## Summary

# Macroeconomics I summary

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Summary "Macroeconomics, European edition, N. Gregory Mankiw, Mark P. Taylor" CH2-CH4.1, CH5-CH14.2

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Tim Eijkenaar

May 2020

1

,Contents

1 GDP Accounting 4

1.1 GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Expenditure components of GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Price indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Unemployment 5

2.1 Measuring joblessness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The natural rate of unemployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Economic growth 6

3.1 Production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Cobb-Douglas production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Solow growth model 8

4.1 The production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Steady state level of capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.3 Golden rule level of capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.4 The effect of efficiency and population growth . . . . . . . . . . . . . . . . . . . . . . 12

5 Money and banking 13

5.1 Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 The quantity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3 The demand for money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.4 The Fisher effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.5 The cost of holding money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.6 Exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Aggregate demand and supply 16

6.1 Aggregate demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.2 Aggregate supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.3 Aggregate supply = aggregate demand . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.4 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 IS-LM Model 19

7.1 IS-Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.2 Shifting the IS-curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.3 LM -Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.4 Shifting the LM -curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.5 Equilibrium analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.6 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Mundell-Fleming model 24

8.1 IS ∗ -curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8.2 When do we shift the IS ∗ -curve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8.3 LM ∗ -curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2

, 8.4 When do we shift the LM ∗ curve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.5 Equilibrium analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8.6 Fixed exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8.7 Equilibrium analysis: e = e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Three models of aggregate supply 29

9.1 The sticky-price model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9.2 The sticky-wage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9.3 Imperfect information model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9.4 The new SRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10 Phillips curve 31

3

,1 GDP Accounting

1.1 GDP

Let Qi,t denote the quantity of product i at year t and let Pi,t be the price of product i at year t.

Assume the chosen base year is t = 0. Then,

X

Real GDP at year t = Pi,0 ∗ Qi,t

i

X

Nominal GDP at year t = Pi,t ∗ Qi,t

i

1.2 Expenditure components of GDP

The four expenditure components of GDP are given by the following formula: Y = C + I + G + N X

Y = Output = Income = GDP

C = Consumption = C(Y − T )

I = Investment = I(r)

G = Government spending

N X = Net exports = N X(e)

Where we used the fact that output Y is a function of disposable income (Y − T ), investment I is

a function of the interest rate r and N X is a function of the exchange rate e.

We define:

Trade deficit = X − M = Exports − Imports

Capital inflow = −N X

Furthermore we have the following formulas where we assume a closed economy (N X = 0):

Private saving = Y − T − C

Public saving = T − G

Saving = S = (Y − T − C) + (T − G) = I

1.3 Price indexes

Both the Paasche price index and the Laspeyres price index reflect what is happening to the price

level in the economy. They are defined in the following way.

P

Nominal GDP at year t Pi,t ∗ Qi,t

Paasche price index = GDP Deflator = = Pi

Real GDP at year t i Pi,0 ∗ Qi,t

P

P i,t ∗ Qi,0

Laspeyres price index = CPI = P i

i P i,0 ∗ Q i,0

4

,2 Unemployment

2.1 Measuring joblessness

Each adult is placed into one of the following three categories:

1. Employed E

2. Unemployed U

3. Not in the labour force

We define

Labour force = Number of employed + Number of unemployed

Number of employed

Unemployment rate = ∗ 100%

Labour force

Labour force

Labour force participation rate = ∗ 100%

Adult population

2.2 The natural rate of unemployment

It is reasonable to assume that the labour force L is fixed. The unemployment rate is under this

assumption fully determined by the transition of individuals in the labour force. Let s denote the

rate of job separation and let f denote the rate of job finding. Then the number of people that lose

their job sE must be equal to the number of people that find a new job f U .

f U = sE

f U = s(L − U )

U 1

=

L 1 + f/s

Where we used the fact that E = L − U and did some basic algebra steps to derive the last formula.

5

,3 Economic growth

3.1 Production function

Output Y , or GDP, is a function of the amount of capital K and the amount of labour L.

That is, Y = F (K, L).

A production function has the property ’constant returns to scale’ if zY = F (zK, zL) for every real

number z.

The marginal product of labour M P L is the extra amount of output a firm gets given a unit change

in the amount of labour. That is, M P L is the partial derivative of F with respect to L:

∂F (K, L)

MPL =

∂L

Likewise, the marginal product of capital M P K is the change in the amount of output given a unit

change in the amount of capital:

∂F (K, L)

MPK =

∂K

The above gives us the following result: let P be the price of one single unit of output and let W

be the wage corresponding to one unit of labour. Then if we increase L by one unit:

∆Profit = ∆Revenues − ∆Cost

= MPL ∗ P − W

Intuitively, as long as ∆Profit > 0 we want to increase the amount of labour until ∆Profit = 0.

Therefore, every competitive firm will satisfy

∆Profit = ∆Revenues − ∆Cost

0 = MPL ∗ P − W

W

MPL =

P

We can do the same steps for the M P K. This will lead us to the following result:

R

MPK =

P

6

,3.2 Cobb-Douglas production function

Any production function of the form F (K, L) = A K α L1−α with A > 0 and 0 ≤ α ≤ 1 is called a

’Cobb-Douglas production function’.

Note that F has constant return to scale since

F (zK, zL) = A(zK)α (zL)1−α = AK α L1−α z α z 1−α = zA K α L1−α = zF (K, L)

Furthermore for every Cobb-Douglas production function it holds that

Y

M P L = (1 − α)AK α L−α = (1 − α)

L

Y

M P K = αAK α−1 L1−α = α

K

7

,4 Solow growth model

4.1 The production function

Recall that the production function is given by F (K, L). In the Solow growth model we assume

that F has constant return to scale: zF (K, L) = F (zK, zL). Furthermore, we want to analyse

all quantities in the economy relative to the size of the labour force. That means that we are not

interested in output Y itself, but in YL . We now rewrite Y = F (K, L) to know more about YL . We

use the fact that F has constant return to scale. Note that in this case z = L1 :

Y = F (K, L)

Y F (K, L)

=

L L

K L

= F( , )

L L

K

= F( , 1)

L

From now on, we denote every variable relative to the size of the labour force with just the small

letter of the variable:

Y K C

=y =k =c

L L L

We can now rewrite our production function. Note that we can simply forget about the variable ’1’

since it is a constant. We use f instead of F for convenience.

Y K

= F ( , 1)

L L

y = F (k, 1)

= f (k)

8

,4.2 Steady state level of capital

In the Solow growth model output per worker y is divided between consumption per worker c

and investment per worker i. We assume a closed economy (N X = 0) and we ignore government

spending G:

y =c+i

We assume that individuals save a fraction s of their income (s ∈ [0, 1]) and consume a fraction

(1 − s) of their income:

c = (1 − s)y = (1 − s)f (k) i = sy = sf (k)

The capital stock changes over time because of the impact investment i and depreciation δ have on

it. Let i be the investment per worker per year and let δ be the fraction of capital that depreciates

per year. Then

∆k = i − δk = sf (k) − δk

Investment and depreciation balance over time. That is, sf (k) = δk. This implies that ∆k = 0.

The level of k at which ∆k = 0 is called the steady state level of capital per worker and is denoted

by k ∗ . This process is illustrated in the figure below.

We now ask ourselves what happens to k ∗ if we change the saving rate s. Suppose we increase the

saving rate s, this results into an increase in sf (k). Therefore, the sf (k)-curve shifts upward. This

results into a new equilibrium sf (k) = δk at a higher steady state level of capital per worker k ∗ .

9

, 4.3 Golden rule level of capital

The steady state value of k that maximizes consumption is called the Golden Rule level of capital

∗

and is denoted by kgold . Recall that c = y − i = f (k) − sf (k). Since we only look for steady state

levels of capital per worker, we have that

∆k = 0 ⇔ sf (k) = δk

therefore the following holds for all steady state levels of capital per worker:

c∗ = y − i = f (k ∗ ) − sf (k ∗ ) = f (k ∗ ) − δk ∗

∗

Since we have to maximize consumption in order to find kgold we have to solve:

d

f (k ∗ ) − δk ∗ = 0

dk ∗

MPK − δ = 0

MPK = δ

Remember that in the [k, f (k)]-plane the slope of the graph of the f (k)-function equals the MPK.

∗

And the slope of the δk-line equals δ. We can find kgold by drawing the graph of f (k ∗ ) and the

∗

δk-line: the point at which the slope of the graph of f (k ) is equal to the slope of the δk-line is the

∗ ∗

Golden Rule level of capital kgold , since only then we have M P K = δ. We can then insert the kgold

into sf (k) = δk to find the saving rate s. This is also illustrated in the figure below.

√

Example: Let f (k) = k and let δ =

0.1. Find the saving rate s at the Golden

Rule level of capital.

Solution:

MPK = δ

∂f (k)

= 0.1

∂k

1

√ = 0.1

2 k

√

10 = 2 k

∗

kgold = 25

∗

We now insert kgold into sf (k) = δk

∗ ∗

sf (kgold ) = δkgold

√

s 25 = 0.1 ∗ 25

5s = 2.5

s = 0.5

∗ 1

The saving rate s at kgold is given by s = 2

10

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