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MATH96011 Mathematics of Business and Economics

创建日期：2024-05-09 15:37:21

所属分类：数学mathematics

标签： MATH96011

Mathematics of Business and Economics

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MATH96011

Mathematics of Business and Economics
SUBMIT YOUR ANSWERS AS SEPARATE PDFs TO THE RELEVANT DROPBOXES ON
BLACKBOARD INCLUDING A COMPLETED COVERSHEET WITH YOUR CID NUMBER,
QUESTION NUMBERS ANSWERED AND PAGE NUMBERS PER QUESTION.
Date: Friday, 28 May 2021
Time: 09:00 to 11:00
Time Allowed: 2 hours
Upload Time Allowed: 30 minutes
This paper has 4 Questions.
Candidates should start their solutions to each question on a new sheet of paper.
Each sheet of paper should have your CID, Question Number and Page Number on the
top.
Only use 1 side of the paper.
Allow margins for marking.
Any required additional material(s) will be provided.
Credit will be given for all questions attempted.
Each question carries equal weight.

1. (a) State the consumer-side analogue notions to the following:
(i) Profit maximisation problem
(ii) Production function
(iii) Cost function
(iv) Marginal rate of technical substitution
(v) Conditional factor demand function
(vi) Isoquant (6 marks)
(b) Decide if the following statements are true or false. Justify your answers.
(i) The market demand curve for heroin is extremely inelastic. The market is monopolised by
the Mafia and the Mafia is only interested in maximising their profits. The two statements
are consistent with one another. (2 marks)
(ii) A firm may be willing to accept losses in the short-run. (2 marks)
(iii) Ordinal utility is a much weaker notion than cardinal utility. (2 marks)
(c) (i) Briefly explain a U-shaped long-run average cost curve in terms of the scale behaviour.
(3 marks)
(ii) Using your answer from part (c) (i), prove that there is no homogeneous production
function (of any degree k ∈ R) that gives a U-shaped long-run average cost curve.
(5 marks)
MATH96011 Mathematics of Business and Economics (2021) Page 2
2. Consider a firm that produces a single output using two input factors. The production function is
given by:
f(x1, x2) = x1x2.
Assume that the output price p and the input prices w1, w2 are all positive.
(a) Compute the elasticity of scale of f and comment on it. (2 marks)
(b) Compute the conditional factor demand function x∗(w1, w2, y) and the cost function
c∗(w1, w2, y). (3 marks)
(c) Verify Shephard’s Lemma. (1 mark)
(d) Verify that the conditional factor demand function and the cost function in (b) satisfy the
required homogeneity properties. (2 marks)
(e) Compute the optimal output y∗(p, w1, w2) and the profit function pi∗(p, w1, w2). (4 marks)
(f) Check if the conditions for the profit-maximising output are satisfied. (4 marks)
(g) Briefly explain what is the expected shape of the marginal cost curve and the average cost
curve and why it does not agree with the results in the section “geometry of costs”. (You are not
asked to sketch a graph.) (4 marks)
MATH96011 Mathematics of Business and Economics (2021) Page 3
3. (a) (i) Determine the relation between the Weak Axiom of Profit Maximisation and the Weak
Axiom of Cost Minimisation. (2 marks)
(ii) Determine the relation between the utility maximisation and expenditure minimisation.
(2 marks)
(b) Consider the following preference relation on R2: for any x = (x1, x2) ∈ R2 and
y = (y1, y2) ∈ R2 it holds that x y if and only if
x1 ≥ y1 and x2 ≥ y2.
(i) Check whether completeness, transitivity, weak/strong monotonicity, local nonsatiation,
and (strict) convexity are satisfied giving a counterexample or a proof. (8 marks)
(ii) Suppose there is a utility function u : R2 → R representing . Show that u is an injection.
(3 marks)
(c) Consider the following possible assumptions made to solve a cost-minimisation problem:
(i) the second order necessary condition holds at all points,
(ii) the production function is quasi-concave,
(iii) the production function is strictly quasi-concave,
(iv) the second order sufficient condition holds at all points.
Briefly explain the advantage of making assumption (ii) instead of (i); assumption (iii) instead
of (ii); and assumption (iv) instead of (iii). (5 marks)
MATH96011 Mathematics of Business and Economics (2021) Page 4
4. (a) Give examples of economic activities that:
(i) are not included in the Gross Domestic Product. (1 mark)
(ii) are not included in the Gross National Product. (1 mark)
(b) Suppose the economy is described by the five-sector model.
(i) Describe the equilibrium in the model and define the notation used. (2 marks)
(ii) Describe what happens if injections exceed leakages over a given time period.
(4 marks)
(c) Consider the oil market. There are n consumers, each having the same utility function, which
leads to the following market demand (measured in thousands of barrels):
X∗(p) =
n
(
a−p
2
)
, 0 ≤ p < a,
0, p ≥ a,
where p ≥ 0 is the oil price in pounds per barrel and a > 0 constant.
There are also m firms producing oil in a perfect competition and each of the firms has a
long-run cost function (reflecting the actual economic costs) c∗(y) = y(b+ y), where y ≥ 0
is the firm’s output in thousands of barrels and b ∈ (0, a) constant.
(i) Compute the individual supply function and use it to compute the market supply, Y ∗(p).
(Note: You do not have to check the second order condition.) (3 marks)
(ii) Can an individual firm ever have negative profit? Explain briefly. (2 marks)
(iii) Sketch a graph of the market demand curve and the market supply curve and compute
the equilibrium price, p∗ and the equilibrium quantity, q∗. (3 marks)
(iv) Using the graph from part (c) (iii), show graphically and justify what happens to the
market equilibrium (p∗, q∗) when (i) n increases while m is held constant; (ii) m increases
while n is held constant. (It would be preferable to draw one graph for each case.)
(4 marks)
MATH96011 Mathematics of Business and Economics (2021) Page 5
1. (a) State the consumer-side analogue notions to the following:
(i) Profit maximisation problem
(ii) Production function
(iii) Cost function
(iv) Marginal rate of technical substitution
(v) Conditional factor demand function
(vi) Isoquant
SOLUTION: (SEEN)
(i) Utility maximisation problem
(ii) Utility function
(iii) Expenditure function
(iv) Marginal rate of substitution
(v) Hicksian demand
(vi) Indifference curve (6 marks)
(b) Decide if the following statements are true or false. Justify your answers.
(i) The market demand curve for heroin is extremely inelastic. The market is monopolised by
the Mafia and the Mafia is only interested in maximising their profits. The two statements
are consistent with one another.
(ii) A firm may be willing to accept losses in the short-run.
(iii) Ordinal utility is a much weaker notion than cardinal utility.
SOLUTION: (UNSEEN/SEEN/SEEN)
(i) False. A monopolist can only maximise their profits when faced with an elastic market
demand curve. (2 marks)
(ii) True. A firm may be willing to accept losses in the short-run, because there are fixed
costs in the short-run, while in the long-run all inputs may vary. (2 marks)
(iii) True. Ordinal utility is a much weaker notion than cardinal utility because it only requires
that the consumer be able to rank goods in the order of his/her preference. (2 marks)
(c) (i) Briefly explain a U-shaped long-run average cost curve in terms of the scale behaviour.
SOLUTION: (UNSEEN)
A downward-sloping long-run average cost curve shows increasing returns to scale, a flat
long-run average cost curve shows constant returns to scale, and an upward-sloping long-run
average cost curve shows decreasing returns to scale. (3 marks)
(ii) Using your answer from part (c) (i), prove that there is no homogeneous production
function (of any degree k ∈ R) that gives a U-shaped long-run average cost curve.
SOLUTION: (UNSEEN)
A homogeneous function has f(tx) = tkf(x) for some k ∈ R. We differentiate wrt t:
MATH96011 Mathematics of Business and Economics (2021) Page 2
n∑
i=1
∂f(tx)
∂xi
xi = ktk−1f(x)
If we set t = 1, we get:
n∑
i=1
∂f(x)
∂xi
xi = kf(x)⇒ e(x) = k
where e(x) is the local elasticity wrt the scale at x. Since e(x) = k is a constant, we either
have k > 1, which implies increasing returns to scale everywhere, or k = 1, which implies
constant returns to scale everywhere or k < 1, which implies decreasing returns to scale
everywhere, so none of these gives a typical U-shaped long-run average cost curve.
(5 marks)
MATH96011 Mathematics of Business and Economics (2021) Page 3
2. Consider a firm that produces a single output using two input factors. The production function is
given by:
f(x1, x2) = x1x2.
Assume that the output price p and the input prices w1, w2 are all positive.
(a) Compute the elasticity of scale of f and comment on it.
SOLUTION: (SEEN SIMILAR) Let (x1, x2) ∈ R2≥0. The partial derivatives are given by:
∂1f(x1, x2) = x2 , ∂2f(x1, x2) = x1.
Hence, the elasticity of scale of f at (x1, x2) is given by:
e(x1, x2) =
〈∇f(x1, x2), (x1, x2)〉
f(x1, x2)
= ∂1f(x1, x2)x1 + ∂2f(x1, x2)x2
f(x1, x2)
= 2 .
This coincides also with the degree of homogeneity and it shows us that f has increasing returns
to scale.
(2 marks)
(b) Compute the conditional factor demand function x∗(w1, w2, y) and the cost function
c∗(w1, w2, y).
SOLUTION: (SEEN SIMILAR) We determine the minimiser of w1x1 + w2x2 subject to
x1x2 = y. For y = 0, we clearly have x∗1(w1, w2, 0) = x∗1(w1, w2, 0) = 0. For y > 0, we
must have x1, x2 > 0. Hence, the constraint yields that x1 = y/x2. Substituting into the cost
function yields
w1x1 + w2
y
x1
.
This is a convex function in x1. So it suffices to consider the first order condition only. This yields
x∗1(w1, w2, y) =
(
y
w2
w1
)1/2
.
Similarly, one obtains
x∗2(w1, w2, y) =
(
y
w1
w2
)1/2
.
Finally, the cost function is given by
c∗(w1, w2, y) = w1x∗1(w1, w2, y) + w2x∗2(w1, w2, y) = 2
(
yw1w2
)1/2
.
(3 marks)
MATH96011 Mathematics of Business and Economics (2021) Page 4
(c) Verify Shephard’s Lemma.
SOLUTION: (SEEN SIMILAR) We get indeed that:
x∗1(w1, w2, y) =
∂
∂w1
c∗(w1, w2, y), x∗2(w1, w2, y) =
∂
∂w2
c∗(w1, w2, y) .
(1 mark)
(d) Verify that the conditional factor demand function and the cost function in (b) satisfy the
required homogeneity properties.
SOLUTION: (SEEN SIMILAR) We should verify that the conditional factor demand functions
are homogeneous of degree 0 in w and the cost function is homogeneous of degree 1 in w. We
get indeed that:
x∗1(tw1, tw2, y) =
(
y
tw2
tw1
)1/2
= x∗1(w1, w2, y), x∗2(tw1, tw2, y) =
(
y
tw1
tw2
)1/2
= x∗2(w1, w2, y)
c∗(tw1, tw2, y) = 2
(
t2
)1/2(
yw1w2
)1/2
= tc∗(w1, w2, y)
(2 marks)
(e) Compute the optimal output y∗(p, w1, w2) and the profit function pi∗(p, w1, w2).
SOLUTION: (UNSEEN) The profit at output y is given by:
pi(p, w1, w2, y) = py − y1/2 2(w1w2)1/2 .
We can see that – as the sum of two convex functions – it is a convex function in y. For p > 0 it
has a global minimum at the critical point y0 = w1w2p2 and diverges to infinity as y →∞. Thus,
y∗(p, w1, w2) =∞, pi∗(p, w1, w2) =∞
(4 marks)
(f) Check if the conditions for the profit-maximising output are satisfied.

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BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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