1. Functions of Two or More Independent Variables
Chapter 14

2. 14.2 Two Independent Variables
Examples of functions of two independent variables:
General form of a function of two variables:

3. Graphing a Two-Variable Function
Fix z=0. The function z=f(x,y) now becomes f(x,y)=0, which may or may not describe an implicit function y=g(x).
Same holds for x=0 and y=0.

4. Two-Variable Functions as Surfaces
Graphically, a two-variable function describes a surface.
For instance, z=ax+by describes a plane.
Point R in the graph is called a projection of point P on the xy-plane.

5. Extension to Many Variables
The general form of a function of n variables:
Geometrically, a function of n variables describes a hypersurface.
Graphical representation is difficult, but we can always fix n-2 variables at some values to come up with a section of the form
Where means a fixed value of variable i , e.g.

6. 14.3 Examples of Functions of Two Variables
Linear two-variable functions describe planes.
This plane can be projected on xy, xz, and yz.

7. Sections
By fixing the value of one of the three variables x, y, and z we create sections.
Definition. A section of a two-variable function z=f(x,y) is a relationship obtained by doing one of the three following things:
Fix

8. Projected Sections
Definition. A projected section is a section whose graph lies entirely in one of the xy, yz, or xz planes.
Projected sections are obtained by a parallel shift of the original section to one of the three planes.

9. Iso Sections
Definition. A section is also referred to as an iso-section.
“Iso” means “same” or “equal” in Old Greek.
Example Consider a function
Fixing produces an iso-z section describing a circle in the plane
We can say that is a “collection” of circles of radius

10. Iso Sections Iso-y sections are parabolas. Iso-z sections are circles.
The surface described by the two-variable function is a cone with its vertex at the origin.

11. Sections through the Cone
The function is symmetrical in x and y since its iso-x and iso-y sections look exactly the same.

12. 14.4 Partial Derivatives
Definition. Consider a function Suppose y is fixed at The partial derivative of function f with respect to variable x is defined as:
Note. Partial derivatives can be thought of as derivatives of iso-sections. Since each iso-section is a function of just one variable, finding partial derivatives is straightforward.

13. Partial Derivative: Geometrical Meaning

14. Examples Find partial derivatives of the following functions:
Consider a Cobb-Douglas production function:
Find partial derivatives with respect to capital K and labor L.
What economic meaning do these partial derivatives have?

15. 14.6 Second-Order Partial Derivatives
Consider an iso-y section of , namely,
The partial derivative is itself
a function of x, which we can differentiate again.
We denote this second direct derivative as

16. Evaluation of Direct Second-Order Partial Derivatives
Direct second-order partial derivatives can be obtained in two steps:
Find a first-order derivative of an iso-section of the original function
Differentiate the function obtained at step 1)
Example.

17. Cross-Partial Derivatives
Change of the slope of an iso-y section as y changes from to
Definition. A first-order derivative with respect to one variable of a first-order partial derivative taken with respect to another variable is called partial derivative of the original function with respect to these two variables.
Notation:

18. Cross-Partial Derivative: an Example
What if we changed the order of variables when computing a cross-partial derivative?
The two cross-partial derivatives are equal to each other!

19. Young’s Theorem
Theorem. Consider a function Suppose all of its second-order partial derivatives are continuous. Then all of the mixed second-order partial derivatives are equal to each other.

20. Case of Many Variables Consider a function .
We have three first-order partial derivatives which can be differentiated as well.
As a result, we have three direct second-order partial derivatives
We also have mixed cross-partial derivatives
The equalities here are due to Young’s theorem.

21. Alternative Notation

22. 14.7 Production Function
Definition. A mapping from the set of production inputs to output is called a production function.
Examples of inputs are capital K and labor L.
Output is often denoted as Y or Q.
Capital, labor and output are flow rather than stock variables, i.e. they are defined in terms of quantity spread over a period of time. We speak of liters per week, worker-hours per month, machine-hours per day.
Simplifying assumptions.
No component parts are used
Only capital and labor are production inputs
Production function is a purely engineering relationship, i.e. no inefficiencies are involved

23. Neoclassical Assumptions
Consider a production function
The neoclassical economic theory assumes the following:
Q, K, and L are infinitely divisible, and the production function f is smooth and continous
If either K or L is zero, production Q is also zero: you need at least some of both inputs to produce something.
An increase in K or L will increase Q.
The ‘law’ of diminishing marginal product holds at all levles

24. 14.8 Shape of the Production Function
Q, K, and L are infinitely divisible, and the production function f is smooth and continous
If either K or L is zero, production Q is also zero: you need at least some of both inputs to produce something.
An increase in K or L will increase Q.
The ‘law’ of diminishing marginal product holds at all levles

25. Isoquants
Definition. A locus of input combinations (K,L) that result in the same level of output Q is called an isoquant.
Note. According to our previous terminology, isoquants can be defined as iso-q lines.

26. Marginal Rate of Substitution
Definition. The marginal rate of substitution is the amount of one input released when one more unit of another input is used while keeping output constant.

27. Law of Diminishing MRS
The law of diminishing MRS says that substitution of one input for another becomes technically more difficult as we employ more of that other input.
Corollary. The slope of the isoquants gets flatter as we use disproportionately more of one of the inputs.
Graphically, the law of diminishing MRS results in convex isoquants.

28. MRS and Partial Derivatives
Consider a movement from D to G.
Output is the same (Q=60) at both points.
At G we employ one more unit of labor.
At G we employ less units of capital.
The marginal rate of substitution of labor for capital is
This value is a difference quotient, i.e. the slope of chord DG, that in the limit becomes the slope of tangent at D to isoquant defined by Q=60:

29. Capital Intensity
Definition. The ratio of capital to labor is called capital intensity.
Note. Sometimes capital intensity is defined in terms of capital per unit of output,
In general, more advanced economies are capital-intensive.
As we move down the isoquant and if labor is on the horizontal axis, our production technique becomes more labor-intensive.

30. Short-Run Production Function
Definition. An iso-K section for a production function is called a short-run production function.
Economic intuition: Capital stock is difficult to adjust in the short run, e.g. build a new factory.

31. Marginal Product of Short-Run Production Functions

32. Diminishing Marginal Product
Also known as “diminishing marginal productivity.”
Holding capital input constant, additional increases in labor result in smaller increases in output.
Economic explanation
Capital is fixed. Workers cannot produce without capital.
Hiring more workers means spreading capital over more people.
As a result, each worker produces less output than before, and productivity falls.
Cross-partial derivatives
Slope of RS is steeper than slope of LM: a small increase in K causes MPL to grow, hence

33. Average Product of Labor
Definition. The ratio is called average product of labor.
Average product of labor is also known as labor productivity.
Average product of labor is the slope of rays OA, OA’, OP, OP’.
MPL<APL
APL decreases with Q
Caution. There are production functions for which MPL>APL for small Q.

34. Quasi-Short-Run Production Function
Definition. Iso-L sections of a production function are called quasi-short-run production functions.
The word “quasi” means that in reality, it is much easier to vary labor than capital.
However, consider the following situation: you rent your tools daily, but your labor is protected by the labor code.
Or the following question: what if you want to use more machines keeping the number of workers unchanged?

35. Quasi-Short-Run Production Function
Marginal product of capital:
MPK diminishes with more labor.
Intuition: using more machines you decrease the number of worker per one machine, which decreases average output per worker.
Cross-partial derivative:

36. Average Product of Capital
Definition. The ratio is called average product of capital.
Average product of capital is measured by the slope of ray OR or OR’.
Normally, MPK<APK.

37. Cobb-Douglas Production Function
Definition. A production function of the form is called Cobb-Douglas production function.
A is a technology parameter are positive real numbers whose sum is often equal to 1.
Isoquants of a Cobb-Douglas production function are hyperbolas:
In case , an isoquant equation becomes , which is a rectangular hyperbola.

38. Short-Run Cobb-Douglas
Consider a Cobb-Douglas production function
To obtain the short-run production function, set K=100.
Our production function becomes

39. Marginal Product of Labor
Fix capital at K=49, then
Marginal product of labor is diminishing:

40. Average Product of Labor
For this short-run production function the average product of labor is equal to
Compare with
We know that if MP*<AP*, AP* must be falling (* stands for any variable).
Indeed, as we hire more workers, the average productivity is falling as well as does the marginal productivity.

41. Cross-Partial Derivative
For

42. Quasi-Short-Run Cobb-Douglas
For the quasi-short-run Cobb-Douglas production function is
for L=64.
The marginal product of capital
Average product of capital is:
MPK<APK for all K>0.
Cross-partial derivatives:

43. General Form of Cobb-Douglas
Specification:
Isoquants:
Marginal product of labor:
Second derivative wrt K:

44. 14.10 Alternatives to Cobb-Douglas
At low levels of output Q the MPL is increasing.
At point P MPL is at its maximum, where it starts decreasing with increases in L.
MPK at S follows the same pattern.
Points P and R are inflection points.
Economic intuition. Some machines cannot be operated efficiently by one or two people. Say, the optimal number of crew is five. Then MPL will be greatest at L=5, then it starts decreasing because “too many cooks spoil the broth.”

45. MPL and APL for Alternative Shape
The law of diminishing marginal product of labor does not hold at all levels of output.
Same for MPK.
The concept of diminishing MPK and MPL is also known as the law of diminishing returns.
However, it should not be confused with diminishing returns to scale! This will be discussed later.

46. Utility Function
Definition. A function of the form that maps the consumption of goods X and Y into a measure of the consumer’s happiness U is called utility function.
Notes.
Utility cannot be measured objectively
Most of the time, utility levels cannot be compared between two or more individuals, so we call these utility functions ordinal
When inter-personal comparison of utility levels is possible, the utility function is called cardinal
Utility functions allow for a comparison of vectors, which is not always possible in a logically consistent way

47. 14.12 Shape of Utility Function
Neoclassical assumptions:
U,X, and Y are infinitely divisible
Utility function is smooth and continuous
“More is better”, or non-satiation: an increase in either X or Y or both increases utility
Decreasing marginal utility in consumption

48. Indifference Curves
Definition. Given a utility function U=f(X,Y), its iso-U sections are called indifference curves.
Consumer receives the same satisfaction from combinations of X and Y at A, G, J, and C.
The exact meaning of U=200 for these points is not important
However, U=200>U=100 so A is preferred to H by this consumer.

49. Indifference Curves
Transitivity: Preferences are transitive in the sense that circular preferences are excluded.
Circular preferences: I prefer A to B, I prefer B to C, but I prefer C to A.
Indifference curve may not cross each other.
Non-satiation implies that the slope of indifference curves is negative, so we have substitution in consumption.

50. Convexity of Indifference Curves
Indifference curves are convex from below.
Convexity reflects the idea of a balanced consumption.

51. Decreasing Marginal Rate of Substitution in Consumption
Definition. For a utility function of the form U=f(X,Y) the marginal rate of substitution (MRS) in consumption is defined as for some level of utility function U.
Economic intuition. As consumption of one good grows disproportionately high compared to that of another good, consumers derives less additional satisfaction from a one-unit increase in the consumption of that good.

52. Marginal Utilities
Consider iso-X and iso-Y sections of some utility function:
Definition. For any utility function U=f(X,Y) the marginal utilities of goods X and Y, denoted as and , are defined as the partial derivatives and , respectively.

53. Diminishing Marginal Utility
The slope of a utility function is decreasing with increased consumption of either one of the goods, which is known as the law of diminishing marginal utility.
This law implies that
Economic intuition: when Imelda Marcos increased her stock of shoe pairs from 2 to 3, her utility increased more than when she increased her stock of shoe pairs form 2999 to 3000.
Appetite grows with eating: this is the case if marginal utility actually grows with consumption!

54. Cobb-Douglas Utility Function
General form:
Note. We don’t have here an analogue to the technological factor we had for the production function because utility functions are used for comparisons, not for exact measurement purposes.
Indifference curves:
Slopes of indifference curves:
Second derivative: