Modeling and analyzing Consumer Theory using Graphs

I. Model
1. Specify two goods. For example food F and clothing C
2. Specify a consumer’s utility function. For example 𝑢(𝐹, 𝐶) = 𝐹0.5𝐶0.5.

II. Analysis
1. Given the above model, solve the following problems.
a. Draw an indifference curve by connecting multiple consumption bundles.
b. Derive the marginal substitution rate (MRS) from the utility function.
c. Given the prices, 𝑃 f, 𝑃 c, and the income 𝐼, express and draw the budget constraint.
d. Solve the optimal consumption bundle as functions of 𝑃f, 𝑃c, and 𝐼. 𝐹𝐶
e. Derive and draw the individual demand curve for each good.
f. Illustrate the substitution and income effect for each good caused by an increase in its own price.
g. Consider a certain and uncertain income with the same expectation, for example, 1000. In particular, be clear about the distribution of the uncertain income. Specify the prices of the two goods. Solve the expected utility for each income under the associated optimal consumption choice. Compare the two expected utilities. How much is the difference? Is the consumer risk-averse, risk-neutral, or risk-loving?