Calculating a Giffen Good

This paper provides a simple example of the utility function with two consumption goods which can be calculated by hand to produce a Giffen good. It is based on the theoretical result by Kubler et al. (Am Econ Rev 103:1034–1053, 2013). Using a model of portfolio selection with a risk-free asset and a risky asset, they showed that there always exists a parameter set which assures that the risk-free asset becomes a Giffen good if the utility function belongs to the HARA (hyperbolic absolute risk aversion) family with decreasing absolute risk aversion (DARA) and decreasing relative risk aversion (DRRA). This paper investigates their result further in a usual microeconomic setting where the risk-free asset and the risky asset are changed to the first and second consumption goods, respectively. It is organized as follows. First, a standard utility maximization problem of a consumer is directly solved to obtain the conditions for the first good to be a Giffen good. Second, the same problem is analyzed by means of decompositions of the price effect due to Slutsky and Sasakura (Italian Econ J 2:258–280, 2016). As is well known, the former decomposition consists of the substitution effect and the income effect, while the latter implies the decomposition into the ratio effect and the unit-elasticity effect. Lastly these analyses are compared and summarized. It should be added that the utility function proposed in this paper can also be used for the analysis of a normal good mutatis mutandis.