Problem 1- Suppose U(x, y) = 4x² + 3y².

a. Calculate ∂U/∂x, ∂U/∂y.

b. Evaluate these partial derivatives at x = 1, y = 2.

c. Write the total differential for U.

d. Calculate dy/dx for dU = 0-that is, what is the implied trade-off between x and y holding U constant?

e. Show U= 16 when x = 1, y = 2.

f. In what ratio must x and y change to hold U constant at 16 for movements away from x = 1, y = 2?

g. More generally, what is the shape of the U = 16 contour line for this function? What is the slope of that line?

Problem 2- Suppose a firm's total revenues depend on the amount produced (q) according to the function

R=70q - q².

Total costs also depend on q:

C=q²+30q+100.

a. What level of output should the firm produce in order to maximize profits (R - C)? What will profits be?

b. Show that the second-order conditions for a maximum are satisfied at the output level found in part (a).

c. Does the solution calculated here obey the "marginal revenue equals marginal cost" rule? Explain.

Problem 3- Suppose that f (x, y) = xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways: by substitution and by using the Lagrangian multiplier method.

Problem 4- The dual problem to the one described in Problem 3 is

minimize x + y

subject to xy = 0.25

Solve this problem using the Lagrangian technique. Then compare the value you get for the Lagrangian multiplier to the value you got in Problem 3. Explain the relationship between the two solutions.

Problem 5- The height of a ball that is thrown straight up with a certain force is a function of the time (t) from which released given by f(t) = -0.5gt² + 40t (where g is a constant determined by gravity).

a. How does the value of t at which the height of the ball is at a maximum depend on the parameter g?

b. Use your answer to part (a) to describe how maximum height changes as the parameter g changes.

c. Use the envelope theorem to answer part (b) directly.

d. On the Earth g = 32, but this value varies somewhat around the globe. If two locations had gravitational constants that differed by 0.1, what would be the difference in the maximum height of a ball tossed in the two places?

Problem 6- A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3x feet long. Now cut a smaller square that is t feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a traylike structure with no top.

a. Show that the volume of oil that can be held by this tray is given by

V = t(x - 2t)(3x - 2t) = 3tx² - 8t²x + 4t³.

b. How should t be chosen so as to maximize V for any given value of x?

c. Is there a value of x that maximizes the volume of oil that can be carried?

d. Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation 3x² - 4t² = 1,000,000 (because the builder can return the cut-out squares for credit). How does the solution to this constrained maximum problem compare to the solutions described in parts (b) and (c)?

Problem 7-  Consider the following constrained maximization problem:

maximize   y = x₁ + 5 ln x₂

subject to  k - x₁ - x₂ = 0,

where k is a constant that can be assigned any specific value.

a. Show that if k = 10, this problem can be solved as one involving only equality constraints.

b. Show that solving this problem for k = 4 require that x₁ = -1.

c. If the x's in this problem must be nonnegative, what is the optimal solution when k = 4?

d. What is the solution for this problem when k = 20? What do you conclude by comparing this solution to the solution for part (a)?

Note: This problem involves what is called a "quasi-linear function." Such functions provide important examples of some types of behavior in consumer theory-as we shall see.

Problem 8- Suppose that a firm has a marginal cost function given by MC(q) = q + 1.

a. What is this firm's total cost function? Explain why total costs are known only up to a constant of integration, which represents fixed costs.

b. As you may know from an earlier economics course, if a firm takes price (p) as given in its decisions then it will produce that output for which p= MC(q). If the firm follows this profit maximizing rule, how much will it produce when p = 15? Assuming that the firm Is PI breaking even at this price, what are fixed costs?

c. How much will profits for this firm increase if price increases to 20?

d. Show that, if we continue to assume profit maximization, then this firm's profits can be expressed solely as a function of the price it receives for its output.

e. Show that the increase in profits from p = 15 to p = 20 can be calculated in two ways: (i) directly from the equation derived in part (d); and (ii) by integrating the inverse marginal cost function [MC^-1 (p) = p - 1] from p = 15 to p = 20. Explain this result intuitively using the envelope theorem.