1. (18 marks) Consider general models and implicit functions:
a. State the three conditions of the Implicit Function Theorem for n functions of the
form FI (1, ..., Yn; X1, ., Xm) Where y; is endogenous for all j € {1,2, ...,n} and x; is
exogenous for all i € {1,2,..., m}. That s, for a point (y19, ..., Yno; X10, ---» Xmo), State
the three conditions by which there exists a neighbourhood around this point fo
which y; = f4(xy, ..., Xm) is an implicitly defined and continuously differentiable
function, and where F/ (yy, ..., Yu; X1, -.., Xm) = 0 is an identity, for all j € {1,2, ...,n}.
. Assuming the conditions refe
ed to in Part (a) hold, take the total differential of
FI (yy, cee) Ys X41, oe, Xp) = O for each j € {1,2, ...,n} to derive the Implicit Function
Derivative Rule
a _ lil
ax;
for allj € {1,2, ...,n} where J is the Jacobian matrix
[Ft OF]
y1 Yn
I=]:
oF aF™
9y1 Yn
and J; is the matrix J wherein the Jj" column is replaced with the negative of the
arm”
adient vector [2
8! oxi’ ox
c. Apply the concepts involved in Parts (a) and (b) to the basic national income model
Y=C+1Io+Go
C=a+pY-T)
T=y+d6Y
to verify the comparative static
ac _ pa-9
a 1-pa-9 °°
where 8,6 € (0,1), a,y > 0 and I; and G,, are exogenous. You may assume without
verification that the conditions of the Implicit Function Theorem are satisfied for the
equili
ium point of the model.
2. (12 marks) Find the critical and inflection points of f(x) = 2sin® x + 3sinx for x € [0,7],
making sure to classify the critical points.
3 45 a7 x9 gl
3. (4 marks) Apply Taylor's Theorem to prove that (i) sinx = x — 5 + = - =z + = - o> +
at as as sr 7 1
ii STE de
and (i) InG+ 1) =x — 24 ZZ FT _2
aud diy iia tA —/ 4A
. (4 marks) Let the exponential E(x) be the inverse of the logarithm L(x). Use the properties
of logarithms to prove that (i) E(a + b) = E(a)E (b) and (ii) E’ (x) = E(x)/L'(1).
(12 marks) Suppose a developer owns a parcel of land worth V(t) = Vok2VE (measured in
dollars) when developed at its highest and best use at time t = 0 (measured in years from
today), where V, > 0 is the value of the parcel today and K > 1 is a parameter. Assume the
only cost to the developer of owning the parcel is foregone interest, at an annual rate of r
0, on the capital tied up in the parcel. Letting A(t) be the present value of the parcel, the
developer seeks to maximize A(t) by choosing the time at which the parcel is developed at
its highest and best use.
a. Derive the optimal time t* at which to develop the parcel at its highest and best use,
and then check the second-order condition to confirm that t* indeed provides for a
elative maximum.
. Use the result of Part (a) to evaluate t* for the case of K = fe and r = 5%, where e
is Euler's number.