# Intermediate Macroeconomics Ps1

Document Type
Essay
Pages
4 pages
Word Count
740 words
Intermediate Macroeconomics
Juan Rubio-Ram´ırez
Problem Set 1 - Suggested Solutions
Math Review
1. Show that the following expressions can be written as log-polynomials.
For this question, we would need to apply dierent formulae of logarithms.
(a) Z=BMN1.
ln Z=ln BMN1
(b) M=eµA
eµD!.
ln M=ln eµA
eµD!
=ln (eµA)ln eµD!
2. Show that the growth rate of variable x,gx, can be approximated as the dierence of the log-level of the
variable.
Let gx
tbe the growth of variable xat time t. We have
gx
t=xtxt1
xt1
3. Calculate the first and second derivative of the following functions:
(a) f(c)=ln c.
(b) u(c)=c1
1.
f(c)=c.
1
(c) g(n)=3x34n2+1n.
4. Calculate all the first, second, and cross derivatives of the following functions:
(a) F(K, N)=zKN1.
FK=zK1N1.
(b) f(k)=zk.
f(k)=zk1.
(c) u(c, l)=eln c+ln(l).
uc=
celn c+ln(l).
ucc =0.
(d) u(x, z)=x1
1(1z)1+
1+.
ux=1
2x1
21
(1)12.
2
5. Using the method we covered in class, solve the following constrained maximization problem:
max
x,y U=ln x+ln y1
The first step is to assume the constraint binds (holds with equality) and solve it for one of the variables.
Doing that for ygives us y=mx
!. Now our problem can be written as:
max
xU=ln x+ln mx
!1
.(1)
=ln x+(1)ln mx
!.(2)
Now that we transformed the two-variable constrained problem into an unconstrained problem with only one
variable we just need to take the first order condition in 2, i.e.:
@U
@x=0
Now, substituting in the optimal choice for xin our original constraint we obtain:
m
+!y=m
6. Evaluate:
(a) 3
j=03j=30+31+32+33=40.
7. Write these using the Sigma notation:
(a) xt+xt+1+xt+2++xt+T=T
i=txt+i.
8. Calculate/expand the following:
(a)
4
j=1
j3=13+23+33+43=100.
(b)
5
t=1
x3
t=x3
1+x3
2+x3
3+x3
4+x3
5.
3
5
9. Show that:
(a) i(Xi+Yi+Zi)+iXiiYiiZi
iXi=2
i(Xi+Yi+Zi)+iXiiYiiZi
iXi=i(Xi+Yi+Zi+XiYiZi)
Xi
(b) i(X2
i+2XiYi+Y2
i+Z2
i)i(X2
i2XiYi+Y2
i+Z2
i)
i12XiYi=1
3
i(X2
i+2XiYi+Y2
i+Z2
i)i(X2
i2XiYi+Y2
i+Z2
i)
i12XiYi=i(X2
i+2XiYi+Y2
i+Z2
iX2
i+2XiYiY2
iZ2
i)
i12XiYi
10. Explain the dierence between ix2
iand (ixi)2. Construct an example to show that, in general, these
quantities are not equal.
One example:
Let x=[1,5,2], that is, x1=3,x
2=5,x
3=2. Therefore,
4

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