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MATH96011
BSc, MSci and MSc EXAMINATIONS (MATHEMATICS)
May-June 2020
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Associateship of the Royal College of Science
Mathematics of Business and Economics
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Each question carries equal weight.
1. (a) A bicycle is assembled out of a bicycle frame and two wheels.
(i) Write down a production function of a firm that produces bicycles out of frames and
wheels. The firm requires no assembly, so labour is not an input in this case. Draw the
isoquant that shows all combinations of frames and wheels that result in producing 100
bicycles. (4 marks)
(ii) Suppose that initially the price of a frame is £100 and the price of a wheel is £50.
Compute the cost level and draw the corresponding isocost line on the graph you drew
in part (i). (3 marks)
(iii) Repeat the part (ii) if the price of a frame rises to £200, while the price of a wheel
remains £50. (3 marks)
(b) Consider a consumer whose preferences for a pair of goods can be represented by the following
utility function:
u1(x1, x2) =
3
2 log(x1) + log(x2)
(i) If the goods are priced at p = (p1, p2) = (3, 4) and the consumer’s total budget for both
goods is m = 100, compute the bundle that maximises the consumer’s utility. (The
second-order condition doesn’t need to be checked.)
(5 marks)
(ii) Explain briefly the dierence between Marshallian demand and Hicksian demand and thus
decide whether the bundle computed in part (i) is the consumer’s Marshallian demand
or Hicksian demand. (3 marks)
(iii) Suppose a second consumer has the same budget for the same two goods and that their
utility function is given by:
u2(x1, x2) = x
3
2
1 x2
Assuming the same prices for the goods as in part (i), write down the bundle that
maximises the second consumer’s utility and justify your answer. (2 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 2
2. Consider a firm that produces a single output using two input factors. The production function is
given by:
f(x1, x2) = 4
Ô
x1x2
(a) Show that f is a homothetic function. (3 marks)
(b) Compute the elasticity of scale of f . (4 marks)
(c) Does f exhibit increasing, decreasing or constant returns to scale? Justify your answer.
(2 marks)
(d) Compute the marginal rate of technical substitution (MRTS) of f and show that it is positively
homogeneous of degree 0. (4 marks)
(e) Compute the conditional factor demand function xú(w1, w2, y) and the cost function
cú(w1, w2, y), where w1, w2 > 0 are the input prices. (4 marks)
(f) Compute the average cost function and the marginal cost function. (2 marks)
(g) Using the scale behaviour, explain whether the results in part (f) seem reasonable.
(1 mark)
MATH96011 Mathematics of Business and Economics (2020) Page 3
3. (a) Consider a firm that produces a single output using two input factors.
(i) Prove that the Weak Axiom of Profit Maximisation (WAPM) implies the Weak Axiom
of Cost Minimisation (WACM). (5 marks)
(ii) Which of the following datasets:
(i) satisfies the WACM, but not the WAPM?
(ii) satisfies both WAPM and WACM?
(iii) violates both WAPM and WACM?
Show your work and justify your answers.
Dataset A:
t p w1 w2 x1 x2 y
1 4 1 1 1 1 1
2 12 1 2 4 4 2
Dataset B:
t p w1 w2 x1 x2 y
1 4 1 1 1 99 1
2 4 1 2 9 9 3
Dataset C:
t p w1 w2 x1 x2 y
1 4 1 1 1 1 1
2 4 1 2 9 9 3
where t is the time point, p is the output price, wi, i = 1, 2 are the prices for the input
factors, xi, i = 1, 2 are the levels of input factors and y is the level of output.
(9 marks)
(b) Let f : RnØ0 æ RmØ0 be a dierentiable, non-decreasing and quasi-concave production
function.
(i) Prove that the profit function fiú is convex. (4 marks)
(ii) Using (i) prove that:
ˆyúi (p, w)
ˆpi
Ø 0, i = 1, . . . , n
where p is the vector of output prices and w is the vector of input prices.
(2 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 4
4. (a) Consider the market for electricity. Suppose market demand for electricity is given by
Xú(p) = 2000 ≠ 200p, where p Ø 0 is the price in pounds per MWh (megawatt hour)
and the quantity demanded is in thousands of MWh.
We have the following information about the market supply:
• Market research shows that for an increase in price by £1/MWh, quantity supplied
increases by 100,000 MWh.
• Additionally, when the price is £1/MWh, 900,000 MWh are supplied.
(i) Use the above information to determine the market supply Y ú(p) as a linear function of
the price p. (2 marks)
(ii) Determine the equilibrium price and the equilibrium quantity. (3 marks)
(iii) Draw a graph (quantity on the horizontal axis, price on the vertical axis) and depict
the market supply, market demand, equilibrium price and equilibrium quantity as well as
producers’ and consumers’ surplus. (3 marks)
(iv) Compute the producers’ surplus, the consumers’ surplus and the community surplus.
(3 marks)
(b) (i) Define the Gross Domestic Product (GDP). (2 marks)
(ii) State the three approaches used to compute GDP. (3 marks)
(iii) Define the Gross National Product (GNP). (2 marks)
(iv) A British citizen is working for the European Commission in Belgium. Describe the
contribution of his income in the above products. (2 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 5
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BSc, MSci and MSc EXAMINATIONS (MATHEMATICS)
May – June 2020
MATH96011 Mathematics of Business and Economics
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Mathematics of Business and Economics
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© 2020 Imperial College London MATH96011 Page 1 of 11
1. (a) A bicycle is assembled out of a bicycle frame and two wheels.
(i) Write down a production function of a firm that produces bicycles out of frames and
wheels. The firm requires no assembly, so labour is not an input in this case. Draw the
isoquant that shows all combinations of frames and wheels that result in producing 100
bicycles.
Solution: (UNSEEN)
The production function is a Leontief production function given by:
f(x1, x2) = min
Ó
x1,
1
2x2
Ô
where x1 denotes the number of frames and x2 denotes the number of wheels.
The isoquant will be L-shaped: ZheelVfUameV 00100 iVRTXanW
(4 marks)
(ii) Suppose that initially the price of a frame is £100 and the price of a wheel is £50.
Compute the cost level and draw the corresponding isocost line on the graph you drew
in part (i).
Solution: (UNSEEN) ZheelVfUameV 200100 iVRTXanW200 400IVRcRVW OiQe…20,000
(3 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 2
(iii) Repeat the part (ii) if the price of a frame rises to £200, while the price of a wheel
remains £50.
Solution: (UNSEEN) ZheelVfUameV 200100 iVRTXanW200 400IVRcRVW OLQe…20,000 IVRcRVW OLQe…30,00010 00
(3 marks)
(b) Consider a consumer whose preferences for a pair of goods can be represented by the following
utility function:
u1(x1, x2) =
3
2 log(x1) + log(x2)
(i) If the goods are priced at p = (p1, p2) = (3, 4) and the consumer’s total budget for both
goods is m = 100, compute the bundle that maximises the consumer’s utility. (The
second-order condition doesn’t need to be checked.)
Solution: (SEEN SIMILAR)
We look to maximise u1(x) subject to the constraint 3x1 + 4x2 = 100.
First, we define the Lagrangian:
L(x1, x2,⁄) = u1(x)≠ ⁄(px,m) = 32 log(x1) + log(x2)≠ ⁄(3x1 + 4x2 ≠ 100)
First-order conditions for maximisation are therefore:
ˆ
ˆ⁄
L(x1, x2,⁄) = 0∆ px = m∆ 3x1 + 4x2 = 100
ˆ
ˆx1
L(x1, x2,⁄) = 0∆ 32x1 = 3⁄
ˆ
ˆx2
L(x1, x2,⁄) = 0∆ 1
x2
= 4⁄
which we can solve to get xú1(p,m) = 20, xú2(p,m) = 10.
(5 marks)
(ii) Explain briefly the dierence between Marshallian demand and Hicksian demand and thus
decide whether the bundle computed in part (i) is the consumer’s Marshallian demand
or Hicksian demand.
MATH96011 Mathematics of Business and Economics (2020) Page 3
Solution: (SEEN)
The Marshallian demand is the quantity of each good that a consumer requires in order
to maximise their utility for a fixed budget. In contrast, the Hicksian demand is the
quantity of each good that the consumer requires in order to minimise their expenditure
for a fixed level of utility.
The bundle computed in part (i) is the consumer’s Marshallian demand for the two goods.
(3 marks)
(iii) Suppose a second consumer has the same budget for the same two goods and that their
utility function is given by:
u2(x1, x2) = x
3
2
1 x2
Assuming the same prices for the goods as in part (i), write down the bundle that
maximises the second consumer’s utility and justify your answer.
Solution: (UNSEEN)
Since u2 is a positive monotonic transformation of u1, both consumers have the same
preferences, and thus the same Marshallian demand:
xú1(p,m) = 20, xú2(p,m) = 10.
(2 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 4
2. Consider a firm that produces a single output using two input factors. The production function is
given by:
f(x1, x2) = 4
Ô
x1x2
(a) Show that f is a homothetic function.
Solution: (SEEN SIMILAR)
Let g : RØ0 æ RØ0, g(z) = 4z 12 and h : R2Ø0 æ RØ0, h(x1, x2) = x1x2.
Then g is (strictly) increasing, h is positively homogeneous (of degree 2) and f = g ¶ h.
(3 marks)
(b) Compute the elasticity of scale of f .
Solution: (SEEN SIMILAR)
Let (x1, x2) œ R2Ø0. Applying the chain rule, we obtain the partial derivatives:
ˆ1f(x1, x2) =
2Ôx2Ô
x1
, ˆ2f(x1, x2) =
2Ôx1Ô
x2
.
Hence, the elasticity of scale of f at (x1, x2) is given by:
e(x1, x2) =
ÈÒf(x1, x2), (x1, x2)Í
f(x1, x2)
= ˆ1f(x1, x2)x1 + ˆ2f(x1, x2)x2
f(x1, x2)
= 1 .
(4 marks)
(c) Does f exhibit increasing, decreasing or constant returns to scale? Justify your answer.
Solution: (SEEN SIMILAR)
f exhibits constant returns to scale because the elasticity of scale of f is 1.
Alternative answer: f exhibits constant returns to scale because the sum of the exponents of
x1 and x2 is 1.
(2 marks)
(d) Compute the marginal rate of technical substitution (MRTS) of f and show that it is positively
homogeneous of degree 0.
Solution: (SEEN SIMILAR)
Let (x1, x2) œ R2Ø0, x1 > 0. Then the MRTS of f at (x1, x2) is given by:
MRTS(x1, x2) = ≠ˆ1f(x1, x2)
ˆ2f(x1, x2)
= ≠x2
x1
.
For any t > 0 MRTS(tx1, tx2) = MRTS(x1, x2).
(4 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 5
(e) Compute the conditional factor demand function xú(w1, w2, y) and the cost function
cú(w1, w2, y), where w1, w2 > 0 are the input prices.
Solution: (SEEN SIMILAR)
We determine the minimiser of w1x1 + w2x2 subject to 4
Ô
x1x2 = y. For y = 0, we clearly
have xú1(w1, w2, 0) = xú1(w1, w2, 0) = 0. For y > 0, we must have x1, x2 > 0. We can either
consider the first-order conditions or use the marginal rate of technical substitution (part d)
that coincides with the economic rate of substitution and get:
≠x2
x1
= ≠w1
w2
Solving the above equation for x2 and substituting into the constraint yields:
xú1(w1, w2, y) =
y
4
3
w2
w1
41/2
.
By symmetry:
xú2(w1, w2, y) =
y
4
3
w1
w2
41/2
.
Finally, the cost function is given by:
cú(w1, w2, y) = w1xú1(w1, w2, y) + w2xú2(w1, w2, y) =
y
2
1
w1w2
21/2
.
(4 marks)
(f) Compute the average cost function and the marginal cost function.
Solution: (SEEN SIMILAR)
The average cost function is given by:
AC(y) = c
ú(w1, w2, y)
y
= 12 (w1w2)
1/2.
The marginal cost function is given by:
MC(y) = ˆ
ˆy
cú(w1, w2, y) =
1
2 (w1w2)
1/2.
(2 marks)
(g) Using the scale behaviour, explain whether the results in part (f) seem reasonable.
Solution: (UNSEEN)
The results seem reasonable, because f exhibits constant returns to scale and that scale
behaviour implies constant marginal cost and constant average cost.
(1 mark)
MATH96011 Mathematics of Business and Economics (2020) Page 6
3. (a) Consider a firm that produces a single output using two input factors.
(i) Prove that the Weak Axiom of Profit Maximisation (WAPM) implies the Weak Axiom
of Cost Minimisation (WACM).
Solution: (UNSEEN)
Indeed, WAPM asserts that for all s, t:
ptyt ≠ wt1xt1 ≠ wt2xt2 Ø ptys ≠ wt1xs1 ≠ wt2xs2 . (1)
Now consider a pair s, t such that ys Ø yt. Then (1) implies that:
≠wt1xt1 ≠ wt2xt2 Ø pt(ys ≠ yt)¸ ˚˙ ˝
Ø0
≠wt1xs1 ≠ wt2xs2 Ø ≠wt1xs1 ≠ wt2xs2 .
Therefore:
wt1x
t
1 + wt2xt2 Æ wt1xs1 + wt2xs2 .
(5 marks)
(ii) Which of the following datasets:
(i) satisfies the WACM, but not the WAPM?
(ii) satisfies both WAPM and WACM?
(iii) violates both WAPM and WACM?
Show your work and justify your answers.
Dataset A:
t p w1 w2 x1 x2 y
1 4 1 1 1 1 1
2 12 1 2 4 4 2
Dataset B:
t p w1 w2 x1 x2 y
1 4 1 1 1 99 1
2 4 1 2 9 9 3
Dataset C:
t p w1 w2 x1 x2 y
1 4 1 1 1 1 1
2 4 1 2 9 9 3
where t is the time point, p is the output price, wi, i = 1, 2 are the prices for the input
factors, xi, i = 1, 2 are the levels of input factors and y is the level of output.
Solution:
(i) (UNSEEN) Dataset C satisfies the WACM, but not the WAPM. Indeed, we have
that y2 > y1. Hence we only need to check that:
2 = w11x11 + w12x12 Æ w11x21 + w12x22 = 18 .
MATH96011 Mathematics of Business and Economics (2020) Page 7
On the other hand, the WAPM is violated since at time point 2 one has a profit of
12≠ 27 = ≠15. However, if they had produced with the inputs and outputs at time 1,
they could have had a profit of 4≠ 3 = 1.
(3 marks)
(ii) (UNSEEN) Dataset A satisfies both WACM and WAPM. We have that y2 > y1
and
2 = w11x11 + w12x12 Æ w11x21 + w12x22 = 8 .
Moreover, at time point t = 1 the firm makes a profit of:
p1y1 ≠ w11x11 ≠ w12x12 = 4≠ 2 = 2 .
If they had used the output and input at time s = 2 with the prices at time t = 1, they
would have obtained:
p1y2 ≠ w11x21 ≠ w12x22 = 8≠ 8 = 0 < 2 .
(3 marks)
(iii) (SEEN SIMILAR) Dataset B violates both WACM and WAPM. Clearly, the costs
at time point 1 are 100 with an output of y1 = 1. At time point 2, there is a higher
output of y2 = 3. Using prices at time 1, that would have caused costs of 18. Since
we have already established the implication WAPM implies WACM, WAPM cannot be
satisfied.
(3 marks)
(b) Let f : RnØ0 æ RmØ0 be a dierentiable, non-decreasing and quasi-concave production
function.
(i) Prove that the profit function fiú is convex.
Solution: (SEEN)
The profit function at prices (p, w) œ RmØ0 ◊ RnØ0 is defined as:
fiú(p, w) = max
xœRnØ0
fi(x, p, w) .
Let (p, w), (pÕ, wÕ) œ RmØ0 ◊ RnØ0, ⁄ œ [0, 1]. Define (pÕÕ, wÕÕ) = (1 ≠ ⁄)(p, w) +
⁄(pÕ, wÕ). Then:
fiú(pÕÕ, wÕÕ) = pÕÕf
1
xú(pÕÕ, wÕÕ)
2€ ≠ wÕÕxú(pÕÕ, wÕÕ)€
= (1≠ ⁄)
5
pf
1
xú(pÕÕ, wÕÕ)
2€ ≠ wxú(pÕÕ, wÕÕ)€6
+ ⁄
5
pÕf
1
xú(pÕÕ, wÕÕ)
2€ ≠ wÕxú(pÕÕ, wÕÕ)€6
Æ (1≠ ⁄)
5
pf
1
xú(p, w)
2€ ≠ wxú(p, w)€6
+ ⁄
5
pÕf
1
xú(pÕ, wÕ)
2€ ≠ wÕxú(pÕ, wÕ)€6
= (1≠ ⁄)fiú(p, w) + ⁄fiú(pÕ, wÕ) .
(4 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 8
(ii) Using (i) prove that:
ˆyúi (p, w)
ˆpi
Ø 0, i = 1, . . . , n
where p is the vector of output prices and w is the vector of input prices.
Solution: (UNSEEN)
The convexity of the profit function fiú implies that:
ˆ2fiú(p, w)
ˆp2i
Ø 0
and from Hotelling’s Lemma, we have that:
ˆfiú(p, w)
ˆpi
= fi(xú(p, w)) = yúi (p, w).
(2 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 9
4. (a) Consider the market for electricity. Suppose market demand for electricity is given by
Xú(p) = 2000 ≠ 200p, where p Ø 0 is the price in pounds per MWh (megawatt hour)
and the quantity demanded is in thousands of MWh.
We have the following information about the market supply:
• Market research shows that for an increase in price by £1/MWh, quantity supplied
increases by 100,000 MWh.
• Additionally, when the price is £1/MWh, 900,000 MWh are supplied.
(i) Use the above information to determine the market supply Y ú(p) as a linear function of
the price p.
Solution: (UNSEEN)
We consider the linear function Y ú(p) = a + bp, where a = 800, b = 100, p Ø 0 is the
price in pounds per MWh and the quantity supplied is in thousands of MWh.
(2 marks)
(ii) Determine the equilibrium price and the equilibrium quantity.
Solution: (SEEN SIMILAR)
The equilibrium price is obtained by equating Xú(p) = Y ú(p). This yields an equilibrium
price of pú = £4/MWh and an equilibrium quantity of Xú(pú) = Y ú(pú) = 1200
thousands of MWh.
(3 marks)
(iii) Draw a graph (quantity on the horizontal axis, price on the vertical axis) and depict
the market supply, market demand, equilibrium price and equilibrium quantity as well as
producers’ and consumers’ surplus.
Solution: (SEEN SIMILAR)
(3 marks)
(iv) Compute the producers’ surplus, the consumers’ surplus and the community surplus.
MATH96011 Mathematics of Business and Economics (2020) Page 10
Solution: (SEEN SIMILAR)
The producers’ surplus at pú = 4 is given by:
PS(pú) =
⁄ pú
0
Y ú(p)dp =
⁄ 4
0
(100p+ 800)dp = 4000.
Alternatively, the producer’s surplus can be found from the area of the trapezoid:
PS(pú) = 12(1200 + 800)4 = 4000
The consumers’ surplus at pú = 2 is given by:
CS(pú) =
⁄ 10
pú
Xú(p)dp =
⁄ 10
2
(2000≠ 200p)dp = 3600
Alternatively, it can be found from the area of the triangle:
CS(pú) = 12(1200)(10≠ 4) = 3600
The community surplus is the sum of the two, that is, 7600.
(3 marks)
(b) (i) Define the Gross Domestic Product (GDP).
Solution: (SEEN)
The Gross Domestic Product (GDP) measures the nominal gross value of all goods and
services produced in a certain country in a certain period of interest.
(2 marks)
(ii) State the three approaches used to compute GDP.
Solution: (SEEN)
Production approach
Expenditure approach
Income approach
(3 marks)
(iii) Define the Gross National Product (GNP).
Solution: (SEEN)
The Gross Domestic Product (GNP) measures the nominal gross value of all goods and
services produced by all citizens of a certain nationality in a certain period of interest.
(2 marks)
(iv) A British citizen is working for the European Commission in Belgium. Describe the
contribution of his income in the above products.
Solution: (UNSEEN)
Contribution to the GNP of United Kingdom and the GDP of Belgium.
(2 marks)
MATH96011 Mathematics of Business and Economics (2020) Page 11
ExamModuleCode QuestionNComments for Students
MATH96011 1
Question 1 was generally answered well. Part (a) (i) required to spot that wheels and frames must
be used in fixed proportions (Leontief production function); Parts (a) (ii) and (iii) a notable number
of students did not compute the cost level as required and did not mark the intercepts in their
drawings; Part (b) (i) mostly well done, although some students did not state the optimisation
problem; Part (b) (ii) the question was not always answered with the appropriate level of detail;
Part (b) (iii) some students did not spot that u2 is a positive monotonic transformation of u1 and
thus did a lot of extra work.
MATH96011 2
Question 2 was generally answered well. Students have seen questions similar to parts (a)‐(f), most
of the errors were algebraic and some students did not state the optimisation problem in part (e);
Part (g) was intended to be challenging and students were required to spot that the average cost
and the marginal cost are constant. Marks were awarded for a different sensible interpretation, i.e.
link to zero maximised profit.
MATH96011 3
Question 3: part (a)(i) required a proof that the WAPM implies the WACM from the definitions.
There were a few logical fallacies, but this was mostly answered well. Part (a)(ii) required some
calculations on three simple datasets to find out if the WAPM and WACM are satisfied. The main
errors here were thoroughly checking the conditions of each axiom; in some cases not all conditions
were fully checked. Part (b) required a proof that the profit function is convex. This was mostly
answered well, the main mistake being overlooking that the profit function is a function of prices p
and w (some students just proved convexity in p, for example). In part (ii), this was usually
answered correctly or not at all, but always make sure to reference results (e.g. Hotelling’s Lemma)
when you use them.
MATH96011 4
Question 4: part (a)(i) was straightforward and generally answered correctly, likewise for part (a)(ii).
Part (a)(ii) was mostly answered well, aside from some graphs being too imprecise. The market
supply was occasionally drawn incorrectly. Part (iv) required the computation of two integrals (or
calculation of areas in the graph), and this is a part where many students lost marks, either because
of incorrectly evaluating the integrals, or omitting the community surplus from the answer. All parts 
of part (b) were generally answered well.
If your module is taught across multiple year levels, you might have received this form for each level of the module. You are only required to fill this out once for each question.
Please record below, some brief but non‐trivial comments for students about how well (or otherwise) the questions were answered. For example, you may wish to comment on common errors and misconceptions, 
or areas where students have done well. These comments should note any errors in and corrections to the paper. These comments will be made available to students via the MathsCentral Blackboard site and should
not contain any information which identifies individual candidates. Any comments which should be kept confidential should be included as confidential comments for the  Exam Board and Externals. If you would like 
to add formulas, please include a sperate pdf file with your email. 

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