level surfaces can be used to visualize single-real-valued trivariate functions, \(f: X \subset \mathbb{R}^3 \rightarrow Y \subset \mathbb{R}\).

Fact (Level surface)

A surface \(S\) in \(\mathbb{R}^3\) is called a level surface of \(f(x_1,x_2,x_3)\) if the value of \(f\) on every point of \(S\) is some fixed \(c\).

Example: Level surface

\(x^2 + y^2 - z = c\) is the level surface of \(w=x^2 + y^2 - z\).

This can of course be generalized to higher order functions, \(f: X \subset \mathbb{R}^n \rightarrow Y \subset \mathbb{R}\)

Fact (Level sets)

A manifold \(M\) in \(\mathbb{R}^n\) is called a level set of \(f(x_1,x_2,\dots,x_n)\) if the value of \(f\) on every point of \(M\) is some fixed \(c\).

The slope of an isoquant curve

Suppose that \(f: \mathbb{R}_{+}^{2} \rightarrow \mathbb{R}_{+}\) is a production function.

\[ Q = f(K,L). \]

The level curve of the production function represent a relationship between the capital (K) and labor (L) inputs that keeps the production constant at \(c\)

\[ f(K,L) = c. \]

Given the specified production function we might be able to find an explicit or implicit function for the isoquant curve, \(L=L(K)\). In both cases we can use the implicit function theorem to calculate the slope of the isoquant curve.

\[\begin{split} \frac{\partial f(K,L)}{\partial K} + \frac{\partial f(K,L)}{\partial L} \frac{\partial L(K)}{\partial K} = 0 \\ \Leftrightarrow \frac{\partial L(K)}{\partial K} = -\frac{\tfrac{\partial f(K,L)}{\partial K}}{\tfrac{\partial f(K,L)}{\partial L}} \end{split}\]

The slope of the isoquant curve is referred to as the marginal rate of technical substitution

\[ \text{MRTS}_{LK} = -\frac{\tfrac{\partial f(K,L)}{\partial K}}{\tfrac{\partial f(K,L)}{\partial L}}. \]

It measures how much of one input would be needed to compensate for a one-unit loss of the other while keeping the production at the same level.

Marginal rate of technical substitution between input \(x_i\) and \(x_j\) of a multivariate function is defined as

\[ \text{MRTS}_{x_i,x_j} = -\frac{\tfrac{\partial f(x_1,x_2,\dots,x_n)}{\partial x_j}}{\tfrac{\partial f(x_1,x_2,\dots,x_n)}{\partial x_i}}. \]

Example: \(f(K,L,T)=c\)

Consider the level surface of the Cobb-Douglas function of three inputs (capital, labor, and land)

\[ A C^{\alpha} L^{\beta} T^{\delta} = c \]

Calculate the marginal rate of substitution between capital and labor

\[\begin{split} \alpha A C^{\alpha-1} L^{\beta} T^{\delta} \frac{\partial C(L,T)}{\partial L} + \beta A C^{\alpha} L^{\beta-1} T^{\delta} = 0 \\ \Leftrightarrow \frac{\partial C(L,T)}{\partial L} = -\frac{\beta A C^{\alpha} L^{\beta-1} T^{\delta}}{\alpha A C^{\alpha-1} L^{\beta} T^{\delta}} = \frac{\beta}{\alpha} \frac{C}{L} \end{split}\]

Calculate the marginal rate of substitution between capital and land

\[\begin{split} \alpha A C^{\alpha-1} L^{\beta} T^{\delta} \frac{\partial C(L,T)}{\partial T} + \delta A C^{\alpha} L^{\beta} T^{\delta-1} = 0 \\ \Leftrightarrow \frac{\partial C(L,T)}{\partial L} = -\frac{\delta A C^{\alpha} L^{\beta} T^{\delta-1}}{\alpha A C^{\alpha-1} L^{\beta} T^{\delta}} = \frac{\delta}{\alpha} \frac{C}{T} \end{split}\]

Calculate the marginal rate of substitution between labor and land

\[\begin{split} \beta A C^{\alpha} L^{\beta-1} T^{\delta}\frac{\partial L(C,T)}{\partial T} + \delta A C^{\alpha} L^{\beta} T^{\delta-1} = 0 \\ \Leftrightarrow \frac{\partial L(C,T)}{\partial T} = -\frac{\delta A C^{\alpha} L^{\beta} T^{\delta-1}}{\beta A C^{\alpha} L^{\beta-1} T^{\delta}} = \frac{\delta}{\beta} \frac{L}{T} \end{split}\]

Homogeneous functions

Consider a function \(f: S \longrightarrow \mathbb{R}\), where \(S \subseteq \mathbb{R}^{n}\). Let \(\lambda>0\) be a positive real number.

Definition

The function \(f\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is said to be homogeneous of degree \(r\) if

\[ f\left(\lambda x_{1}, \lambda x_{2}, \cdots, \lambda x_{n}\right)=\lambda^{r} f\left(x_{1}, x_{2}, \cdots, x_{n}\right) \]

for all \(\left(x_{1}, x_{2}, \cdots, x_{n}\right) \in S\) and all \(\lambda>0\).

Example

The function

\[ 3x^5 y + 2x^2 y^4 - 3 x^3 y^3, \]

is homogenous of degree 6

\[\begin{split} 3(\lambda x)^5 (\lambda y) + 2(\lambda x)^2 (\lambda y)^4 - 3 (\lambda x)^3 (\lambda y)^3 &= 3\lambda^5 x^5 \lambda y + 2\lambda^2 x^2 \lambda^4 y^4 - 3 \lambda^3 x^3 \lambda^3 y^3 \\ &= 3\lambda^6 x^5 y + 2\lambda^6 x^2 y^4 - 3 \lambda^6 x^3 y^3 \\ &= \lambda^6 \left(3 x^5 y + 2 x^2 y^4 - 3 x^3 y^3 \right). \end{split}\]

The function

\[ 3x^{3/4} y^{1/4} + 6x + 4, \]

is not homogenous

\[ 3(\lambda x)^{3/4} (\lambda y)^{1/4} + 6(\lambda x) + 4 = 3 \lambda x^{3/4} y^{1/4} + 6\lambda x + 4. \]

Definition

A function that is homogeneous of degree one is said to be linearly homogeneous.

Definition

Let \(z=f(x)\) be a \(C^1\) function. If \(f\) is homogenous of degree \(r\), its first order partial derivatives are homogenous of degree \(r-1\).

By definition

\[ f(\lambda x_1, \lambda x_2, \dots, \lambda x_n) = \lambda^r f(x_1,x_2,\dots,x_n). \]

Use the chain rule wrt x_i on both sides of the equation

\[ \frac{\partial f(\lambda x_1, \lambda x_2, \dots, \lambda x_n)}{\partial x_i} \lambda = \lambda^r \frac{\partial f(x_1,x_2,\dots,x_n)}{\partial x_i}, \]

and divide both sides by \(\lambda\)

\[ \frac{\partial f(\lambda x_1, \lambda x_2, \dots, \lambda x_n)}{\partial x_i} = \lambda^{r-1} \frac{\partial f(x_1,x_2,\dots,x_n)}{\partial x_i}. \]

Fact

Let \(q=f(x)\) be a \(C^1\) homogenous function on \(\mathbb{R}^n_+\). The tangent planes to the level sets of \(f\) have constant slope along each ray from the origin.

Consider the homogenous production function on \(\mathbb{R}^2_+\). We want to show that the MRTS is constant along rays from the origin. Let \((\lambda L_0,\lambda K_0)=(L_1, K_1)\). The MRTS between input \(L\) and \(K\) at \((L_1, K_1)\) equals

\[\begin{split} \frac{f_{K}(L_1, K_1)}{f_{L}(L_1, K_1)} &= \frac{f_{K}(\lambda L_0,\lambda K_0)}{f_{L}(\lambda L_0,\lambda K_0)} \\ &= \frac{\lambda^{r-1} f_{K}( L_0, K_0)}{\lambda^{r-1} f_{L}( L_0, K_0)} \\ &= \frac{f_{K}(L_0, K_0)}{f_{L}( L_0, K_0)} \\ \end{split}\]

Euler’s theorem

Fact (Euler’s theorem)

Suppose that the function \(f\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is homogeneous of degree \(r\). In this case we have

\[ \sum_{i=1}^{n} x_{i}\frac{\partial f}{\partial x_{i}}\left(\mathbf{x} \right)=r f\left(\mathbf{x}\right). \]

Recall the definition of a homogenous function of degree \(r\)

\[ f(\lambda \mathbf{x}) = \lambda^{r} f(\mathbf{x}). \]

Differentiate both sides wrt \(\lambda\)

\[ \sum_{i=1}^{n} x_{i}\frac{\partial f}{\partial x_{i}}\left(\lambda \mathbf{x} \right) = r \lambda^{r-1} f(\mathbf{x}). \]

Set \(\lambda=1\).

Returns to scale

Suppose that an \(n\) input and one output production technology can be represented by a homogenous production function of the form

\[ Q=f\left(L_{1}, L_{2}, \cdots, L_{n}\right) . \]

The production technology is said to display:

• constant returns to scale: \(f\left(\lambda L_{1}, \lambda L_{2}, \cdots, \lambda L_{n}\right)=\lambda Q\).

• decreasing returns to scale: \(f\left(\lambda L_{1}, \lambda L_{2}, \cdots, \lambda L_{n}\right)<\lambda Q\).

• increasing returns to scale: \(f\left(\lambda L_{1}, \lambda L_{2}, \cdots, \lambda L_{n}\right)>\lambda Q\).

Cobb-Douglas production functions

Suppose that an two input and one output production technology can be represented by a production function of the form

\[ Q=f(L, K)=A L^{\alpha} K^{\beta} . \]

This type of production function is known as a Cobb-Douglas production function.

Suppose that \(\lambda>0\). Note that

\[\begin{split} \begin{aligned} f(\lambda L, \lambda K) & =A(\lambda L)^{\alpha}(\lambda K)^{\beta} \\ & =A \lambda^{\alpha} L^{\alpha} \lambda^{\beta} K^{\beta} \\ & =\lambda^{\alpha+\beta} A L^{\alpha} K^{\beta} \\ & =\lambda^{\alpha+\beta} f(L, K) \end{aligned} \end{split}\]

Thus we know that the Cobb-Douglas production function is homogeneous of degree \((\alpha+\beta)\).

• If \((\alpha+\beta)<1\), then the Cobb-Douglas production function displays decreasing returns to scale;

• If \((\alpha+\beta)=1\), then the Cobb-Douglas production function displays constant returns to scale; and

• If \((\alpha+\beta)>1\), then the Cobb-Douglas production function displays increasing returns to scale.

Homothetict function

Definition

Let \(I\) be an interval on the real line. Then, \(g: I \rightarrow \mathbb{R}\) is a monotonic transformation on \(I\) if \(g\) is a strictly increasing function on \(I\).

Definition

If \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\), write

\[ \mathbf{x} \geq \mathbf{y} \text{ if } x_i \geq y_i \text{ for all i=1,2,...,n,} \]

\[ \mathbf{x} > \mathbf{y} \text{ if } x_i > y_i \text{ for all i=1,2,...,n.} \]

A function \(u: \mathbb{R}^n_+ \rightarrow \mathbb{R}\) is monotone if for all \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\),

\[ x \geq y \Rightarrow u(\mathbf{x}) > u(\mathbf{y}). \]

The function is strictly monotone if for all \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\)

\[ x > y \Rightarrow u(\mathbf{x}) > u(\mathbf{y}). \]

Definition

A function \(v: \mathbb{R}_{+}^n \rightarrow \mathbb{R}\) is called homothetic if it is a monotone transformation, \(g(z)\), of a homogenous function \(u: \mathbb{R}_{+}^n \rightarrow \mathbb{R}\)

\[ v(x) = g\left(u(\mathbf{x})\right) \]

for all \(\mathbf{x}\) in the domain.

Example: \(G(x,y,z)=c\)

The two functions

\[\begin{split} v_1(x,y)=x^3y^3 + xy \\ v_2(x,y)=xy + 1, \end{split}\]

are homothetic functions with \(u(x,y)=xy\) and the monotonic transformations

\[\begin{split} g_1(x,y) = z^3 + z \\ g_2(x,y) = z + 1, \end{split}\]

respectively.

Definition

Let \(u: \mathbb{R}_{+}^n \rightarrow \mathbb{R}\) be a strictly monotonic function. Then, \(u\) is a homothetic if and only if for all \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbb{R}^n_+\),

\[ u(\mathbf{x}) \geq u(\mathbf{y}) \Leftrightarrow u(\alpha \mathbf{x}) \geq u(\alpha \mathbf{y}) \text{ for all } \alpha > 0. \]

Definition

Let \(u\) be a \(C^1\) function on \(\mathbb{R}^n_+\). If \(u\) is homothetic then the slopes of the tangent planes of the level sets of \(u\) are constant along each ray from the origin.

Some economic applications

Often in economic analysis the partial derivative has a straight forward interpretation

• production function: marginal product

• cost function: marginal cost

• profit function: marginal profit

• utility function: marginal utility

• demand function: marginal change in demand

Marginal products of production inputs

Suppose a firm’s production is described by a Cobb-Douglas production function, where labor (L) and capital (K) are the inputs:

\[ Q = f(L, K)=A L^{\alpha} K^{\beta} . \]

The marginal product of labour is simply the partial derivative with respect to labor. Thus we have

\[\begin{split} \begin{aligned} M P_{L}(L, K) & =\frac{\partial f(L, K)}{\partial L} \\ & =\frac{\partial\left(A L^{\alpha} K^{\beta}\right)}{\partial L} \\ & =\alpha A L^{\alpha-1} K^{\beta} \end{aligned} \end{split}\]

The marginal product of capital is simply the first-order derivative with respect to capital. Thus we have

\[\begin{split} \begin{aligned} M P_{K}(L, K) & =\frac{\partial f(L, K)}{\partial K} \\ & =\frac{\partial\left(A L^{\alpha} K^{\beta}\right)}{\partial K} \\ & =\beta A L^{\alpha} K^{\beta-1} \end{aligned} \end{split}\]

Elasticities

An elasticity measure the fractional response of the dependent variable, \(y=f(x_1,x_2,\dots,x_n)\), to a fractional change in an independent variable, e.g. \(x_k\), which can be written as

\[\begin{split} \varepsilon_{x_{k}} &= \left(\frac{f(x_1,\dots,x_k + h, x_{k+1},\dots,x_n) - f(x_1,\dots,x_k, x_{k+1},\dots,x_n)}{f(x_1,\dots,x_k, x_{k+1},\dots,x_n)} \right) / \left( \frac{h}{x_k} \right), \\ &= \frac{f(x_1,\dots,x_k + h, x_{k+1},\dots,x_n) - f(x_1,\dots,x_k, x_{k+1},\dots,x_n)}{h}\frac{x_k}{f(x_1,\dots,x_k, x_{k+1},\dots,x_n)}, \end{split}\]

where \(h\) is the change in \(x_k\). As \(h \rightarrow 0\), we can write this in terms of the partial derivative

\[ \varepsilon_{x_{k}} = \frac{\partial f(\mathbf{x})}{\partial x_k} \frac{x_k}{f(\mathbf{x})}. \]

Similar to partial derivatives, elasticities measure how sensitive the dependent variable is to changes in the independent variables. Unlike partial derivatives elasticities are unit-free.

Elasticities of production with respect to inputs

Suppose a firm production is described by a Cobb-Douglas production function:

\[ f(L, K)=A L^{\alpha} K^{\beta} . \]

We know from earlier that the partial derivatives of the Cobb-Douglas production function are

\[\begin{split} \frac{\partial f(L, K)}{\partial L} = \alpha A L^{\alpha-1} K^{\beta}. \\ \frac{\partial f(L, K)}{\partial K} = \beta A L^{\alpha} K^{\beta-1}. \\ \end{split}\]

Insert the partial derivate with respect to labor into the definition of elasticities

\[\begin{split} \varepsilon_{L} &= \tfrac{\partial f(L,K)}{\partial L}\tfrac{L}{f(L,K)}, \\ &= \alpha A L^{\alpha-1} K^{\beta}\frac{L}{A L^{\alpha} K^{\beta}}, \\ &= \alpha. \end{split}\]

Insert the partial derivate with respect to capital into the definition of elasticities

\[\begin{split} \varepsilon_{K} &= \tfrac{\partial f(L,K)}{\partial L}\tfrac{L}{f(L,K)}, \\ &= \beta A L^{\alpha} K^{\beta-1}\frac{K}{A L^{\alpha} K^{\beta}}, \\ &= \beta. \end{split}\]

Elasticities of demand

Suppose that an individual’s demand function for commodity \(k\) is given by

\[ D_{k}\left(\mathbf{p},y\right) = D_{k}\left(p_{1}, p_{2}, \dots, p_{n}, y\right), \]

where \(p_{i}\) is the price of commodity \(i\) and \(y\) is the consumer’s income.

• The own-price elasticity of demand for commodity \(k\) for this consumer is

\[ \begin{aligned} \varepsilon_{k}^{k}\left(\mathbf{p},y\right) & =\left(\frac{p_{k}}{D_{k}\left(\mathbf{p},y\right)}\right)\left(\frac{\partial D_{k}\left(\mathbf{p},y\right)}{\partial p_{k}}\right) . \end{aligned} \]

• The cross-price elasticity of demand for commodity \(k\) with respect to the price of commodity \(l\) for this consumer is

\[ \begin{aligned} \varepsilon_{l}^{k}\left(\mathbf{p},y\right) & =\left(\frac{p_{l}}{D_{k}\left(\mathbf{p},y\right)}\right)\left(\frac{\partial D_{k}\left(\mathbf{p},y\right)}{\partial p_{l}}\right). \end{aligned} \]

• The income elasticity of demand for commodity \(k\) for this consumer is

\[ \begin{aligned} \varepsilon_{y}^{k}\left(\mathbf{p},y\right) & =\left(\frac{y}{D_{k}\left(\mathbf{p},y\right)}\right)\left(\frac{\partial D_{k}\left(\mathbf{p},y\right)}{\partial y}\right) . \end{aligned} \]

We can use these elasticities to classify the types of commodities that are being considered:

• If \(\varepsilon_{y}^{k}>0\), then commodity \(k\) is a normal good.

• If \(\varepsilon_{y}^{k}<0\), then commodity \(k\) is an inferior good.

• If \(\varepsilon_{l}^{k}>0\), then commodities \(k\) and \(l\) are substitutes.

• If \(\varepsilon_{l}^{k}<0\), then commodities \(k\) and \(l\) are complements.

• The demand curve for most commodities will usually slope down. As such, we would usually expect \(\varepsilon_{k}^{k}<0\).

• However, there are circumstances in which the demand curve for a commodity can slope up over some range of prices (at least in theory). Such commodities are known as Giffen goods. In such circumstances, we would have \(\varepsilon_{k}^{k}>0\) over the relevant range of prices.

Cournot aggregation

Suppose that a consumer’s preferences over bundles of \(L\) commodities. Assume that the budget constrain holds exactly

\[ \sum_{l=1}^{L} p_{l} x_{l}(p, y)=y, \]

where \(p=\left(p_{1}, p_{2}, \dots, p_{n}\right)=\left(p_{k}, p_{-k}\right)\) is the price vector, \(y\) is the consumer’s income and \(x_{l}(p, y)\) is the consumer’s Marshallian demand for good \(l\).

Note that this can be rewritten as

\[ p_{k} x_{k}\left(p_{k}, p_{-k}, y\right)+\sum_{l \neq k} p_{l} x_{l}(p, y)=y \]

Partially differentiating both sides of this equation with respect to the price of commodity \(k\), we obtain

\[ \left\{(1) x_{k}(p, y)+p_{k}\left(\frac{\partial x_{k}(p, y)}{\partial p_{k}}\right)\right\}+\sum_{l \neq k} p_{l}\left(\frac{\partial x_{l}(p, y)}{\partial p_{k}}\right)=0 . \]

This can be simplified to obtain

\[ x_{k}(p, y)+\sum_{l=1}^{L} p_{l} \frac{\partial x_{l}(p, y)}{\partial p_{k}}=0 \]

This can be rearranged to obtain

\[ \sum_{l=1}^{L} p_{l} \frac{\partial x_{l}(p, y)}{\partial p_{k}}=-x_{k}(p, y) \]

This can be rewritten as

\[ \sum_{l=1}^{L}\left(\frac{p_{k}}{p_{k}}\right)\left(\frac{x_{l}(p, y)}{x_{l}(p, y)}\right)\left(\frac{y}{y}\right) p_{l} \frac{\partial x_{l}(p, y)}{\partial p_{k}}=-x_{k}(p, y) . \]

This can be rearranged to obtain

\[ \sum_{l=1}^{L}\left(\frac{p_{l} x_{l}(p, y)}{y}\right)\left(\frac{p_{k}}{x_{l}(p, y)}\right)\left(\frac{\partial x_{l}(p, y)}{\partial p_{k}}\right)=-\left(\frac{p_{k} x_{k}(p, y)}{y}\right) . \]

This can be rewritten as

\[ \sum_{l=1}^{L} s_{l} \varepsilon_{k}^{l}=-s_{k} \]

where

\[ s_{l}=\frac{p_{l} x_{l}(p, y)}{y} \]

is the budget share of commodity \(l\) and

\[ \varepsilon_{k}^{\prime}=\left(\frac{p_{k}}{x_{l}(p, y)}\right)\left(\frac{\partial x_{l}(p, y)}{\partial p_{k}}\right) \]

is the \(k\) th commodity-price elasticity of demand for commodity \(l\).

The above formula is a result known as Cournot aggregation. It provides a relationship between the \(k\) th price elasticities of demand for the various commodities.

Engel aggregation

Suppose that a consumer’s preferences over bundles of \(L\) commodities are locally non-satiated in the neighbourhood of any potentially feasible commodity bundle. Then we know that budget exhaustion (which is sometimes called Walras’ law for the individual) must hold for the consumer. This ensures that

\[ \sum_{l=1}^{L} p_{l} x_{l}(p, y)=y \]

where \(p=\left(p_{1}, p_{2}, \cdots, p_{n}\right)\) is the price vector, \(y\) is the consumer’s income and \(x_{l}(p, y)\) is the consumer’s Marshallian demand for good \(I\).

Partially differentiating both sides of this equation with respect to income, we obtain

\[ \sum_{l=1}^{L} p_{l}\left(\frac{\partial x_{l}(p, y)}{\partial y}\right)=1 \]

This can be rewritten as

\[ \sum_{l=1}^{L}\left(\frac{y}{y}\right)\left(\frac{x_{l}(p, y)}{x_{l}(p, y)}\right) p_{l}\left(\frac{\partial x_{l}(p, y)}{\partial y}\right)=1 \]

This can be simplified to obtain

\[ \sum_{l=1}^{L}\left(\frac{p_{l} x_{l}(p, y)}{y}\right)\left(\frac{y}{x_{l}(p, y)}\right)\left(\frac{\partial x_{l}(p, y)}{\partial y}\right)=1 \]

This can be rewritten as

\[ \sum_{l=1}^{L} s_{l} \varepsilon_{y}^{l}=1 \]

where

\[ s_{l}=\frac{p_{l} x_{l}(p, y)}{y} \]

is the budget share of commodity \(l\) and

\[ \varepsilon_{y}^{l}=\left(\frac{y}{x_{l}(p, y)}\right)\left(\frac{\partial x_{l}(p, y)}{\partial y}\right) \]

is the income elasticity of demand for commodity \(l\).

The above formula is a result known as Engel aggregation. It provides a relationship between the income elasticities of demand for the various commodities. –>