# Intermediate Macroeconomics Ps1

Document Type

Essay

Pages

4 pages

Word Count

740 words

School

Emory University

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N/A

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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 di↵erent formulae of logarithms.

(a) Z=BM↵N1−↵.

ln Z=ln �BM↵N1−↵�

(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 di↵erence of the log-level of the

variable.

Let gx

tbe the growth of variable xat time t. We have

gx

t=xt−xt−1

xt−1

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)=�3x3−4n2+1�✓−�n.

4. Calculate all the first, second, and cross derivatives of the following functions:

(a) F(K, N)=zK↵N1−↵.

FK=↵zK↵−1N1−↵.

(b) f(k)=zk↵.

f′(k)=↵zk↵−1.

(c) u(c, l)=eln c+ln(l).

uc=

celn c+ln(l).

ucc =0.

(d) u(x, z)=�x1−

1−−✓�(1−z)1+

1+.

ux=�1−

2�x1−

2−1

(1−)1�2.

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=m−x

!. Now our problem can be written as:

max

xU=ln x↵+ln �m−x

!�1−↵

.(1)

=↵ln x+(1−↵)ln �m−x

!�.(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 x∗in 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)+∑iXi−∑iYi−∑iZi

∑iXi=2

∑i(Xi+Yi+Zi)+∑iXi−∑iYi−∑iZi

∑iXi=∑i(Xi+Yi+Zi+Xi−Yi−Zi)

∑Xi

(b) ∑i(X2

i+2XiYi+Y2

i+Z2

i)−∑i(X2

i−2XiYi+Y2

i+Z2

i)

∑i12XiYi=1

3

∑i(X2

i+2XiYi+Y2

i+Z2

i)−∑i(X2

i−2XiYi+Y2

i+Z2

i)

∑i12XiYi=∑i(X2

i+2XiYi+Y2

i+Z2

i−X2

i+2XiYi−Y2

i−Z2

i)

∑i12XiYi

10. Explain the di↵erence 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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