Problem Set #5: Production and Cost
1) Consider the following production function for your company
Q  25L0.35 K 1.45
a.
suppose the labor input is increased by 20 percent next year (with K held constant), determine
the approximate percentage change in output (20%*0.35)
b.
What type of returns to scale characterizes this production function? Increasing returns to
scale. How do you know? Elasticities (Labour = 0.35 & capital=1.45) add up to more than 1
c.
Based on the production function, determine an expression for the marginal product of labour.
Take the derivative with respect to labor. MPL = 25(0.35)L-0.65 K1.45
Q.2.
Explain the difference between law of diminishing returns and the diminishing returns to scale?
Answer: Diminishing returns means that, in the short run, marginal product is falling because
each worker is producing less than the one before him, due to the fact that capital (or some
other factor of production) is fixed. Decreasing returns to scale means that, in the long
run, average costs are rising, usually due to coordination problems. There is no connection
between these things
Another way to answer this question is: Only one factor (the variable factor) changes under
‘diminishing returns” and marginal output is decreasing. Under diminishing returns to
scale, all factors are variable and output is decreasing with increasing use of inputs.
Q.3
Having just completed a class in “production and costs”, Tian, the owner of Tian Industries, decided to pay her
workers a wage equal to the value marginal product (w =MPL*P). She found out that her labour employment is
at a point where MPL>APL. In what stage of production is she operating? Is she maximizing profits at this point?
Explain fully.( 4 lines)
Tian is in Stage 1 which is not maximizing profits. She is paying her workers more than her
Total revenue despite the fact that w=MPL*P.
Algebraically, if w=MPL*P then w/p=MPL. If MPL > APL then w/p>APL and APL=Q/L then
wL>PQ  payroll (wage bill) is greater than revenue (Price*Q)
Q.4
Suppose your firm produces small garden tractors requiring tires. Suddenly the price of tires falls
because of extra supply of tires. The reduction in costs induces your firm to increase production and
thus increase the purchase of tires. However, a complementary input, steel wheels, is also necessary
(tires will not work without steel wheels). Should your firm make a relatively large purchase of tires if
the supply of wheels is inelastic? What happens if steel wheels have an easy and available substitute,
aluminum wheels?
Be more interested in acquiring tires when the supply of steel wheels is elastic, as this allows using
the tires with the steel wheels. However, if the supply of steel wheels is inelastic ( it is difficult to
find wheels), then the interest to buy tires will still exist if an easy substitute for steel wheels (like
aluminum wheels) can be found. Thus a reduction in the price of tires will lead to a greater
purchase of tires the easier it is to find a substitute (aluminum wheels) for a complementary input
(steel wheels)
Q.5
What are “economies of scale”? What are the implications of ‘economies of scale”? What are
economies of scope? If a firm produces two products x and y, do economies of scale imply economies
of scope?
Under ‘economies of scale” long run average costs decline as output increases. Implication: the
higher the economies of scale for a given market size, the lower the number of firms in the
market.
Under “economies of scope”, producing two or more products costs less on average than
producing them separately.
If the long run average cost of x decreases as more x is produced and the long run average cost of
y decreases as more y is produced, then the average of producing x and y together is not
necessarily lower. Conversely, a firm may experience economies of scope but not economies of
scale.
Q6
The management of Waleed Industries is considering a plan to terminate a new employee. The action stemmed
from documented evidence supplied by the firm's accounting department that this new employee did
not add as much to the firm's overall output as did a worker hired two weeks earlier. Based on this
evidence, do you agree that the latest worker hired should be fired? Explain
No. In order to maximize profits, firms should hire workers up to the point where the value
marginal product equals the wage rate in the range of diminishing marginal returns. The data
suggests that the last worker added less to total output that the previous worker, which means that
the firm is indeed operating in the range of diminishing marginal returns, as it should. The
worker should be fired if his or her value marginal product is less than the wage. Unfortunately,
management is not considering this information
Q7
A firm has two plants, one in the United States and one in Mexico, and it cannot change the size of the plants or
the amount of capital equipment. The wage in Mexico is $5. The wage in the U.S. is $20. Given current
employment, the marginal product of the last worker in Mexico is 100, and the marginal product of the last
worker in the U.S. is 500.
a.
Is the firm maximizing output relative to its labor cost? Show how you know.
b.
If it is not, what should the firm do?
Answer: a. Maximizing output between plants requires that (MPL/w)U.S. = (MPL/w)MEXICO.
Since 100/5 (=20) is not equal to 500/20 (=25), the firm is not maximizing output relative to its labor cost.
b. The firm should hire more U.S. workers and fewer Mexican workers, all other things equal.
Q8
In the aftermath of the recent earthquake in Haiti, you took a one-month leave of absence (without pay) from
your $4000 per month job in order to operate a kiosk, in Port au Prince, the capital of Haiti, selling fresh drinking
water. During the month you operated this venture, you paid the government of Haiti $2000 in kiosk rent and
purchased water from a local wholesaler at a price of $1.23 per gallon.
a. Write an equation that summarizes the cost function of your operation as well as equations that
summarize the marginal, average variable, average fixed and average total cost of selling fresh drinking
water at the kiosk. ?
ANSWER: Taking into account both implicit and explicit costs, the total fixed cost from operating the
kiosk is $6,000; the $2,000 in rent plus the $4,000 in forgone earnings. Total variable costs are $1.23
per gallon. The cost function is CQ  6,000  1.23Q . The marginal cost is
dC Q 
 $1.23 ; the wholesale price. The average variable cost is
dQ
C Q  1.23Q
$6000
.
AVCQ  

 $1.23 . The average fixed cost is AFCQ  
Q
Q
Q
MC Q  
Q.11
How would each of the following affect the firm's marginal, average, and average variable cost curves?
a. An increase in wages  increase in variable input priceincrease in AC, AVC and MC
b. A decrease in material costs decrease in variable input price  decrease in AC, AVC MC
c. The government imposes a fixed amount of tax.  increase in fixed cost. Increases only AC. MC
and AVC unchanged
d. The rent that the firm pays on the building that it leases decreases  decrease in only AC. MC and
AVC unchanged
Q.12
The relativity of SWOT analysis: Your store has multiple locations. In a meeting, one of your managers
presents the following lists of “strengths” as part of a SWOT analysis:
a. We are located where customer traffic is high : Location is costly in high traffic locations
b. We carry exclusive high-margin merchandize: Much of the high-margin merchandise does
not sell
c. We have lower costs of labor (salespersons): Firm pays lower salaries but has high labor
turnover, subsidizing competitors
d. We engage in brand partnerships with other firms. High fees have to paid to owners of the
brands the firm partners with
You remember your course in Strategic Management at VIU, particularly the lessons on SWOT
analysis. Evaluate each item in the list and determine the circumstance under which each item could be
a weakness. In other words, explain to your peers the other side of the firm’s purported “strengths”.
Q.13 (we haven’t fully discussed this in class but give it your best shot.)
A multi-product firm’s cost function was recently estimated as
TC (Q1Q2 )  75  0.25Q1Q2  0..1Q12  0.2Q22
a. Are there economies of scope in producing 10 units of product 1 and 10 units of product 2?
Explain
TC (Q1, Q2) = 75 –0.25(10)(10)+0.1(100) + 0.2(100) = 80
TC (Q1) = 75 + 0.1(100) =85
TC (Q2) = 75 + 0.2(100) = 95
Total cost of producing
Q1 and Q2 together (80) is less than producing them separately (85 + 95)
b. Are there cost complementarities in producing products 1 and 2? Explain
Cost complementarities exist if increasing the production of Q1, reduces the MC of Q2
(or the other way round)
MC(Q2) = -0.25Q1 +0.02Q2.
As the production of Q1 increases, the MC of Q2 decreases. Yes there are cost
complementarities in producing products 1 and 2 together
c. Suppose the division selling product 2 is floundering and another company has made an offer
to buy the exclusive rights to produce product 2. How would the sale of the rights to produce
2 change the firm’s marginal cost of producing product 1? Explain
If firm produces only product 1, the MC = 0.2Q1, this is higher than if it produces both
products MC(Q1) = 0.2Q1 – 0.25Q2