The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra

Jorge Marques and Nuno Baeta

CeBER and CISUC University of Coimbra

June 26, 2018, Coimbra, Portugal 7th CADGME - Conference on Digital Tools in Mathematics Education

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Outline Summary

1 The neoclassic consumer model in Economics 2 2 Smooth Preferences on R+ Representation by a utility function Representation by the marginal rate of substitution

3 2 Characterization of Preferences Classes on R+ 4 Graphic Representation on GeoGebra

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra The neoclassic consumer model in Economics

Variables: Quantities and Prices N Let R+ = {x = (x1,..., xN ): xi > 0} be the set of all bundles of N goods Xi , N ≥ 2, and Ω = {p = (p1,..., pN ): pi > 0} be the set of all unit prices of Xi in the market.

Constrained Maximization Problem The consumer is an economic agent who wants to maximize a utility function u(x) subject to the budget constraint pT x ≤ m, where m is their income. In fact, the combination of strict convex preferences with the budget constraint ensures that the ∗ ∗ problem has a unique solution, a bundle of goods x = (xi ) ∗ i such that xi = d (p1,..., pN , m).

System of Demand Functions In this system quantities are taken as functions of their market prices and income.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra The neoclassic consumer model in Economics

Economic Theory of Market Behavior However, a utility function has been regarded as unobservable in the sense of being beyond the limits of the economist’s knowledge. The economic theory has testable consequences, that is, the observations (empirical data) can confirm the theoretical predictions. Assuming that theory is not disproved, can we recover the preferences (or utility) from the observations (prices, quantities and income)? We have an inverse problem: given a differentiable map

N d :Ω × R+ −→ R+ (p, m) 7−→ x = d(p, m)

can one find a utility function u(x) such that x∗ = d(p, m) is the system of demand functions for the maximization problem?

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra The neoclassic consumer model in Economics

The Problem of Integrability in Utility Theory (Samuelson, 1950) That question of the existence of a utility function is known as the integrability problem in economics.

Ordinal Utility Utility functions are invariant by increasing transformations g.

Economic Assumptions A utility function u must be: Increasing since the consumer prefers more rather than fewer goods. So that the marginal utility of a good is positive (It is measured by the rate of change of u). Strictly quasiconcave by the principle of diminishing marginal utility: If the consumption of one good increases then its marginal utility decreases, holding the consumption of the other goods constant. Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra The neoclassic consumer model in Economics

Mathematical Integration We have to deal with a mathematical problem which consists in solving an ordinary differential equation (ODE) or a system of partial differential equations (PDEs) .

Mathematical Solving If N = 2 then the existence of a utility function is guaranteed since an ODE always admits an integrating factor. Otherwise, it is only possible to find an integrating factor for a system of PDEs provided that it holds the integrability conditions, hence there exists a utility function under appropriate conditions.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Smooth Preferences on R+

2 Consumer’s Preferences on R+ 2 2 Let R+ = {(x, y) ∈ R : x > 0 ∧ y > 0} be the set of all bundles of two goods X and Y . Given any two bundles P1 = (x1, y1) and P2 = (x2, y2), an individual consumer can take one of three choices:

i) He (or she) will prefer P1 to P2, P1 P2

ii) He (or she) will prefer P2 to P1, P2 P1

iii) He (or she) will be indifferent, P1 ∼ P2.

Mathematical Formulation So an individual consumer is assumed to have a preference 2 relation  on R+ represented by a binary relation which satisfies three axioms: reflexivity, transitivity and completeness. The axioms of continuity, monotonicity and strict convexity are usually imposed on  in the differentiable setting (Mas-Colell, 1995). Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Smooth Preferences on R+

Methodology - Distinct Approaches In the neoclassical approach, it is defined a representation that requires the determination of an arbitrary utility function of class C2. A utility function aims to construct a ranking of choices based on the consumer’s preferences relation. In our work we consider a representation using an exchange rate between two goods, called marginal rate of substitution, in order to describe smooth preferences. In our approach, If we use this rate then we can determine a utility representation for consumer’s preferences.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Representation by a marginal rate of substitution

Economic Interpretation It was Hicks who introduced the concept of marginal rate of substitution between two goods as an incremental amount of one good that the consumer is willing to give up in order to gain an incremental increase of consumption of other good.

Marginal Rate of Substitution as a Primitive Concept Suppose now that a utility function of class C2 is unknown. Assuming that Y is a reference good, without loss of generality, then the preferences can be represented by the marginal rate of substitution of Y for X.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Representation by a marginal rate of substitution

Definition (An indifference curve) An indifference curve is defined as the set of all bundles of goods which have the same preference rank (or utility level), that is, given a bundle P = (x0, y0), an indifference curve is the set 2 I = {(x, y) ∈ R :(x, y) ∼ (x0, y0)}

Definition (Marginal Rate of Substitution) 2 Let I be an indifference curve on R+. We say that τ is the marginal rate of substitution of Y for X if there exists a function of class C2 defined by y = f (x) such that

dy = −τ(x, y) , for all (x, y) ∈ I . (1) dx

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Representation by a marginal rate of substitution

Assumptions on Marginal Rate of Substitution In according to Hicks and Allen, it is assumed that: 1 2 (i*) Differentiability: τ is of class C on R+; 2 (ii*) Positivity: τ is positive on R+; ∂τ ∂τ (iii*) Convexity: − τ is negative on R2 . ∂x ∂y + The last assumption expresses expresses strict convexity of every indifference curve because the diminishing MRS principle states that the rate will decrease as Y good is substituted for X along an indifference curve.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Geometric Interpretation 2 The consumer’s preferences on R+ are defined by giving for 2 each point P ∈ R+ the direction orthogonal to an unknown indifference curve at P.

Theorem If τ satisfies (i*), (ii*) and (iii*) then for every indifference curve there exists a function f defined by y = f (x) of class C2 satisfying (1) which is strictly decreasing and strictly convex.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

The Indifference Map It follows that the indifference map is the set of all curves such that: Each indifference curve is the graph of a function 2 f : D ⊂ R+ → R, defined by y = f (x), of class C , positive, strictly decreasing and strictly convex;

Indifference curves cannot intersect, that is, if c1 6= c2 we have

Ic1 ∩ Ic2 = ∅; The further the indifference curve from the origin, the higher the preference rank (or level of utility) c; A reunion of all indifference curves satisfies

[ 2 Ic = R+. c>0

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Definition (Linear Preferences)

Two goods X1 and X2 are called perfect substitutes if marginal rates of substitution are positive constants for all bundles.

Proposition (Representation Theorem) 2 Consumer’s preferences are linear on R+ if and only if the indifference map is represented by a family of lines of the equation y = −kx + c, where k > 0.

Utility Representation Linear preferences are represented by any utility function U(x, y) = g(kx + y), where g is strictly increasing and k > 0.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Definition (Quasi-Linear Preferences) Consumer’s preferences are said to be:

(i) Quasi-linear with respect to X2 if there exists a positive and strictly decreasing function η on R+ such that

2 τ(x, y) = η(x) , for all (x, y) ∈ R+ ;

(ii) Quasi-linear with respect to X1 if there exists a positive and strictly decreasing function φ on R+ such that

2 τ(x, y) = φ(y) , for all (x, y) ∈ R+ .

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Proposition (Representation Theorem)

Consumer’s preferences are quasi-linear with respect to X2 on 2 R+ if and only if the indifference map is represented by a family of curves of equation y = c − F(x), where F is strictly increasing and strictly concave.

Proposition (Representation Theorem)

Consumer’s preferences are quasi-linear with respect to X1 on 2 R+ if and only if the indifference map is represented by a family of curves of equation y = F(c − x), where F is strictly increasing and strictly convex.

Utility Representation It is well known that these preferences admit a particular representation that is additive in the good Xi .

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Definition (Separable Preferences) 2 Consumer’s preferences on R+ are said to be separable if there exist a positive and strictly decreasing function η on R+ and a positive and strictly increasing φ on R+ such that 2 τ(x, y) = η(x)φ(y) , for all (x, y) ∈ R+ .

Proposition (Representation Theorem) 2 Consumer’s preferences are separable on R+ if and only if the indifference map is represented by y = G (c − F(x)), where G is strictly increasing and strictly convex and F is strictly increasing and strictly concave.

Utility Representation Separable preferences are represented by any utility function U(x, y) = g (F(x) + H(y)), where g is strictly increasing, F and H are strictly increasing and strictly concave. Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Definition (Homothetic Preferences) 2 Consumer’s preferences on R+ are said to be homothetic if there exists a positive and strictly increasing function h on R+ y
2 such that τ(x, y) = h x , for all (x, y) ∈ R+ .

Proposition (Representation Theorem) 2 Consumer’s preferences are homothetic on R+ if and only if the c
indifference map is represented by y = xF ln ( x ) , where F is strictly increasing and strictly convex.

Utility Representation Homothetic preferences are represented by any utility function  H y  U(x, y) = g xe ( x ) , where g is strictly increasing and H is strictly increasing and strictly concave.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Proposition (Representation Theorem)

2 Consumer’s preferences are separable and homothetic on R+ if and only if there exist constants a > 0 and δ < 1 such that τ(x, y) = axδ−1y 1−δ.

Separable and Homothetic Preferences Both separable and homothetic are precisely the preferences that admit constant elasticity of substitution (CES) utility functions. This positive constant is given by σ = 1/(1 − δ).

Utility Representation It is well known that these preferences have a particular representation which is homogeneous of degree one.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Definition (Cobb-Douglas Preferences) 2 Consumer’s preferences on R+ are said to be Cobb-Douglas if there exists a constant 0 < α < 1 such that αy τ(x, y) = , for all (x, y) ∈ R2 . (1 − α)x +

Proposition (Representation Theorem) 2 Consumer’s preferences on R+ are Cobb-Douglas if and only if the indifference map is represented by a family of curves of − α equation y = cx 1−α .

Utility Representation Cobb-Douglas utility functions are represented by any U(x, y) = g(xαy 1−α) where g is strictly increasing and 0 < α < 1.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra 2 Characterization of Preferences classes on R+

Definition (CES Preferences) 2 Consumer’s preferences on R+ are said to be CES if there exist constants 0 < α < 1 and δ < 1 such that α y 1−δ τ(x, y) = , for all (x, y) ∈ R2 . 1 − α x +

Proposition (Representation Theorem) 2 Consumer’s preferences on R+ are CES (δ 6= 0) if and only if the indifference map is represented by a family of curves of 1/δ c − αxδ  equation y = . 1 − α

Utility Representation CES utility functions are represented by any U(x, y) = g (αxδ + (1 − α)y δ)1/δ
, g is strictly increasing.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Graphic Representation on GeoGebra

Advances in technology are influencing the learning and teaching of the ODEs in many first courses. There is an increased emphasis on qualitative and numeric approaches to study the behavior of its solutions. Analytic techniques are still important, but will be necessary to make connections between algebraic, graphic and numeric representations. The representation of consumer’s preferences on the space of two goods through marginal rates of substitution is a contextual situation in Economics where we can use a mathematical model represented by ODEs. GeoGebra is a powerful tool to show the indifference map for consumer’s preferences. First, it is plotted the direction field for an ODE; Second, it is fixed a point P; Third, it is determined a particular solution of an ODE and so on. We can draw a set of strictly decreasing and strictly convex curves, written in the explicit functional form. Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Graphic Representation on GeoGebra

We include criteria to find relationships between the sign of τ and properties of monotonicity, the assumption (iii*) and properties of convexity, the infinite limit of τ and vertical slope for indifference curves. The approach based on marginal rates of substitution allows to cover a wider range of consumer’s preferences. We can consider functional forms for some additional preferences whenever the ODE does not has solutions in close form. We would like to remark the strength of our approach as compared to the traditional (neoclassical) approach. It is useful to define marginal rates of substitution as slopes of indifference curves and interpreted them as linear approximations to an unknown utility function. Thereby we derive a utility representation for consumer’s preferences 2 on R+.

Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra Some References

Allen, R. G. D. (1938). Mathematical Analysis for Economists. London: Macmillan and Company Limited. Debreu, G. (1972). Smooth preferences. Econometrica, 40 (4), 603–615. Hicks, J. R. and Allen, G. D. (1934). A Reconsideration of the Theory of Value, II. Economica, New Series, 1 (2), 196–219. Marques, J., (2014). An application of ODEs in economics: modeling consumer’s preferences using marginal rates of substitution. In: F. M. a. M. M. N. Mastorakis, ed. Mathematical Methods in Science and Mechanics: Mathematics and Computers in Science and Engineering. s.l.: Wseas Press, pp. 46-53. Mas-Colell, A., Whinston, M.D., Green, J.R. (1995) Microeconomic theory. New York: Oxford University Press. Samuelson, P. A. (1950). The Problem of Integrability in Utility Theory. Economica, New Series, 17 (68), 355–385. Jorge Marques and Nuno Baeta The indifference curve analysis - An alternative approach represented by ODEs using GeoGebra