Optimal Use of One Input
How do we find the optimal use of one input? (Hint: set what two equations equal to each other)
Set MRP=MC
(True or False) Returns to scale is the percentage change in output for a given percentage change in all inputs.
True
How do we find the optimal use of two inputs? (Hint: set what two equations equal to each other)
Set MPa/Pa=MPb/Pb (set MPa/MCa=MPb/MCb)
How do we find the optimal use of two inputs if they have the same cost? (Hint: Set what two equations equal to each other)
How do you determine which good is more profitable to produce?
By determining which good has a higher marginal profit.
If the cost of a worker is $17 per time period and the MRP per worker is constant at $15, how many workers should the firm hire?
None
What is the returns to scale for this problem:
Q=4L2K^2
Increasing Returns to Scale
A cleanup program uses two types of workers – highschool students (H) and college students (C). They produce feet of clean streets (Q) per day according to the production function
Q = 100H - 5H2 + 50C - 5C2
Highschool students are paid $10 per day, while college students are paid $20 per day. The company has a daily budget of $1000 to hire highschool and college students. What is the optimal number of college students to hire?
37
Assume there are 2,000 barrels of oil that can go to two refineries. The production functions for the two refineries are:
Refinery A: Q = 45MA - .5MA2
Refinery B: Q = 30MB – 2MB2
What would be the constraint equation in this problem?
Ma+Mb=2000
Daisy the cat makes cookies and biscuits. It takes her 3 hours to make a batch of cookies, which she can sell for a $20 profit. It takes her 5 hours to make a batch of biscuits, which she can sell for a $60 profit.
If she has an extra hour to work, what is the best use of her time?
Making biscuits
If a firm's production process is Q = 30+3L2 , and labor costs $144 per worker and output sells for $12, how much labor should the firm hire?
2
What is the returns to scale for this problem:
Q=2L
constant returns to scale
A firm produces output according to the production function:
Q = 50L – 2L2 + 25K – .5K2
The price of L is $12, and the price of K is $8. There is a budget of $450 available to purchase these inputs. How many K inputs should you buy and how many L inputs should you buy?
15.5
Suppose a firm has two factories, North and South, each producing the same product but using different production processes. After analyzing production reports a student intern realizes that the marginal product of labor in factory North is less than the marginal product of labor in factory South. What should he or she recommend as a short-run remedy?
Move labor from the North factory to the South factory.
A prominent artist can sell small paintings for $10,000 and large paintings for $30,000. Should the artist just stick to larger paintings? What other piece of information do we need?
No. We also need information on how long each piece takes to make or the cost of the materials.
Suppose the production function for a firm is Q = 75L – 4L2 . Output sells for $5 and the cost of labor is $10 per hour. How many hours of labor should the firm hire?
9.125
What is the returns to scale for this problem:
Q=5K+10L
Constant returns to scale
Suppose half-time salespeople bring in $225,000 in sales and cost $30,000 per year. Full-time salespeople bring in $600,000 and cost $80,000 per year. Currently, the firm has 6 full-time and 3 half-time salespeople. Is this the optimal labor mix?
No
Daisy the cat manufactures cookies for her family using hours of time as an input. Daisy operates her cookie making in two locations, one on the bed and one on her chair in the laundry room. The production functions for Daisy cookie making are:
Bed: QB = 35H – 0.25H2
Laundry Chair: QL = 50H – 0.5H2
Where QB and QL are the locations that Daisy can make cookies. H is the amount of hours of work she can put in. She has 60 hours available to work.
How much hours should be spent making cookies on the bed?
30
A firm is deciding how to divide up its crude oil between gasoline production and fiber production. The production functions are:
Gasoline: G = 75MG – 1.5MG2
Fiber: F = 90MF – 2MF2
Constraint: MG + MF = 30
Profit is $0.25 per unit of gasoline and $0.50 per unit of fiber. How much crude oil should the firm allocate to gasoline (MG)?
12.27
Suppose the production function for a firm is Q = 50L – 2L2 . Output sells for $25 and the cost of labor is $13 per hour. How many hours of labor should the firm hire?
12.37
What is the returns to scale for this problem:
Q=20L-.5L^2
A firm produces output according to the production function:
Q = 50L – .5L2 + 50K – 1.5K2
The price of L is $2, and the price of K is $5. There is a budget of $250 available to purchase these inputs. What’s the most output that can be produced from this budget?
(round K and L to two decimal places)
1486.49
Assume there are 500 barrels of oil that can go to two refineries. The production functions for the two refineries are:
Refinery A: Q = 45MA - 1.5MA2
Refinery B: Q = 50MB – MB2
How many barrels of oil should go to Refinery A?
199
A firm is deciding how to divide up its human labor between clothing and automobile production. The production functions are:
Clothing: C = 8C – .5C2
Automobile: A = 80A – 2A2
Constraint: MC + MA = 80
Profit is $0.50 per unit of clothing and $75 per unit of automobiles. How much labor should the firm allocate to clothing (MC)?
59.91