Abstract
Chapter 4 analyzes optimization and its implications for utility maximization subject to one or more constraints, and it develops the relationship between money income and the quantity consumed of a good. This chapter addresses the effects of changes in the good’s price. Such a change causes both the relative prices of the goods consumed and the consumer’s real income to change. Initially, analysis is based on the CES utility function. Then the analysis turns to the case of a Giffen good. A large amount of time is spent on the latter, not because of its inherent importance, but because it provides added insights into the nature of the consumer’s reaction to price and income changes.
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Notes
- 1.
More questions would have to be answered, but we use assume and declare to anticipate two questions.
- 2.
The utility function can be written as \({u}^{b} = \frac{1} { \frac{a} {{x}^{b}}+\frac{1-a} {{y}^{b}} }\). Therefore, \(a \cdot {u}^{b} - x = \frac{a} { \frac{a} {{x}^{b}}+\frac{1-a} {{y}^{b}} } - x\). The right-hand side of this term equals \(\{a \cdot {x}^{b} \cdot {y}^{b} - {x}^{b} \cdot [(1 - a) \cdot {x}^{b} + a \cdot {y}^{b}]\}/D\), where \(D = (1 - a) \cdot {x}^{b} + a \cdot {y}^{b}\). The denominator D is positive, so the sign of the numerator determines the sign of a ⋅ u b − x b. Multiplying through by x b and collecting terms shows that the numerator is \(-(1 - a) \cdot {x}^{2\cdot b}\), which must be negative because a < 1. This negative term is multiplied by the negative value (a − 1) ⋅ x, resulting in a positive expression.
- 3.
The price increase to 80 has symmetrical effects: Each of the 8.8 units still bought costs more, and the consumer loses the net gain from buying 4.2 (13–8.8) units at the lower initial price.
- 4.
From the budget constraint, \(m = y + p \cdot x\), with y units defined so that its price is 1. Thus, \(y = m - p \cdot x\).
- 5.
We selected the values for m and p, m0 and po as follows. We started with x = 10 and used the xy1 and xy2 functions to find a range of y values. We selected a value near the middle of this range and determined the mrs as this point. The initial price is approximately that mrs value, and the resulting income level is the one required to fit a budget line through the selected point.
- 6.
Selecting y values closer to the xy1 boundary can cause the graphic representations below to misbehave. Doi et al. modify this function to remove this difficulty, but doing so makes using it for our illustrative purposes more difficult.
- 7.
The requisite number of points can vary. The command has a default value that determines how finely the curve is represented. If the resulting graph is insufficiently smooth, this value can be changed using either of two commands, ip_grid or ip_grid_in. Increasing the number of values of the parameter values for which the functions are to be evaluated can slow processing appreciably.
References
Hausman JA (1981) Exact consumer’s surplus and deadweight loss. Am Econ Rev 71:662–676
Rutherford TF (2008) Calibrated CES utility functions: a worked example, mpsge.org/ calibration.pdf
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Hammock, M.R., Mixon, J.W. (2013). Preferences and Demand. In: Microeconomic Theory and Computation. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9417-1_5
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DOI: https://doi.org/10.1007/978-1-4614-9417-1_5
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