econ hw

Useful Derivative Rules:

Chain Rule: (f (x))y → y (f (x))y-1 δf (x)

Product Rule: f (x) g (x) → f (x) δg (x)

δx + g (x)

δf (x)

δx

Quotient Rule: f (x)

g (x) →

g (x) δf (x)

δx − f (x)

δg (x)

δx

g (x) g (x)

Useful Exponent Rules:

x-α= 1/ xα

xαxβ = xα+β

(xα) β = xαβ

(xy)α = xα yα

( 𝑥

𝑦 ) α = xα/y α

(x + y) α ≠ xα + yα

(x - y) α ≠ xα - yα

Useful Fraction Rules:

𝑎/𝑏

𝑐 =

𝑎

𝑏𝑐

𝑎

𝑏/𝑐 =

𝑎𝑐

𝑏

𝑎/𝑏

𝑐/𝑑 =

𝑎𝑑

𝑏𝑐

𝑎+𝑏

𝑐 =

𝑎

𝑐 +

𝑏

𝑐

𝑎−𝑏

𝑐 =

𝑎

𝑐 -

𝑏

𝑐

𝑎

𝑏+𝑐 ≠

𝑎

𝑏 +

𝑎

𝑐

𝑎

𝑏−𝑐 ≠

𝑎

𝑏 -

𝑎

𝑐

II. Economics Formulas

Aggregate Capital Growth: Kt+1 = (1-d) Kt + It

Aggregate Consumption/Savings: St = s Yt ; Ct = (1-s) Yt ; St + Ct = Yt

Aggregate Investment/Savings: It = St

Aggregate Production Function: Yt = A Kt α Nt

1-α or Yt = At Kt α Nt

1-α

Per Capita Capital Growth: kt+1 = (1-d) kt + it

Per Capita Consumption/Savings: st = s yt ; ct = (1-s) yt ; st + ct = yt

Per Capita Investment/Savings: it = st

Per Capita Production Function: yt = A kt α or yt = At kt

α

Population Growth: Nt+1/Nt = (1+n)

1) Imagine a Solow Growth Model with a standard Cobb-Douglas production function and

the following parameters:

α = 0.15; d = 0.03; A = 200; s = 0.4; n = 0.05

a) Is the Solow Growth Model an exogenous or endogenous growth model? How so?

b) Calculate the rate of capital accumulation (law of motion).

c) Calculate the steady state level of capital?

d) Calculate the steady state level of real output/income?

e) Calculate the steady state level of investment?

f) Calculate the steady state level of consumption?

g) What effect does a higher depreciation rate have on this model?

h) What effect does a higher savings rate have on this model?

i) What effect does a higher population growth rate have on this model?

j) What effect does a higher productivity/technology factor have on this model?

2) Imagine an AK Model with a standard Cobb-Douglas production function and the following

parameters:

d = 0.02; s = 0.2; n = 0.02

a) Is the AK Growth Model an exogenous or endogenous growth model? How so?

b) Calculate the rate of capital accumulation (law of motion).

c) If technology does not stop improving, what will happen in this model?

d) If technology stops improving when A=250, what is the steady state growth rate of capital?

e) If technology stops improving at this point, what is the steady state growth rate of real

output/income?

f) If technology stops improving at this point, what is the steady state growth rate of

investment/savings?

g) If technology stops improving at this point, what is the steady state growth rate of

consumption?

h) What effect does a higher depreciation rate have on this model?

i) What effect does a higher savings rate have on this model?

j) What effect does a higher population growth rate have on this model?

3) Consider two fictional economies: Westview and Asgard. Based on the following

economic statistics, pick one of the two economies and write a paragraph using 5

statistics explaining why their economy is stronger.

Statistic Westview Asgard

GDP (constant PPP $) $100,000,000 $5,000,000,000

GDP per capita (constant PPP $) $1,000,000 $50,000

GDP Growth Rate 5% 2%

GNI (constant PPP $) $90,000,000 $6,000,000,000

GNI per capita (constant PPP $) $90,000 $60,000

Consumer Price Index 125 120

GDP Deflator 115 120

GINI Index 50 30

Central Government Debt (as %

of GDP)

50 0

Foreign Direct Investment

(constant PPP)

-$100,000 $100,000

Poverty Rate (%) 0% 1%

Unemployment Rate 0% 2%

Bonus: Write a paragraph using 5 statistics to argue the other economy is stronger.