1. Acme, Inc. produces widgets (w) and gadgets (g). Its profit function is given by:
TO = 20w -
a) How many widgets and gadgets should Acme sell to maximize profits?
b) Suppose there is a constraint on production such that w + g 8. Compute the optimal
quantity of widgets and gadgets.
2. A firm has revenue and cost functions given by:
R = 1000Q-2Q²
C = Q3-59Q² + 1315Q-2000
Show the profit function. Note that the function is "optimized" when Q = 3 and Q = 35. Find the
profit maximizing level of output and profits by checking the sign of the second derivative of the
profit function with respect to Q.
For five years, an oil drilling company has operated profitably in the state of Alaska (the only place it
operates). Last year, the state legislature instituted a flat annual tax of $100,000 on any company
extracting oil (or natural gas) in Alaska. How would this tax affect the amount of oil the company
extracts? Explain. Suppose instead the state imposes a wellhead tax-that is, oil companies must
pay a tax of $2.00 on each barrel of oil extracted. How would this tax affect the amount of oil the
company extracts? Explain.
4. Suppose you are the manager of a California winery. How would you expect the following
events to affect the price for a bottle of wine (assume that you are a "price-taker")? Use
demand and supply curves to illustrate.
a. The price of comparable French wines decrease.
b. The unemployment rate in the United States decreases.
c. The price of cheese increases.
d. The price of a glass bottle significantly increases due to new government anti-shatter
e. Researchers discover a new wine-making technology that reduces production costs.
The price of vinegar, which is made from leftover grape mash, increases.
g. The average age of consumers decreases, and older people drink less wine.
5. Differentiate the functions defined by the various formulas:
b. 12-2x2+5x3 2 1 X
c. 1-3x7 -
d. x-2(22+1)VE -
e. x+1 Xt/
For this function, determine the value(s) of x where dy/dx = 0:
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