Abstract
The economic analysis of consumer behavior begins with a model of rational behavior. The consumer is assumed to have a well-defined utility function that, coupled with a specification of the constraints facing the consumer, implies a specific action. This chapter develops and illustrates the nature of a utility function. The next two chapters add specifications of the constraints that face the consumer and derive the model’s implications.
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Notes
- 1.
“Established” here means that the utility function does not change due to any thought experiments that we undertake—specifically, income level changes and price level changes. It does not mean that the function cannot change as a result of outside forces like education or persuasion.
- 2.
Let the arbitrarily selected pair (x1,y1) yield a utility level u1, and let the arbitrarily selected pair (x2,y2) yield a utility level u2 for this function. An order preserving transformation of the function would yield different utility values for these two pairs, but the order of the two would remain the same as the order of u1 and u2.
- 3.
The elasticity of substitution is \(s = \frac{1} {1+b}\). We demonstrate this relationship later when the term is formally defined.
- 4.
Note the axis rotation. We executed this input group with draw3d replacing wxdraw3d. This command produces a graph outside wxMaxima that can be rotated. We made note of the rotation and then applied the selected rotation to wxdraw3d. The range over which the explicit functions are graphed begins at x = 0. 01 and y = 0. 01, rather than zero, to avoid division by zero.
- 5.
Recall that Maxima uses the command diff for both derivatives and partial derivatives.
- 6.
The utility function is \(u = {(a \cdot {x}^{-b} + (1 - a) \cdot {y}^{-b})}^{-1/b}\). Therefore, \({u}^{-b} = a \cdot {x}^{-b} + (1 - a) \cdot {y}^{-b}\), or \({u}^{-b} - a \cdot {x}^{-b} = (1 - a) \cdot {y}^{-b}.\) Thus, a positive value of y implies that the term on the left-hand side is positive. That is, 1∕u b > a∕x b, and so x b > a ⋅ u b. We use a placeholder name u0 for utility because the name u has been assigned to the specified utility function.
- 7.
The x list is entered directly into the function ic and so the result is a list.
- 8.
As an exercise, compute the last six items in the y change column based on the printed values of ylist2 (the list immediately above the table). Your answers will differ from the tabled value due to rounding. Use Maxima, y[i] - y[i-1].
- 9.
To save space, the function is not printed.
- 10.
This reflects the fact that the value 56. 002 above refers to a change over the range x = 4. 5355 to x = 5. 5355, and so forth.
- 11.
Chapter 4 shows that the mrs equals the price ratio for the two goods and, therefore, does not depend on the consumer’s choice. In this setting s is the percentage change in \(\frac{y} {x}\) per 1 % change in the price ratio.
- 12.
The value b = 149, implying s = 0. 004, is used because larger values can result in an overflow error.
- 13.
Be aware of the “at the margin” modifier here. The consumer would rather have 30 units of x and 10 units of y than no x and 20 units of y, so x does provide utility in total, but not at the margin.
- 14.
Each of the two references in this chapter is an important part of empirical research in economics. Also, the models developed in these articles are frequently used to illustrate economic principles, as they are used in this chapter.
References
Each of the two references in this chapter is an important part of empirical research in economics. Also, the models developed in these articles are frequently used to illustrate economic principles, as they are used in this chapter.
Arrow KJ, Chenery HB, Minhas BS, Solow RM (1961) Capital-labor substitution and economic efficiency. Rev Econ Stat 43:225–250
Cobb CW, Douglas PH (1928) A theory of production. Am Econ Rev 18(Suppl.):139–165
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Hammock, M.R., Mixon, J.W. (2013). Demand Theory: Preferences. In: Microeconomic Theory and Computation. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9417-1_3
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DOI: https://doi.org/10.1007/978-1-4614-9417-1_3
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