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Assignment 1
Due date: Thu, Sept. 15
Notes: Show all calculations. Draw neat graphs. Total points = 34.
1. (12 points) Consider a firm's cost function: C(Q) = 1.5Q2 + 4Q + 150, where C is total cost
and Q is the amount of output.
a.
b.
c.
d.
e.
(2 points) Obtain the firm's fixed and variable costs. Provide a sketch of each.
(2 points) Obtain the firm's marginal cost function. Provide a sketch.
(2 points) Obtain the marginal cost at Q = 20. What does this tell us?
(2 points) Obtain the firm's average cost function. Provide a sketch.
(2 points) At what output is average cost minimized? [Hint: What is the slope at this
point?] Call this output Q0. Indicate the minimum point on the AC curve.
f. (2 points) At what output are marginal cost and average cost equal to each other? Sketch
MC and AC on the same graph to illustrate.
2. (8 points) Consider a monopolist with the following demand for its product: P(Q) = 500 - 4Q,
where Q is quantity demanded by consumers, and P is the price of a unit of the good.
a. (1 point) Obtain the firm's revenue function R(Q).
b. (2 points) Obtain the marginal revenue function. What does it represent?
c. (2 points) Sketch the demand curve and the MR curve on the same graph. What is the
relationship between the slope of the two curves?
d. (3 points) Sketch the revenue function. Where does it reach a maximum? What is the
value of MR at that point? Explain.
3. (4 points) Consider the implicit function, x2 + 3y2 = 50.
a. (2 points) Totally differentiate to obtain dy/dx (except for y = 0).
b. (2 points) Why is the function not differentiable at y = 0? Plot a sketch to confirm.
4. (10 points) Consider the function g(x) = 24x - 0.25x2.
a. (2 points) Find the first- and second-order derivatives: g'(x) = ______; g"(x) = _______.
b. (2 points) Evaluate the expression and the derivatives at x = 2:
g(2) = _______, g'(2) = _________, g"(2) = ___________.
c. (1 point) Plot g vs. x over x = 0 to 48.
d. (1 point) Plot g' vs. x over x = 0 to 48.
e. (2 points) What is the slope of the function g(x) at x = 5? At x = 6?
f. (2 points) Is the slope of the function increasing--or decreasing--in x? Explain.