Substitution and Income Effects

LEARNING GOALS:

III.C.2: Solve the utility maximization problem for Marshallian demands, the indirect utility function, and the expenditure function, including corner solutions when prompted.
III.C.3: Solve the expenditure minimization problem for compensated demands, the expenditure function, and the indirect utility function, including corner solutions when prompted.
III.C.5: Explain the Slutksy equation and demonstrate that it holds for a given utility function.
III.C.6: Explain the difference between uncompensated (Marshallian) and compensated (Hicksian) demand functions

Consider the following Cobb-Douglas utility function: $u(x_1,x_2)={x_1^{\alpha}}{x_2^{1-\alpha}}$

Derive the Marshallian demand functions given the budget constraint ${p_1}{x_1}+{p_2}{x_2}=I$
Derive the Hicksian demand functions and expenditure function using the expenditure minimization approach, given the constraint $u(x_1,x_2)=\bar{U}$
If the two goods are normal (i.e., the consumer purchases more of each as their income increases), which set of demand functions, Marshallian or Hicksian, do you think are more sensitive to changes in price? Why? Use your intuition, and don’t do any additional algebra yet.

Use the above visualization to help formulate your answers to the following questions. Assume that $\alpha=0.5$, $p_1=2$, $p^\prime_1$ (i.e., the new price) $=4$, $p_2=4$, and $I=16$.

Which transition, X to C or C to Z, represents the income effect? Which represents the substitution effect? Give a justification for your answer.
In terms of the variables shown in the graph, what is the total change in demand from the price change, and what are the income and substitution effects?
What happens to the magnitudes of the income and substitution effects when $\alpha$ increases or decreases? Answer this question using the graph above. You can check your answer for the income effect by differentiating it with respect to $\alpha$. Don’t try this with the substitution effect–The algebra required is nightmarish.
Still assuming $\alpha=0.5$ and $I=16$, use the Marshallian demand function you derived above and the Slutsky equation to solve for the substitution effect. Show that this matches the substitution effect you can derive from your Hicksian demand function.