3. A canonical utility function. Consider the utility function u(c) 1-σ where c denotes consumption of some arbitrary good and ơ (Greek lowercase letter sigma") is known as the "curvature parameter" because its value governs how curved the utility function is. In the following, restrict your attention to the region c> (because "negative consumption" is an ill-defined concept). The parameter σ is treated as a constant. Plot the utility function for ơ-0. Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function? Plot the utility function for ơ marginal utility? Is marginal utility ever negative for this utility function? Consider instead the natural-log utility function u(c)=In(c). Does this utility function display diminishing marginal utility? Is marginal utility for this utility function? Determine the value of σ (if any value exists at all) that makes the general utility function presented above collapse to the natural-log utility function in part c. (Hint: Examine the derivatives of the two functions.) a. b. 2. Does this utility function display diminishing c. ever negative d.

Suppose a consumer has utility function U(X, Y) = X2/3Y1/3. Suppose the consumer has M =$90 to spend and the prices of goods X and Y are PX = $15 and PY = $6 . a. Carefully express the consumer’s choice problem, using the given information (this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it todetermine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether|MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumer’s choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the…

1) Suppose that a person consumes two goods, x and y, in fixed proportions. He or she always consumes 1 unit of x together with 3 units of y no matter what the relative prices are.a) What is the mathematical form for this person's utility function?b) Calculate the Marshallian demand functions for both goods for this person.c) Calculate the indirect utility function and the expenditure function for this person.d) In class we discussed why expenditure functions are concave in prices. Is the expenditure function you calculated in part (c) concave in ???

a. Suppose Fiona’s income is $100 per week, which she allocates between chocolates and books. Chocolates cost $2 each. Books cost $10 each if she purchases between 1 and 5 books. If she purchases more than 5 books in a week, the price falls to $5 for the 6th book and all subsequent books. Draw the budget constraint. Is it possible that Fiona might have more than one utility-maximizing solution? b. Confirm that if a consumer’s utility function is described by U = 2X + Y, and prices are px = 2 and py = 1, there is no unique utility maximizing solution regardless of income level. What does this tell you about X and Y as commodities? (Hint: draw a graph showing a budget constraint and indifference curve using the information provided.)