1. Compensating and Equivalent Variation

Write a question that involves a price increase of $5 on one good in a two good set up and ask what the level of compensating variation and equivalent variation associated with the price decrease is. The prices should all be positive integers less than $100. Restrict yourself to the Cobb-Douglas utility function (1, 2 ) = 1 22. If you find yourself bogged down in algebra, try to skip steps for ample partial credit. For example, if you need to, if you get to one equation with one unknown, you can write "Let (this symbol) be the solution to the above equation. The level of compensating variation is (answer)." or similar. This should include a series of several different subquestions and ask to derive an Engel curve and an income offer curve under the original prices as a part of the answer. Note that you do not have to graph these as a part of the answer; here, you only need to specify the equation for the Engel and income offer curves mathematically. It should also ask if the consumer has diminishing marginal utility as a part of the problem. Next, consider the same exact problem where the same price increases by $10 instead of $5 and solve for whether the compensating and equivalent variation grow linearly with the price increase. Please feel free to use a calculator if the numbers get large.

2. This question has two parts in order to make the math more straightforward. Part I: Specify a cost function for a firm of the form () = 3 2 + + where , , and are positive integers less than or equal to 20 and = 6. Ask to solve for an expression for variable costs, fixed costs, marginal costs, average variable costs, average fixed costs, and average costs. Demonstrate mathematically that marginal costs intersect average variable costs at their minimum. Part II: Now suppose that the cost function is () = 2 + where and are positive integers less than or equal to 20. Identify the regions of output where the firm has increasing, constant, and decreasing returns to scale. Only consider positive values of output.

3. Preferences and Utility Maximization Specify at least 4 utility functions and then for each one ask to solve for the marginal rate of substitution, to check whether the monotonicity assumption holds, and whether the assumptions of either weak or strict convexity hold. At least one utility function should be not weakly or strictly convex at all, at least one should be weakly convex, and at least one should be strictly convex. The numbers 1, 5, 15, and 20 should be used in at least one of these utility functions each. Using one of these numbers in each separate utility function, for example, would work well (e.g. 1 in the first utility function, 5 in the second, etc.). Lastly, ask and solve a utility maximization problem with a utility function that you chose that is strictly convex.

4. Write a profit maximization problem with the production function (1, 2 ) = (1 ) + (2 ) where and are positive integers less than 10.