Reviews on applications
1 Marginal value, elasticity and the change
1.1 Absolute change and relative change
Given an indicator whose value changed from x to x + ∆x.
Absolute change refers to the simple difference in the indicator over two
periods in time:
∆x.
Relative change expresses the absolute change as a percentage of the value
of the indicator in the earlier period:
∆x
.
x
1.2 Marginal value
Given y = y(x1 , x2 , . . . , xn ). Marginal value of y in xi at x = (x1 , x2 , . . . , xn )
is
∂y(x1 , x2 , . . . , xn )
M yi (x) = .
∂xi
Notice: M yi (x) is the absolute change of y when xi increases 1 unit.
M yi (x) ≈ y(x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xn )−y(x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ).
Calculate the absolute change of y when xi changed to xi + ∆xi for all i:
Direct way:
∆y = y(x1 + ∆x1 , x2 + ∆x2 , . . . , xn + ∆xn ) − y(x1 , x2 , . . . , xn ).
Approximate way:
∆y ≈ M y1 (x)∆x1 + M y2 (x)∆x2 + · · · + M yn (x)∆xn .
Example.
1
d(T R)
marginal revenue M R = ,
dQ
d(T C)
marginal cost M C = ,
dQ
dC
marginal propensity to consume M P C = ,
dY
dS
marginal propensity to save M P S = = 1 − M P C.
dY
∂Q
marginal product of labor M PL = , marginal product of capital
∂L
∂Q
M PK = (for production function Q = Q(K, L)),
∂K
∂U
marginal utility M Ui = .
∂xi
Exercise 1. Given the total cost function
C = 1000 + 10Q + 0, 1.Q2
Find the absolute change of C at Q = 20 when Q rises 0,013 units.
Exercise 2. Given the demand function Q = 2000 − 0, 5P12 − 2P22 − P32
and assume that the current prices are P1 = 10, P2 = 15, P3 = 25. Find the
absolute change of demand when
1. P2 rises 1 unit, P1 and P3 do not change.
2. P1 , P2 , P3 all rise 1 unit.
3. P1 and P2 rises 1 unit, P3 falls 0,12 unit.
1.3 Elasticity
Given y = y(x1 , x2 , . . . , xn ). Elasticity of y in xi at x = (x1 , x2 , . . . , xn ) is
xi ∂y(x)
Exyi (x) = .
y(x) ∂xi
2
Notice: Exyi (x) is the relative change of y when xi increases 1%.
y(x1 , . . . , xi−1 , 1, 01 · xi , xi+1 , . . . , xn ) − y(x1 , . . . , xi−1 , xi , xi+1 , . . . , xn )
Exyi (x) ≈ .
y(x1 , . . . , xi−1 , xi , xi+1 , . . . , xn )
Calculate the relative change of y when xi changed to xi + ∆xi for all i:
Direct way:
∆y y(x1 + ∆x1 , x2 + ∆x2 , . . . , xn + ∆xn ) − y(x1 , x2 , . . . , xn )
=
y y(x1 , x2 , . . . , xn )
Approximate way:
∆y ∆x1 ∆x2 ∆xn
≈ Exy1 + Exy2 + · · · + Exyn .
y x1 x2 xn
Example.
P dQ
Elasticity of demand E = × (for the demand function Q = f (P )).
Q dP
P ∂Q
Price elasticity of demand EP = × , cross-price elasticity of
Q ∂P
PA ∂Q Y ∂Q
demand EPA = × , income elasticity of demand EY = ×
Q ∂PA Q ∂Y
(for the demand function Q = f (P, PA , Y )).
Exercise 1. Given the production function Q = θK α Lβ Aγ . Find the
elasticity in each variable.
Exercise 2. Given the demand function
Q = 480 − 0, 1P.
The supply is 280. Find the elasticity of demand function at the equilibrium.
Exercise 3. The demand function of a product is given by
Q = 1, 5Y 0,4 p−0,2 ,
where Y is the income.
1. Calculate price and income elasticity of demand.
2. Calculate the relative change in demand when income rises 1% and
price rises 1%.
3. Calculate the relative change in demand when income rises 1% and
price falls 2%.
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1.4 Substitution
Given y = y(x1 , x2 , . . . , xn ). Marginal rate of substitution of xi in xj at
x = (x1 , x2 , . . . , xn ) is
dxi yx′ j (x)
=− ′ .
dxj yxi (x)
dxi
Notice: is the absolute change of xi when xj increases 1 unit, keeping
dxj
y unchanged.
Elasticity of substitution of xi in xj at x = (x1 , x2 , . . . , xn ) is
xj dxi
Exxji = · .
xi dxj
Notice: Exxji is the relative change of xi when xj increases 1%, keeping y
unchanged.
Example.
dx2 U′
marginal rate of commodity substitution M RCS = − = x′ 1 (for
dx1 Ux2
utility function U = U (x1 , x2 )).
dK Q′ M PL
marginal rate of technical substitution M RT S = − = ′L =
dL QK M PK
(for production function Q = Q(K, L)).
Exercise. Given the utility function
0,5 0,4
U = 50.xA .xB ,
where xA and xB are the amounts of products A and B consumed, respec-
tively.
1. At which amounts can 1 unit of A be substituted for 1 unit of B?
2. At xA = 8, xB = 10, if quantity of A increases 1%, calculate the relative
change in quantity of B such that the utility is unchanged.
Practice. Review problems in Sections 4.3, 4.5, 5.2, 5.3 in the textbook.
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2 Equilibrium models
2.1 Supply and demand
2.1.1 Model for one good
′
QD = f (P ), f < 0,
QS = g(P ), g ′ > 0,
QD = QS ,
where P : price, QD : demand, QS : supply. More general,
′
QD = f (P, Y, . . . ), fP < 0,
QS = g(P, T, . . . ), gP′ > 0,
QD = QS ,
where Y : income, T : tax. FindRthe equilibrium point by solving the system.
Q
Consumer’s surplus: CS = 0 0 f (Q)dQ − P0 Q0 .
RQ
Producer’s surplus: P S = P0 Q0 − 0 0 g(Q)dQ.
2.1.2 Model for two goods
∂f1
QD1 = f1 (P1 , P2 ), ∂P1
< 0,
∂f2
QD2 = f2 (P1 , P2 ), < 0,
∂P2
Q = g (P , P ), ∂g1
S1 1 1 2 ∂P1
> 0,
∂g2
QS2 = g2 (P1 , P2 ), ∂P2
> 0,
QD1 = QS1 ,
QD2 = QS2 .
Notice
substitutable goods: ∂f1
∂P2
> 0 and ∂f2
∂P1
> 0,
complementary goods: ∂f1
∂P2
< 0 and ∂f2
∂P1
< 0.
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2.2 National income
2.2.1 Consumption and saving functions
(
C = C(Y ),
S = Y − C.
Notice
autonomous consumption: C(0),
autonomous savings: S(0) = −C(0),
dC
marginal propensity to consume: M P C = ,
dY
dS
marginal propensity to save: M P S = = 1 − M P C.
dY
2.2.2 Macroeconomic models
T = T (Y ),
Y = Y − T,
d
C = C(Yd ),
Y = C + G + I,
where C: consumption, I: investment, G: government expenditure, T : tax,
Yd = Y − T : disposable income. Find the national income by solving the
system in Y .
2.2.3 IS and LM schedule
C = aY + b,
I = cr + d,
∗
MS = MS ,
L1 = k1 Y,
L2 = k2 r + k3
Y =C +I (IS schedule),
M = L + L
S 1 2 (LM schedule).
Find national income Y and interest rate r by solving the system.
Practice. Review exercises in Sections 1.5, 1.7 in the textbook.
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3 Optimization models
3.1 Optimization with production functions
3.1.1 In the long-term
Given the production function Q = Q(K, L) such that
Q(λK, λL) = λn Q(K, L).
There are three cases
n < 1, the function Q has decreasing returns to scale,
n = 1, the function Q has constant returns to scale,
n > 1, the function Q has increasing returns to scale.
3.1.2 In the short-term
The firm can only change one input. They will maximize the average product:
Q(L)
maximize the average product of labor APL = L
→ max
⇔ M PL = APL ,
Q(K)
maximize the average product of capital APK = K
→ max
⇔ M PK = APK .
3.2 Maximizing the profit
One good: π(Q) = T R(Q) − T C(Q) → max
⇔ M R = M C,
where T R(Q) = P (Q) · Q and T C(Q) = F C + V C · Q.
If the production function is Q = Q(K, L), then T C = PK K +PL L and
we have to maximize: π(K, L) = P · Q(K, L) − PK K − PL L → max.
Two goods: π(Q1 , Q2 ) = T R(Q1 , Q2 ) − T C(Q1 , Q2 ) → max,
where T R(Q1 , Q2 ) = P1 · Q1 + P2 · Q2 .
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3.3 Constrained optimization
3.3.1 Optimizing the output
Maximizing the output subject to a cost constraint
Q(K, L) → max, where PK K + PL L = M.
Minimizing the cost subject to an output constraint
PK K + PL L → min, where Q(K, L) = Q0 .
M PL M PK
Using the method of Lagrange multipliers to find out PL
= PK
for
both problems.
3.3.2 Optimizing the utility
Maximizing the utility subject to a budgetary constraint
U (x1 , x2 ) → max, where P1 x1 + P2 x2 = M.
Minimizing the budgetary subject to an utility constraint
P1 x1 + P2 x2 → min, where U (x1 , x2 ) = U0 .
Ux′ 1 Ux′ 2
Using the method of Lagrange multipliers to find out P1
= P2
for
both problems.
Practice. Review exercises in Sections 2.2, 4.6, 4.7, 5.4, 5.5, 5.6 in the
textbook.
4 Mathematics of finance
4.1 Interest and discounting
4.1.1 Compound interest
If the interest is
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compounded n times per year, then the future value in t years is
r nt
F =P 1+ ,
n
compounded continuously, then the future value in t years is
F = P ert .
4.1.2 Discounting
If the interest is
compounded n times per year, then the present value is
r −nt
P =F 1+ ,
n
compounded continuously, then the present value is
P = F e−rt .
Note 1:
1. To compute annuity, regular saving or payment, we may use the geo-
metric series formula
q n+1 − 1
a + aq + aq 2 + · · · + aq n = a , (q ̸= 1).
q−1
The present value of a continuous
R n revenue stream for n years at an
−rt
annual rate of S per year P = 0 Se dt.
2. To compare interest rates of different periods, we may use APR. Annual
percentage rate (APR) for the interest compounded n times per year is
r n
AP R = 1 + − 1.
n
Annual percentage rate (APR) for the interest compounded continu-
ously is
AP R = er − 1.
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3. To find the real data (adjusted data) from the raw data (nominal data),
we may use the compound interest and discounting formulae, taking
inflation rate as the interest rate (compounded annually).
Note 2: If a value changes from P at t0 to F at t1 , then
the percentage change is F −P
P
× 100%,
the scale factor is F
P
,
the index at t1 , taking t0 as the base, is F
P
× 100,
4.2 Investment appraisal
4.2.1 Net present value (NPV)
The net present value (NPV) of an investment is equal to the present value
of the revenue flow minus the present sum of all related costs.
4.2.2 Internal rate of return (IRR)
The internal rate of return (IRR) is the annual interest rate for which the
NPV equals zero.
4.2.3 Investment appraisal
For the same cost, the project with higher NPV is better.
For any cost, the project with higher IRR is better.
Practice. Review exercises in Chapter 3 of the textbook.
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