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Lecture 4

MGEB02H3 Lecture Notes - Lecture 4: Tesseract, Isoquant, List Of Bus Routes In Queens

Economics for Management Studies
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MGEB02F Problem Set 4 Professor Michael Krashinsky
1. You are given a Cobb-Douglas production function: q = K1/3L2/3
a) Graph the isoquant for q=36, placing L on the horizontal axis [Hint: use the production
function to express K in terms of q and L]. One point on the isoquant is (36,36). At that point,
compute the marginal rate of technical substitution (the MRTS) in different ways:
i) measure the slope of the isoquant
ii) use calculus on the production function
iii) increase L to 36.001 and compute the value of K on the isoquant
iv) derive the marginal products of labour and capital and take the ratio
b) Demonstrate that this production function exhibits constant returns to scale two different
ways: i) using algebra ii) by numerical example
c) Do the following production functions exhibit increasing, constant, or decreasing returns to
scale?: i) q = L + 3K
ii) q = L + 3K + 2
iii) q = LK
iv) q = 0.4K + 3L2
v) q = L.5K.7
vi) q = L.5K.3
vii) q = L.3K.7
viii q = (L1/2 +K1/2)2
ix) q = (2L1/2 +K1/2)2
x) q = (L1/2 +K1/2)1.5
xi) q = (L1/2 +K1/2)2.5
d) Can a production function simultaneously exhibit increasing returns to scale and diminishing
marginal productivity?
2a) Given the production function q = K1/3L2/3 from the previous problem, suppose that you
decide to produce 36 units of output. Using the isoquant diagram from the previous problem set,
draw isocost lines to determine what combinations of inputs you should use and what your total
costs will be if:
i) the price of capital (PK) is $2.70 and the price of labour (PL) is $12.80
ii) PK = $8 and PL = $6.75
iii) PK = $3 and PL = $6
iv) PK = $4 and PL = $1
b) Confirm your results using calculus and cost minimization in each case.
c) Consider two other production functions:
i) q = (1/9)(K1/2 +2L1/2)2
ii) q = 3(K−1 + 2L−1)−1
In each case draw the isoquant for q = 36 and confirm that cost minimization occurs at the same
point as the previous production function when PK = $3 and PL = $6. Demonstrate that input
substitution is more dramatic when the relative prices change if the isoquant is flatter, and less
dramatic when the isoquant is “bendier”. Show this on your diagrams when prices change from
PK = $3, PL = $6 to PK = $4, PL = $1.
d) Over time, Canadians have evolved a "throw-away" technology in which consumer goods
(toasters, radios, etc.) are replaced when they malfunction rather than being repaired. Model this
by imagining that "goods" and "repair services" are two inputs into a household production

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function that we use to produce output ("toast", "music", etc.). Ignoring technological change,
use this model to explain why we have altered our "repair services to goods" ratio over time.
3. Tesseracts are produced by capital and labour. You are given, in figure 5.1, four isoquants for
producing 10, 20, 30, and 40 tesseracts. The following points - shown as (L,K) - lie on the
Output Input combinations on the isoquant
10 (1,29),(3,14),(8.8),(12,4),(15,3),(21,2),(39,1)
20 (9,34),(11,25),(15,14),(21,8),(28,5),(39,3),(57,2)
30 (21,50),(25,36),(30,25),(33,20),(39,14),(45,11),(56,8),(87,6)
40 (39,55),(45,42),(53,32),(60,25),(66,20),(72,17),(83,14)
You are given that the price of one unit of labour is $2 and the price of one unit of capital is $3.

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a) Show graphically, using the isoquant diagram, the optimal way to produce 10, 20, 30, and 40
tesseracts. Draw the expansion path at these prices.
b) Using the results obtained above, derive a table for the long run costs of producing 10, 20, 30,
and 40 tesseracts. Your table should include q, TC, AC, and MC (show MC midway between the
two levels of output).
c) Now assume that the manufacturer has already built the optimal plant for producing 30
tesseracts. In the short run, show how the manufacturer would produce 10, 20 and 40 tesseracts.
Use this information to derive a table for the short run costs of the various levels of production of
tesseracts (for q = 0, 10, 20, 30, and 40). Your table should include q, TC, AC, AVC, and MC.
d) On a piece of graph paper, draw, on one graph, both the long run and short run average cost
curves facing the manufacturer. Also draw in the long run and short run marginal cost curves
(again show MC midway between the two levels of output). You might use different colours for
the long run and the short run.
e) On the graph, why does the short run average cost curve never cut below the long run average
cost curve? Why do they touch where they do? Notice that the SRAC curve is not at its
minimum at Q = 30. How do you explain this? Is there any significance to where the SR and
LR MC curves intersect?
4a) Suppose that you are told that a firm is producing where PK = $2, MPK = 10, PL = $3, and
MPL = 20. Explain verbally how this firm could produce at a lower cost and why changing input
ratios would cut costs. Now show the cost reduction suing an isocost-isoquant sketch.
b) Repeat if a firm produces where PK = $2, MPK = 10, PL = $3, and MPL = 5.
c) Repeat if a firm produces where PK = $2, PL = $3, and RTS = 1.
5a) During a period of restraint, the government imposes a freeze on all hiring by government
departments. Shortly thereafter, one department faces an increased workload, which it carries
out by significantly increasing its use of outside personnel (consultants and secretarial services).
Assuming that the department had been using the two types of labour - inside and outside
personnel, which are imperfect substitutes - efficiently before the freeze and the increase in
workload, comment, using diagrams, on the efficiency of production.
b) Mail service is achieved by combining labour and machines. An agreement between the
postal worker's labour union and the post office restricts layoffs. Suppose that technological
change occurs and suppose that production was efficient before that change. Show why the
union agreement will result in higher than necessary costs.
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