Cobb Douglas Utility Functions and Optimal Bundles

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.7: Determine if a good is normal, borderline, inferior, regular/ordinary, Giffen, luxury or necessity.
III.C.8: Define and recognize whether goods are complements or substitutes.

For the following exercise, use this diagram:

Consider the Cobb-Douglas utility function $u(x_1,x_2) = x_1^{0.7}x_2^{0.3}$
Let the budget constraint be $p_1x_1 + p_2x_2 = I$ where $p_1$ and $p_2$ are the prices of good $x_1$ and $x_2$ respectively, and $I$ denotes the income.

Write the Lagrangian for this utility maximization problem.
Solve the first-order conditions to find the demand functions for both good $x_1$ and $x_2$
In the optimal consumption bundle, how much money is spent on $x_1$ ? How much on $x_2$ ? Because this is a Cobb-Douglas utility function you can (and should) express your answers as proportions of the total income $I$.

Now assume $p_1 = 2$, $p_2 = 1$, and $I = 40$.

Find the optimal choices using the visualization above. Show that this is consistent with the demand functions you derived.
Suppose the price of $p_2$ increases. How does the optimal consumption bundle change? Draw the graph at the original given price, and then two more graphs with higher prices.
Suppose instead that the price of good $x_1$ increases to 6. How does the optimal consumption bundle change? That is, what are the new $x_1^*$ and $x_2^*$?
Are the goods complements or substitutes?
Suppose instead that the income doubles. What is the new optimal consumption bundle? How does the spending proportion change? Are the goods normal? Draw the graph before and after the increase in income.