Deriving Marshallian Demand from Stone-Geary Utility Function PDF

Preview

Summary

Lecture ...

Description

Stone-Geary Utility Maximisation Harold Walden The corresponding video can be found at:!https://youtu.be/qHZHAJozLc8 The Stone-Geary unction is often used to model problems involving subsistence levels of consumption. In these cases, a certain minimal level of some good has to be consumed, irrespective of its price or the consumer’s income – for example; food and water. In the two-good case, consumers will first set aside a subsidence level of consumption of the two goods; A and B of 𝛼 for good A and 𝛽 for good B. The Stone Geary utility function is based on the traditional Cobb-Douglas utility function 𝑈 𝐴, 𝐵 = 𝐴( 𝐵)*( where 𝛾 is the proportion of each of good A and B consumed (consequently, 𝛾 ∈ ℝ; 0 ≤ 𝛾 ≤ 1 ) As a result the Stone-Geary utility function for the two-good case can be presented as the following; 𝑈 𝐴, 𝐵 = 𝐴 − 𝛼 ( 𝐵 − 𝛽 )*( or more simply;3 𝑢 𝐴, 𝐵 = ln 𝑈 𝐴, 𝐵 = 𝛾 ln( 𝐴 − 𝛼) + (1 − 𝛾) ln 𝐵 − 𝛽 The derivation of consumer demand involving the subsistence levels of consumption involves the same method as the regular Cobb-Douglas (maybe with a sexier standard of algebra) however, in its derivation we will yield some interesting results. To begin with let us first set up the consumer utility optimisation problem and the resulting Lagrangian function. Maximise: 𝑢 𝐴, 𝐵 = 𝛾 ln( 𝐴 − 𝛼) + (1 − 𝛾) ln 𝐵 − 𝛽 Subject to: 𝑀 = 𝑃< 𝐴 + 𝑃= 𝐵 Hence, ℒ = 𝛾 ln( 𝐴 − 𝛼) + (1 − 𝛾) ln 𝐵 − 𝛽 + 𝜆 𝑀 −...