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Advanced Macroeconomics I,UPF Professor Andrea Caggese PROBLEM SET 1-suggested answers to selected problems

1. A)

¶y 2 ฀ = ¶฀k 3

-฀1 / 3

B)Plugging in numbers for k we find for the MPK: 2/3 (k=1), 0.53 (k=2) and 0.46 (k=3). C)Take derivative of MPK w.r.t. k to see that MPK is decreasing in k.

D)

¶2y 2 ฀ = -฀ k 4 / 3 < 0 2 ¶฀k 9

¶ p฀ 3 ฀ -฀1 / 4 = p฀= k 1 / 2 ¶฀y 4 . First replace y via the production function . It follows then that

¶ p฀ 1 ฀ = ¶฀k 2

-฀1 / 2

.

1฀ ¶ p฀ = -฀ -฀3 / 2 2 4 ¶฀k 2

E)Second derivative is negative so profits are decreasing in k,

2

•

¶฀ln X (t ) X (t ) Make use of = X (t ) ¶฀t A and B) When we are interested in variables that grow over time, it is more natural to work with the (natural) logarithm of these variables, especially when growth is approximately constant. This is for the simple reason that (as you have seen in 1.2.A) when Z(t) grows at a constant rate ln Z(t ) grows linearly, and (as you can convince yourself) if Y(t) and Z(t) ( Y (t ) ¹ ฀ Z(t ) grow at the same proportional rate, ln Y (t) - ฀ ln Z(t )is constant, while Y(t ) - ฀ Z(t )is not. D In every case, just take logs and differentiate with respect to time at both sides of the equation using the provided basic properties of the logarithm.

3 •

•

•

A)! Given that k (t) = −7 + 2k(t), k (t ) = 0 if and only if k(t)=7/2. Any higher k will cause k (t ) to be •

higher, and any lower k will cause k (t ) to be negative, hence the equilibrium is UNstable. •

k (t) −7 The growth rate is + 2 . It is clearly decreasing as k approaches to the steady state value. = k(t) k(t)

B) •

•

•

Given that k (t) = −3k(t) + 2 , k (t ) = 0 if and only if k(t)= 2/3. Any higher k will cause k (t ) to be •

negative, so k will decrease to the steady state. Any lower k will cause k (t ) to be positive, hence •

2 k (t) = −3+ the equilibrium is stable. The growth rate is . It is still clearly decreasing as k k(t) k(t)

approaches to the steady state value.

4) A profit maximizing firm in a perfectly competitive setting has the following problem: max F (K , AL ) -฀RK -฀ wL

K ³฀ 0 ,L³฀ 0

Which implies:

¶ F K AL฀ ( =R ¶฀K

¶ F K AL฀ ( =w ¶฀L

Given the CD-Production function this yields,

¶ F ( K , AL ) ฀ = a฀Y K = R ¶฀K

¶ F( K, AL) ฀ = -฀a฀ Y L = w ¶฀L

1.7.B.As in lecture slides, labor market equilibrium is where labor demand and supply curves intersect. Analogue for capital market.

6)

i) Assuming Cobb-Douglas, in Steady State: This implies

Since

:

In particular if

,

ii) If capital can move, it will move until

Since

is monotonically decreasing, this implies

so

will be the same only if there’s no differences in A....