Basic macroeconomics bmaka qualification course for fa and ba students at the vietnamese german university (dr le van ha)

Macro econom i c Measurem ent 5 1.1 GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 The three approaches to measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Real vs. Nominal GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Other important macroeconomic gauges: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 GNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 PPPbased GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 The Long Run Macro economy 9 2.1 The Solow Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 The capital intensive form of the SGM . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 The dynamics of labor and technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 The dynamics of the capital stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.4 Some steady state results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.5 The golden rule of saving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.6 Problemset exercises on the SGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.7 Problem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.8 Problem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.9 Problem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.10 Problem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.11 Problem 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 The Short Run Macro economy 29 3.1 A Simple Theory of Household Consumption C(YT,r) . . . . . . . . . . . . . . . . . . . . . 29 3.2 A Simple Theory of Firm’s Investment I(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Government Spending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 The Trade Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.1 The Interest Rate Parity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.2 The Balance of Trade Function Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.6 The IS Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6.1 Exercises on the IS curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.7 ISTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7.1 Monetary Policy Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7.2 The lending rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7.3 The TR curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7.4 The ISTR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7.5 Quantitative analysis of the ISTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 Exercises on the ISTR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8.1 Problem 3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8.2 Problem 3.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8.3 Problem 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8.4 Problem 3.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9 The Extended ISTR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9.2 Problemset Exercises on the Extended ISTR Model . . . . . . . . . . . . . . . . . . . 53 3 Downloaded by EBOOKBKMT VMTC (nguyenphihung1009gmail.com) lOMoARcPSD|2935381 4 CONTENTS 4 The Macro economy i n t he Medi um Run 57 4.1 The Supply Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 The Firm’s Pricing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.2 Wage bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.3 The Supply Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 The Demand Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 The ADAS model in the normal time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Problemset Exercise on the Normal Time Medium Run . . . . . . . . . . . . . . . . . . . . . 61 4.4.1 Problem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 The ADAS Model During Deep Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5.1 Problem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5.2 The Nonconditioned π Z L B . . . . . . . . . . . . . . . . . . .

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Vietnamese - German University Faculty of Economics and Management

Lecture Summary and Problem Set Manual

Prepared for FA2015 Students

Basic Macroeconomics - BMAK

A qualification course for FA and BA students at the Vietnamese - German University

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This is a preliminary version which will be updated continuously Strictly meant to be used withinFA2015 BMAK Class.

• Issues thoroughly discussed in the lecture slides will not be repeated in this note

•

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1.1 GDP 5

1.1.1 The three approaches to measurement 5

1.1.2 Real vs Nominal GDP 5

1.2 Other important macroeconomic gauges: 6

1.2.1 GNP 6

1.2.2 PPP-based GDP 7

2 The Long Run Macroeconomy 9 2.1 The Solow Growth Model 10

2.1.1 The capital intensive form of the SGM 11

2.1.2 The dynamics of labor and technology 11

2.1.3 The dynamics of the capital stock 11

2.1.4 Some steady state results 12

2.1.5 The golden rule of saving 13

2.1.6 Problemset exercises on the SGM 14

2.1.7 Problem 1.5 19

2.1.8 Problem 1.5 23

2.1.9 Problem 1.6 26

2.1.10 Problem 1.7 27

2.1.11 Problem 1.8 27

3 The Short Run Macroeconomy 29 3.1 A Simple Theory of Household Consumption - C(Y-T,r) 29

3.2 A Simple Theory of Firm’s Investment - I(r) 31

3.3 Government Spending 33

3.4 The Trade Balance 33

3.4.1 The Interest Rate Parity Theory 33

3.4.2 The Balance of Trade Function Form 34

3.5 Exercises 34

3.6 The IS Curve 39

3.6.1 Exercises on the IS curve 42

3.7 IS-TR 45

3.7.1 Monetary Policy Rate 45

3.7.2 The lending rate 45

3.7.3 The TR curve 45

3.7.4 The IS-TR model 46

3.7.5 Quantitative analysis of the IS-TR 47

3.8 Exercises on the IS-TR model 47

3.8.1 Problem 3.14 47

3.8.2 Problem 3.15 48

3.8.3 Problem 3.16 48

3.8.4 Problem 3.17 51

3.9 The Extended IS-TR Model 51

3.9.1 The Model 51

3.9.2 Problemset Exercises on the Extended IS-TR Model 53

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4 The Macroeconomy in the Medium Run 57

4.1 The Supply Curve 57

4.1.1 The Firm’s Pricing Problem 58

4.1.2 Wage bargaining 58

4.1.3 The Supply Curve 59

4.2 The Demand Curve 59

4.3 The AD-AS model in the normal time 60

4.4 Problemset Exercise on the Normal Time Medium Run 61

4.4.1 Problem 4.1 61

4.5 The AD-AS Model During Deep Crisis 63

4.5.1 Problem 4.2 63

4.5.2 The Non-conditioned πZLB 67

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Chapter 1

Macroeconomic Measurement

The measurement of economic performance is the prelude to sensible economic analysis which provides uswith the understanding of how the economy works Understanding how the economy works helps businessleaders (I mean you!) making the right decisions given the changes in major macroeconomic variables such

as unemployment, inflation, economic growth, stock market indexes, and the exchange rate of the domesticcurrency against major trade partner currencies

Good measurement of the development of main economic indicators is also of vital importance for the ernment to make good policy responses to the ailments of the economy, thus avoiding painful and prolongedperiods of economic downturns For politicians, having a good team of economic advisers is also the key tokeep the economy’s performance at an acceptable level, thus raising their chance of being in power

1.1.1 The three approaches to measurement

Gross domestic product, or GDP, is the market value of all final goods and services produced within acountry in a given period of time (recall your OVWL discussions) Although GDP is not a perfect measure

of economic well being1, higher GDP is strongly correlated with higher living standards, longer longevity,better health care, more quality education as well as a larger number of other services that people enjoy.Three approaches to measuring GDP are: Product approach (measuring the contribution of each stage ofthe production process, answering the question What is being produced); Income approach (how the profit

of the production process is divided among owners of the factors of production in the economy, answeringthe question (Who gets the income generated by the production mentioned in the product approach); andExpenditure approach answering the questionWho buys the goods/services created in the production processmentioned in the product approach Regardless of which approached being employed, the same value of GDPshould be obtained In practice, however, there often exist the differences between these approaches due tothe problem of statistical discrepancies

Talking of GDP, two salient features arise: 1) In most if not all countries, GDP increases over a longperiod of time (increasing trend); however, 2) the development of GDP is far from smooth, meaning thatactual GDP fluctuates around this trend These observations give rise to the study of the economy in thelong run, and the short run, as we will proceed in this courses

i in year t

· qit

quantity of good/service

i in year t

,

When comparing values of nominal GDP across different years, this generally involves both changes inquantities and prices, and thus does not allow to infer about changes in the standard of living To allow forsuch inference, real GDP adjusts for price changes and reflects the sum of the value added produced by all

1 Suffers from such limitations as excluding home production, disregarding environmental degradation, not taking income distribution and other indicators of happiness into account.

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firms in the country in terms of only one set of prices, namely those of the so-called base year.

• The Laspeyres approach to computing real GDP involves using initial year prices as the base-yearprices:

(Real GDP)Laspeyrest = YtLaspeyres=

N

X

i=1

pi0× qit, t = 0, 1, , T,implying the growth rate of real GDP:

• The Paasche approach to computing real GDP involves using final year prices as the base-year prices:

(Real GDP)Laspeyrest = YtLaspeyres=

N

X

i=1

piT × qit, t = 0, 1, , T,implying the growth rate of real GDP:

• The Annual Chain Weight approach aims to overcome these (opposing) biases of the Laspeyres andPaasche approaches and computes real GDP using the geometric average of results for the Laspeyresand Paasche approaches, with annually updated base years:

1 + gAnnual Chain Weight

vuuuuu

N

P

i=1

pi,t−1× qit N

(Real GDP)Annual Chain Weight

0 = YAnnual Chain Weight

YAnnual Chain Weight

t = (1 + gAnnual Chain Weight

Y t ) × YAnnual Chain Weight

t−1

GNP = GDP + Net Factor Income From Abroad (NFIA)

• NFIA = Values of (exports - imports) of factor services

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• Export of factor services:

– A Vietnamese person working in Japan on a labor export contract for 3 years who earns an annualincome of 10,000 USD

– Viettel Group invests in Myanmar and made a profit of 10 million USD in 2017

• Import of factor services:

– Dr Michel Toulouse, a Canadian, working at VGU as the coordinator of the Computer Scienceprogram His income is an example of the import of labor (a factor of production of educationservice)

– Samsung Vietnam posted 5 billion USD in profit from their smart phone business in Vietnam in2017

How to compare the levels of GDP per capita across countries, which are measured in different currencies?This task is complicated by the fact that prices of traded goods relative to those of non-traded goods tend

to be sizably higher in "poor" countries than in "rich" countries, and that pricing of market exchange rates

is more closely related to the pricing of traded than of non-traded goods Purchasing power parity (PPP)exchange rates use a standardized basket of traded and non-traded goods and services across the countries

in question, estimating the market value of the basket in the rich countries currency, Vrich, using the richcountry’s currency; and the market value of the same basket in the poor country using the poor country’scurrency, Vpoor The PPP-based exchange rate is computed as: EXppp =Vpoor

V rich.The PPP-based GDP is calculated using the EXppp instead of the market exchange rate As illustrated

in the class, using market exchange rate the GDP per capita in Vietnam is about 2,200 USD However,using the PPP-based exchange rate, it’s about 7,000 USD The idea of the purchasing power parity is toask: to live like a typical Vietnamese, how much would an American has to pay? Normally, the prices ofnon-tradable goods and services in Vietnam are lower than in the US, thus PPP-based GDP per capita inVietnam is higher than the GDP per capita in current GDP

The story will be reversed if we compare a country like Qatar and the USA In Qatar, most services aremore expensive than in the USA, thus PPP-based GDP per capita for Qatar is lower than the GDP percapita in current USD

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Chapter 2

The Long Run Macroeconomy

Figure 2.1: Long-run Economic Growth

• Figure (1.1) presents the graph of GDP per capita (measured in purchasing power parity dollar) from theyear 0001 to 2010 for some countries, including Vietnam1 Of course with such critical VGU minds, you allmay simultaneously utter some objection: how do they measure income from such a far distance in history?

I can’t tell you All I know is that some prestigious professors have spent all of their life devising methods toestimate the mankind’s living standards over the last two millennia Still, they may have made some grossmistakes and all the data created are nothing more than the guesswork True! It is hard to accept but most

of what we know about our own history is guesswork (and the rest is pure prejudice) So if we all for thismoment contend with what we have at hand, Figure (1.1) tells some interesting story

Figure 2.2: Vietnam’s GDP Per Capita Since 1820

We, mankind as one, have lived most our life in the last two thousand years around the subsistence level,i.e we just had barely enough to eat and hardly enough to cover our body In developed countries, theliving standard only experienced a sustainable pick up sometime at the middle of the 18th century Yeah,

if you happen to like the history subject during the high-school time, you may also recall that mid-18th

1 The data can be downloaded from here: http://www.ggdc.net/maddison/oriindex.htm

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century also marked the introduction of new technologies that had never existed before, namely the steamengines, and together with its, came the steamships, steam spinning, steam trains and many other ensuinginventions These newly invented technologies lifted people’s lives above the subsistence level People insome Western countries suddenly found themselves unable of eating everything they had made So theysaved those redundant products for future use The future life, thus, became better thanks to the fact thatpeople would be able to consume both the saving now with the production of the future As the story goes,humankind has become richer and richer over the last two centuries.

• Figure (1.2) in combination with Figure (1.1) tells the story of Vietnam Like the rest of the world,our ancestors spent all of their lives living at the subsistence level Things only started to look a little bitbrighter in the 1990s Before that, we suffered decades of wars, war against the US, war against China, andwar against ourselves Each war was followed by a sharp drop in our parents’ and grandparents’ hard-earnedstandard of living

• It’s beyond the scope of this manual, to say anything less superficial than that when it comes to thehistory of economic development However, this manual as part of the great BMAK course, will equip youwith the models so that once mastering them, you are able to tell a much deeper story in your own way.Before we start, let’s send our special thank to Nobel Laureate Robert Solow, who put up such a powerfulanalytical framework we know as the Solow Growth Model

The Solow model is powerful in explaining why humankind has, over the last two centuries, and Vietnamhas, over the last three decades, experienced unprecedented improvement in our living standards In ourworld today, it seems that production and distribution of goods and services involves a lot of things: humanminds, human hands, machines, tools, equipments, factories, roads, ports, know-hows, recipes, and so on.Solow suggested that we bundle them into three categories only, namely labor, capital and technology Thesethree ingredients combine available inputs in certain way to produce all the goods and services we enjoy inour daily life A simple, but yet, powerfully way some summarizing this process is to define a productionfunction as:

Yt= F (Kt, At, Lt) = Ktα(AtLt)1−αwhere Kt, Lt, and At are capital, labor, and technological progress, respectively At any point in time,

we have Kt units of capital, Lt of labor, and At units of know-hows ∂Y

• This is the very point I reiterated during the lectures, students are workers working so hard to increase

the A factor within themselves If you do not read the text by Burda and Wyplosz, and other accompanying

documents, not to mention dozing over during the lectures, blame yourself for the fact that Vietnam is poorerthan Germany even though the two countries are comparable in terms of the number of human beings Let

us dream of a day when one Vietnamese worker is valued as 15 Chinese, and we will forever escape theagonizing feeling of living next to a giant and cruel neighbor

• Similar to the case of capital, the salary paid to an effective unit of labor is its marginal product,

∂Y

t(AtLt)−α, and so the total wages and salaries paid to workers are AtLt·(1−α)Kα

t(AtLt)−α=(1 − α) · Kα

t(AtLt)1−α = (1 − α) · Yt In other words, if α = 0.3 and Vietnam’s GDP is USD 240 billion,USD 80 billion will be paid to capital owners in the form of rent and depreciation, while USD 160 billionwill be paid to the workers (and their effectiveness factors) in the form of wages and salaries

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• One last point, the production function Y = Kα· (AL)βis specified in the manner that α + β = 1 This

is interpreted as the production function exhibits the constant returns to scale That is,

F (λ · Kt, λ · (At, Lt)) = (λ · Kt)α(λ · AtLt)1−α= λ · Kα

t(AtLt)1−α= λ · Yt

This constant returns to scale implicitly assumes two underlying assumptions What are they?

2.1.1 The capital intensive form of the SGM

For the ease of analysis (remember the simple Solow model?), we transform the SGM into the capital intensiveform as:

where κt= Kt

A t ,L t

2.1.2 The dynamics of labor and technology

Labor is assumed to grow at a constant rate of n, that is Lt+1 = (1 + n) · Lt Similarly, technology alsogrows at a constant rate g, At+1 = (1 + g) · At We find ourselves smarter each period (i.g each year) by a

factor of (1+g) Do you believe that your g is positive? For the society as a whole, we assume that n and g

are positive, but rather small in magnitude Empirical evidence shows that g ≈ 0.015 in the US in the last

150 years For Vietnam, the current n ≈ 0.01

2.1.3 The dynamics of the capital stock

Vietnamese people are known for our thriftiness For the given amount of income Yt, i.e everything weproduce, we consume only part of it, and save s · Yt= s · Kα

t(AtLt)1−α This saving is reinvested to coverthe wears and tears of the existing capital, known as depreciation (δ · Kt) and also to increase the capitalstock, i.e the number of structures, machines, and tools, of the economy in the next period, Kt+1 Thedynamics of the capital stock can be described as:

The net change in κ is

∆κt+1= ( 1−δ

1+n+g− 1)κt+ s

1+n+gκα t

∆κt+1= −δ+n+g1+n+gκt+ s

1+n+gκα t

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Figure 2.3: When κt< κ∗

Figure 2.4: When κt> κ∗

At the steady state, κt+1= κt= κ∗, and therefore, ∆κt+1= 0, or

δ+n+g 1+n+gκ∗= s

2.1.4 Some steady state results

• The growth rate of GDP, Yt:

gic

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At the steady state, κ is constant, that means the growth rate of Ytis equal to (n+g), which is the sum

of the growth rate of At, g, plus the growth rate of Lt, n In words, the total economy becomes bigger

if we have more hands to work (yet, also more mouths to feed!), and if each pair of hands become moreand more skillful (bravo!)

• What happens if s is becomes larger? Please take notice that κ∗ = s

δ+n+g

1 1−α

> 0 An increase in the saving rate will lead to an increase in investment, bringingthe economy to a new level of steady state capital per effective worker κ∗ Higher κ∗ means higher

• The growth rate of GDP per worker, Y t

∆YtLt

Yt Lt

= ∆At

A t + α ·∆κκ∗∗

The growth rate of GDP per capita at the steady state is equal to the growth rate of technology Inother words, in the long run we can only have a better life if we train ourselves to have more skillfulpairs of hands and smarter heads

2.1.5 The golden rule of saving

The result in the above equation states that the economy will settle down at a steady state level of capital

per effective worker, which is determined by the exogenous parameters δ, n, and g; and the rate of saving,

s Given the parameters, each choice of s will give us one steady state, with just a temporary change in

the Yt/Lt growth In the end, we don’t care much about Y t

L t, but rather the amount left for consumption(1 − s) ·Yt

L t One natural question is: How much to save so that we can maximize the amount of consumption

• At each steady state corresponding to a saving level s, the consumption per effective worker, also means

per worker, is:

and take notice from Figures (1.3) and (1.4) that at the steady state, s · κ∗α = (δ + n + g) · κ∗ Using this

result, the equation becomes:

c∗t = κ∗α− (δ + n + g) · κ∗Choosing the rate of saving is at the same time choosing the capital stock per effective worker It is easier

to maximize c∗t with respect to κ∗ Therefore, taking the derivative of c∗t w.r.t κ∗ in (1.8) yields:

∂c ∗ t

∂κ ∗ = −(δ + n + g) + ακα−1

Setting the derivative ∂c

∗ t

p r o b l

e m

1 3

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Noting that a = (1a)−1, we get:

and the above result, it isapparent that at the optimum, α = s That is the golden rule to save is s = α It is best to save the incomeshare of capital to invest back to capital When α is high, the marginal product of capital is also high, so it

is worth investing a higher share of income to build a larger stock of capital

Another interpretation of this theoretical result is that capital per effective worker is accumulated to thepoint when the marginal cost of holding capital per effective worker is equal to the marginal product ofcapital per effective worker

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• This simple model only features capital as the only input in the production process Therefore, there is

no need to derive the intensive form Let’s go straight to the dynamics of capital:

When Ktis relatively small, δ · Kt< (s − m) · Kα

t, the capital stock is increasing because net investment ispositive Ktkeeps increasing until δ · Kt≈ (s − m) · Kα

t, the level of capital stock will stay constant, say at

K∗ and the economy enters the steady state

To solve for K∗, let

b) Derive how the steady-state level of income depends on the coefficient m, reflecting the current account

surplus,0 < m < 1 Provide the economic reasoning for your results

• Answer: First notice that ∂K ∗

∂m == −δ(1−α)1 s−mδ 1−αα , which indicates that the level of capital stockdepends negatively on the level of current account If the country is a net lender, the current account ispositive, the domestic capital stock at the steady state is lower than the level of capital that would otherwiseaccumulated in a close economy Reversely, if the current account CA < 0, the country is a net borrower,the capital stock at the steady state is higher than the closed economy case because domestic investment ishigher than domestic saving With a lower level of capital stock, the output Y∗= K∗α is also lower

Figure 2.5: Natural disaster destroys half of the economy’s capital stock

c) Suppose that some years after the economy had reached steady state, a natural disaster destroys half

of the economy’s capital stock Describe both graphically and verbally what will happen in this economyafter the natural disaster has struck

Answer: When half of the capital stock is destroyed, the economy find itself at the level of capital that is

lower than the capital stock at the steady state, gross investment, (s − m) · Yα

t is larger than depreciation,

δKt The level of capital stock will keep increasing until it reaches the steady state level This process isdemonstrated in Figure (1.5)

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Problem 1.3

A country’s production function is given by Yt= At·K0.65

t ·L0.35

t In the year 2016 we observed Kt= 10, 000;

Lt= 100; and Yt= 10, 000 Suppose that from 2016 to 2017, income (Yt) grew by 2.6%, the capital stock(Kt) by 1.2%, and employment (Lt by 0.5% What is the rate of technological progress between 2016 and2017? What is the level of technology in 2017?

• Answer: Plugging the giving information to the production function 10000 = At· 100000.65· 1000.35 yields

A2016= 5.011872336 Next taking the natural logarithm of the production function yields:

log(Yt) = log(At) + 0.65 · log(Kt) + 0.35 · log(Lt)

Taking the total differentiation of both sides of the previous equation yields:

Plugging the information on the growth rate yields∆A t

A t = 0.01645, thus the growth rate of Atis 1.65% from

2016 to 2017 The level of technology progress in 2017 is (1 + 0.01645) · 5.011872336 = 5.094317636

Problem 1.4

Let us suppose that income per worker in Mexico in 2017 is equal to $ 16,000 Also suppose that theeconomy-wide rate of saving s = 0.15 (with a balanced government budget and net exports, net factorpayments from abroad as well as net unilateral transfers equal to $ 0), the rate of worker growth n = 0.01,the rate of depreciation of capital δ = 0.03, the rate of technological progress g = 0.02, and the Cobb-Douglasproduction function

Yt= ·Kα

t · (AtLt)1−α,

with α = 0.4 Assume that capital per effective worker in Mexico in 2017 is in steady state a) If no further

information is given, what do you predict regarding the level of income per worker in Mexico in 2020?

• Answer: At the steady state, the income per worker growth rate is the same as the growth rate of

technological progress As a result, the income per worker in Mexico in 2020 is given by:

L t)2020= (Y t

L t)2017· (1 + 0.02)3= 16979.328

b) Now take into account the following additional information: from 2017 to 2018 onward, the rate of

technological progress was to decrease to g = 0.01, What is your new prediction regarding the level ofincome per worker in Mexico in 2020? Provide detailed economic rationale for the change in your prediction

• Answer: At the 2017 steady state, the level of capital per effective worker is given by: κ∗

new= s

δ+n+g 2018

1 1−α

To make the prediction of the level of income per worker by 2020, we compute the level of capital per effective worker in 2017 and the following years, as well as the level of technology

in 2017 and the following years This information will allows the prediction of income per capita in 2020.

• The level of technology in 2017 is computed as:

κ α 2017

Edit

ed

by Foxi

t Rea der

For Eval uati

on Onl y.

g d p

p e

r c a p it a

le v

el o

f t e c h

= A

t le v

el o

f la b o u r

= L

t le v

el o

f c a pi t

al

= K t

Trang 17

A2017= 4.60503937316000 0.4 = 8686.136373

• We need first to derive the formula of κ∗at the steady state For the sake of completeness, I also

include the derivation here Please do this in similar exam questions

∆κt+1= (1+n+g1−δ − 1)κt+ s

1+n+gκα t

δ+n+g 1+n+gκ∗= 1+n+gs κ∗α

(δ + n + g)κ∗= sκ∗α

κ ∗

κ ∗α = s δ+n+g

δ+n+g

κ∗= s δ+n+g

1 1−α

• Now the level of capital per intensive worker for 2017 can be computed as:

κ∗

2017= s

δ+n+g 2017

1 1−α

• Notice that the new level of income per capita is lower because of the lower technology progress growth

c) For the year 2017, derive the golden rule level of consumption per effective worker If technological

progress for 2017 and all following years was equal to g = 0.02, what is the time path of consumption perworker implied by this golden rule level of consumption per effective worker?

l e a r n

b y

h e a rt

l e a r n

b y

h e a rt

Edit

ed

by Foxi

t Rea der

For Eval uati

on Onl y.

Trang 18

• Answer (for the sake of completeness, I include the full answer here, which means a tition of what has been said in the summary):

repe-• At each steady state corresponding to a saving level s, the consumption per effective worker is:

∂κ = −(δ + n + g) + ακ∗(α−1)Setting the derivative ∂c

∗ t

.Noting that a = (1a)−1, we get:

δ+n+g

1 1−α

combining the general steady state level of capital per effective worker, which is κ∗ = s

δ+n+g

1 1−α

,with this golden rule steady state saving rate, it is apparent that at the optimum, α = s That is thegolden rule to save is s = α It is best to save the income share of capital to invest back to capital

• Similar to question b), a change in the saving rate put the economy on a new growth path toward anew steady state The new level of capital per worker is determined as:

κ∗= s 0.4

δ+n+g

1 1−α

• From the previous question κ2017= 4.605039373 A2017 = 8686.136373; A2018= A2017· (1 + 0.02) =8859.859101; A2019= A2018· (1 + 0.02) = 9037.056283; A2020= A2019· (1 + 0.02) = 9217.797408;; and

κtcan be computed recursively as:

• Following the golden rule of saving, the time path of consumption per worker, Cpw, between 2017 and

2020 for Mexico is:

t Rea der

For Eval uati

on Onl y.

Trang 19

C2020pw = (1 − s2020) · Y2020

L 2020 = 0.4 · A2020· κ0.4

2020= 11285.51235

• Which Factor is More Relevant to economic growth?

To this point, we have discuss the golden rule, and the apparent indication is that ico should save more, increasing its saving from 15% to 40% This policy prescription is painful and impossible in practice because people are shortsighted and asking them to give

Mex-up 17.64% of their current consumption for the optimal consumption in the future sounds nothing more than a crazy idea.

Figure 2.6: Golden Rule Vs Techonological Progress

An alternative policy prescription could be that Mexico should go on with the current consumption,however, there are measures to boost the skills of its people so that the country can increase itstechnological progress by 1 more %, so g increases from 0.02 to 0.03 (saving stays the same) Figure(1.6) illustrates these alternative policy prescriptions by graphing the development of consumption percapita for the base scenario discussed in question a), the golden rule scenario, and the 3% technologicalprogress

1 CPC stands for consumption per capita

2 CPC lines are measured on the left vertical axis

3 The red long-dash line is the difference between CP Cs=0.15,g=0.03and CP Cs=0.4,g=0.02, measured

in the right vertical axis

• It can be seen that the red line became positive in 2017 due to the sudden increase in saving tomeet the golden rule while the 3% growth in technology scenario does not require any adjustment inconsumption The difference narrows gradually and reaches its lowest point in some 30 years but stillpositive After that, the difference starts to go to infinity

• The discussion in this section emphasizes the overwhelmingly important role of technologicalprogress in general, and human capital investment in particular Only by investing in human can

we sustain the improvement in our standard of living A nation is doomed to fail if its young

generations are not provided with the proper education to increase their wealth of human capital.

2.1.7 Problem 1.5

Consider an economy, in which labor input grows by 1% per year and technology improves by 3% per year.Every year, 3% of the capital stock wears out The households save a constant fraction of 35% of theirincome The elasticity of output with respect to capital in the country’s standard Cobb-Douglas productionfunction is 0.5

a) Determine the steady state value of capital per effective worker.

Trang 20

• Answer: Given the standard Cobb-Douglas production function in the context of the Solow growth

model, the production function is:

Yt= Kα

t(AtLt)1−αThe elasticity of output with respect to capital is defined as:

1+n+gκα t

1 1−α

Plugging the model parameters into this formula: n = 0.01; g = 0.03; δ = 0.03, and s=0.35 yields:

0.01+0.03+0.03

2

= 25b) Suppose the government could provide incentives for households to either increase or reduce the savingrate If the government’s aim was to maximize consumption per effective worker, should the households savemore or less of their output?

Trang 21

• Answer: Firstly, we need to find the optimal saving rate for this country.

At each steady state corresponding to a saving level s, the consumption per effective worker is:

to maximize c∗

t with respect to κ Therefore, taking the derivative of c∗

t w.r.t κ in yields:

∂c ∗ t

.Noting that a = (1

a)−1, we get:

δ+n+g

1 1−α

combining the general steady state level of capital per effective worker, which is κ∗ = s

δ+n+g

1 1−α

, withthis golden rule steady state saving rate, it is apparent that at the optimum, s = α=0.5 The governmentshould encourage households to save more

c) Given the result you found in (b) what are the consequences for output, investment, capital,

consump-tion per effective worker following a change of your policy recommendaconsump-tion Provide a detailed economicrational on how the outcomes arise

• Answer:

• Capital per effective worker:

Economic Reasoning: Due to the change in the saving rate from s=0.35 to s1= 0.50, gross investmentincreases from It= 0.35 · Yt to It= 0.50 · Yt, the net investment per capita

Trang 22

∆κt+1= −δ+n+g1+n+gκt+ s1

1+n+gκα t

becomes positive, the level of capital per effective workers start to increase from the former steadystate level of 25 to the new steady state level of 51.02040816

• Output per effective worker:

Economic Reasoning: Initially, the economy is at the steady state and output per effective worker

was constant and equal to κ∗α

1 With the new saving rate, the level of capital per effective workersstarts growing and the economy converges to the new steady state k∗ Output per effective workeralso converges to the new level κ∗α

2

• Investment per effective worker:

Economic Reasoning: Investment was constant at 35% of the constant output per effective worker

initially The jump in the saving rate to 50% leads to a jump in the investment per effective worker

As the economy converges to the new steady state level, investment per capita also converges to thenew constant steady state level of 0.5 · κ∗α

2

Trang 23

• Consumption per effective worker:

Economic Reasoning: Initially, consumption was constant at 0.65 · κ∗α

1 With the increase in savingimplied by the golden rule, consumption dropped 1 to 1 with the increase in saving Over time,the economy grows due to more capital per effective worker, leading to the growth in consumptionper effective worker The increase in consumption is more than enough to compensate for the initialdrop because the new saving rate is the optimal choice However, it takes almost 40 years for theconsumption to reach the initial level following the increase in saving, which is painfully slow

2.1.8 Problem 1.5

Suppose an economy’s output per worker was $50,000 and its labor force had a of size 1 million people in

2015 The saving rate was 20%, labor force grew at a rate of 0.5%, capital depreciated at a rate of 3% whiletechnology improved by 1% Further the capital share of production in the economy was roughly 0.35

a) Suppose that the economy could be modeled using the standard Cobb-Douglas production function

and the market is perfectly competitive, compute the level of technology for this economy in 2015 given theeconomy is in steady state

Trang 24

Given that the market is perfectly competitive, firms are free to maximize their profit by choosing theoptimal level of labor and capital without influencing the market price, interest rates, and wages (non com-patitive market structure such as monopoly,oligopoly, and monopsony have to take the market price/wagesinto account) The profit maximization problem is:

∆κt+1= (1+n+g1−δ − 1)κt+1+n+gs κα

t

1+n+gκα t

Trang 25

κ∗1−α= s

δ+n+g

κ∗= s δ+n+g

1 1−α

Plugging the model parameters into this formula: n = 0.005; g = 0.01; δ = 0.03, and s=0.2 yields:

κ∗=0.01+0.03+0.0050.2

1 0.65

κ ∗ α = 9.92298374350,000 0.35 = 22, 394.6978031362

b) Assume the economy stays on its balanced growth path What is your prediction for output per

effective worker, output per worker and output in 2030 for this economy Also comment on the growth rate

Taking the logarithm of both sides yields:

log(yt) = log(κ∗) + log(At)Taking the total differential of both sides with respect to time yields:

T ≥ t Plugging t = 2015, and T = 2030 to get y2030= y2015· (1 + 0.01)2030−2015= 58048.44777

Output: At the steady state, Yt= Y t

A t L t· At· Lt= κ∗· At· Lt Taking the logarithm of both sides yields:log(Yt) = log(κ∗) + log(At) + log(Lt)

Taking the total differential of both sides with respect to time yields:

∆Y t

Y t =∆κκ∗∗ +∆At

A t +∆Lt

L t

At the steady state, the level of capital per effective worker is constant, and therefore, output grows

at the rate of technological progress plus the rate of growth in the labor force Yt+1 = Yt(1 + n + g),

Yt+2 = Yt(1 + n + g)2 and YT = Yt+1(1 + n + g)T−t, for T ≥ t Plugging t = 2015, T = 2030, and

Y2015= 50, 000 · 1, 000, 000 to get Y2030= Y2015· (1 + 0.01 + 0.005)2030−2015= 62, 511, 603, 332.72

c) Now suppose the growth rate of the labor force was to decrease to -1% What is your prediction for

consumption per effective worker, consumption per worker and consumption in 2030?

• Answer:

Using the results from question a), the level of capital per effective worker at the steady state is:

Trang 26

κ∗= s

δ+n+g

1 1−α

When the rate of labor force grows changed from n=0.5% to n1= −1%, the steadystate level of capital per effective worker changed to:

κ∗=δ+ns1+g

1 1−α

Plugging the new values of the parameters into this formula yields:

κ∗=0.01+0.03+(-0.01)0.2

1 0.65

= 18.51620196

From question a) the 2015 value of capital per effective worker was: 9.922983743 and the level of

technol-ogy progress was 22,394.6978031362 Due to the change in the population growth rate, capital per effectiveworker started growing until it reaches the new steady state level of 18.51620196 Using the results from

question a), we can compute the value of capital per effective worker recursively as:

pro-• Answer:

The capital per effective worker in the rich country at its steady state is:

κ∗ rich= ( s rich

δ+n+g)1−α1

The capital per effective worker in the poor country at its steady state is:

κ∗ poor= ( spoor

(δ+n+gspoor ) 1−α1 = (s rich

s poor)1−α1 = (30.125

5 )1−0.31 = 13.00841426Furthermore, at any point in the steady state:

Trang 27

13.00841426 = 3.843666029The technological progress level is about 3.84 times higher in the rich country in comparison with that

of the poor country

em-of middle income countries like China and Thailand because em-of environmental problems (type B); and theadvanced technologies that go with the modern machines and equipment from countries such as Germany,Japan and the USA (type G) Suppose further that type B equipment and machines drag the technologicalprogress in Vietnam down by 0.5% per year, while type G equipment and machinery boost the technologicalprogress by 1% In the context of the standard Solow Growth Model:

a) Please explain the consequences on GDP, GDP per worker in Vietnam in the next 15 years if this

country chooses to import type B technologies only Please draw the Solow model diagram and the timepaths of output, capital, investment, and consumption per effective worker for this scenario

b) What is your prediction on GDP, GDP per worker in Vietnam in the next 15 years if instead this

country imports the type G technology Please draw the Solow model diagram and the time paths of output,capital, investment, and consumption per effective worker for this scenario

Trang 28

Trang 29

Chapter 3

The Short Run Macroeconomy

• In the long run, price and wages fully adjust, the output in the economy is purely determined by thesupply side of the economy (i.e how much capital, labor that we have; how fast is the growth of technology;and how obstructing government and government regulations are) We discuss these issues in depth usingthe insights from the Solow Growth Model, and the Solow-Romer Model

• In the short run, price and wages are sticky, firms have the opportunity to adjust their production withoutraising costs (at least substantially) because wage contracts, input supply contracts and the level of capitalare fixed in the short run At the same time, demand is volatile because the economy faces constant shocks:consumer confidence, domestic government policies, foreign demands and exchange rate policies, etc

• As apparent in equation (3.1), the interest rate, the cost of current consumption or the return on saving,

is a determinant of three out of 4 demand components Also, the exchange rate works, together with theinterest rate, to determine the last component, i.e imports and exports Luckily, the central bank has thepower to set the interest/exchange rates, and therefore, capable of influencing the demand in the short run,even though to what extent such policy can improve economic outcome is still open to debate

• A general short-run macroeconomic model is of the form:

• Thus, it is important for us to understand the mechanism through which firms and households make

their consumption/investment decisions, governments decide their spending levels, and how the trade balance

varies with the changes in the interest rates and the exchange rate Such understanding will pave the wayfor us to build suitable short run macroeconomic models capable of explaining the short-run fluctuations,thus enabling us to devise suitable countering policies In the followings, we will step by step discuss all ofthe four components of demand

By the household, we mean a typical household in the economy, which is endowed with certain level of resources For simplicity, we assume that the household only lives in two periods: now and the future There

are three sources of wealth the household may possess, namely accumulated wealth at the beginning of period

1 (F W1); labor income during period 1 (Y1); and labor income during period 2 (Y2) If the household does

not have the motive to save at the end of the future, they will consume everything they have in period 2.

Instead, if they want to give some bequest F W2 to their future generation, they will leave certain amount

of wealth at period 2 Suppose that the real interest rate is r, the present value of all resources that thehousehold possesses is:

P V R = (1 + r)F W1+ Y1+ Y2

1 + r −

F W2

1 + r29

Trang 30

• Given this resources, the household face the problem of allocating consumption between now and thefuture so that its level of utility obtained is maximized Let’s denote C1and C2as the level of consumption

at the current and future period respectively The decision by the household is constrained by the level ofresources, which could be formally expressed as:

• The allocation facing the household is:

V ≡ max

C 1 ,C 2

U (C1, C2) = U (C1) +U (C2)

1 + ρs.t C1= P V R − C 2

∂C 2 = 0, we obtain the optimal condition as:

• This equation is known as the Euler Equation Some further discussion here is guaranteed:

• U′(C1) and U′(C2) are the marginal utility obtained by consuming one more unit of C1, and one moreunit of C2, respectively; and (1 + r) and (1 + ρ) are the marginal cost of consuming one more unit of

C1 and C2, respectively Intuitively, equation (3.2) states that the price of utility should be equalize,i.e U1+ρ′

(C 2 ) = 1+r

U ′

(C 1 ) Let’s imagine that you buy two chickens, weighing 1.5 and 2 kg, respectively;you pay 30,000 VND for the smaller one and 40,000 for the bigger The price per kilogram should bethe equal, i.e 30,0001.5 = 40,0002 Think of U′(C1) and U′(C2) as two chickens

• The optimal condition does not depend on Y1and Y2, which means that households smooth out theirconsumption between now and the future, given their present value of resources and the ability toborrow freely In other words, what is important is the PVR, not when the PVR is earned The

allocation of consumption between now and the future depends on the subjective discouting rate ρ and the interest rate r Given the same level of PVR, some households choose to be borrowers if their

subjective discount rate is higher than the interest rate (ρ > r), others choose to be lenders, when theirsubjective discount rate is lower than the interest rate (ρ < r)

Trang 31

Figure 3.1: Optimal Consumption: Borrower

Figure 3.2: Optimal Consumption: Lender

• Some other points to remember:

– Households may face credit constraints, preventing them from maximizing utility by borrowingagainst the future This leads to the empirical finding that consumption is closely related tocurrent disposable income

– When utility function is strictly concave, consumption smoothing implies that a temporary crease in income will lead household to consume only part of the increase and save part of it Apermanent increase will lead the household to increase their consumption proportionately

in-The aggregate consumption function is, therefore, specified as:

Intercept:

capturing all factors other than Y,

The purpose of this section is to highlight the key principle of firms’ investment, equalizing marginal costand benefit of investment Consider a two-period model, the periods being the current and a future period,

in which a firm considers at the beginning of the current period to purchase one additional unit of capital

so as to:

- Increase the amount of output produced in the future period (we assume that capital purchased inthe current period can only be used to produce output in the future period (it takes time to installcapital)), and

Trang 32

- Sell that portion of the additional capital that has not depreciated at the end of production in thefuture period.

To keep notation as simple as possible, we also assume that the prices of:

- One unit of capital, and of

- One unit of output in the current and future periods are equal to one

The firm will purchase the additional unit of capital if the marginal (extra) revenue (MR) from doing soexceeds the marginal cost (MC):

M R

in current period values

current period

of future revenue

of output

portion of additional capital that has not depreciated

current period

+

1

1 + r

current value of additional unit of future cost

future period

The firm will purchase the additional unit of capital if:

M R = M P K+1−δ1+r ≥ 1 = M C

M C = T obin′sq = M P K+1−δ1+r ≥ 1This idea can be represented graphically as:

An increase in the interest rate r lower the present value of the marginal revenue of the additional vestment unit, raising the opportunity cost of investment, leading firms to cut back on their purchases of

in-additional capital This idea can be formalized by taking the derivative of the the Tobin’s q with respect to

the interest rate:

The aggregate investment function is, therefore, specified as:

Trang 33

I = I0

Intercept:

capturing all factors other than Y,

On the contrary, as the economy heads into recession, consumer confidence plummets, the government willincrease in spending to stimulate the economy, increase demand with the hope of bringing the economy back

to its long-run growth path Of course, this theory mainly reflects the best practice Government spending

in Vietnam in the last decades has always been increasing, regardless of the business cycle Perhaps, we needsome VGU graduates to join the Ministry of Finance so change could be initiated

• Another important factor influencing government spending is government debt, especially governmentexternal (in foreign currency) debt We are all aware of the Greek public debt crisis, while the threat toVietnam is widely discussed on the mass media recently However, to keep things simple, we will ignoregovernment debt for now and will come back in later sections

As is apparent from the lecture, the trade balanced is the difference between export and import Exportrepresents foreign demand, which depends on how much income foreigners have to buy our products Import,

on the contrary, depends on how much income domestic consumers have and how fancied of foreign (by this Imean Japanese, American or German) made products On top of that, both import and export are stronglyinfluenced by the level of the exchange rate The income you have at this point of your life mainly comes fromyour parents’ subsidy, mostly in VND of course The current exchange rate is 22,800 VND/USD If you want

to buy an Iphone priced at 599 USD, you have to pay 13,657,200 VND The price will be 14,076,500 VND,14,975,000, and 17,970,000 VND if the exchange rate increase to 23,500 VND/USD, 25,000 VND/USD, and30,000 VND/USD respectively Given your parents’ income are mostly in VND, and thus yours, the price

of the Iphone keeps increasing even though its price in USD stays constant The Iphone is becoming moreexpensive because the VND depreciates The reverse is true to an American who loves Vietnamese Phonoodle

• To capture such insights, the trade balanced in this course is modeled as:

T B = T B0+ T Bex· (Y∗− T∗) − T Bim· (Y − T ) + T Bǫ[ǫ0− ǫr(r − r∗)]

The parameters T Bex, T Bim, and T Bǫrespectively captures the impact of an increase in foreign disposableincome (on export (+)), domestic disposable income (on import(+)), and the real effective exchange rate(on both import (-), and export (+))

3.4.1 The Interest Rate Parity Theory

While the impact of (Y∗− T∗) and (Y − T ) is obvious, it is important to understand the mechanism throughwhich we postulate the relationship between r and r∗, and the real effective exchange rate Let’s start with

a question constantly asked by financial investors: should I invest in my own country’s financial market andearn the return of 1 + rt or should I invest in the international market to earn 1 + r∗

t, and then convertthe earnings back to my own country’s currency to earn the investment return of ǫ1t · (1 + r∗

t)ǫt+1 At any

Trang 34

given point in time, if 1 + rt> ǫ1t · (1 + r∗

t)ǫt+1, investors will sell their assets in the international marketand buy the domestic assets, raising the demand for domestic currency The domestic currency appreciates,depressing the ǫ1t term in ǫ1t · (1 + r∗

t)ǫt+1 The reverse is true if 1 + rt< ǫ1t · (1 + r∗

t)ǫt+1 The process ofadjustment will keep going until:

1 + rt

Equation (3.5) states that the current real effective exchange rate ǫtdepends on the domestic and tional interest rate, and the expectation of future exchange rate If the central bank increase the interestrate rt, ǫtbecomes smaller, the domestic currency appreciates Keep replacing ǫs by ǫs+1recursively to get:

interna-ǫt= 1 + r

∗ t

1 + rt

ǫt+1

ǫt= 1 + r

∗ t

1 + rt

1 + r∗ t+1

1 + rt+1

ǫt+2

ǫt= 1 + r

∗ t

1 + rt

1 + r∗ t+1

1 + rt+1

1 + r∗ t+2

1 + rt+2

ǫt+3

ǫt=1 + r

∗ t

1 + rt

1 + r∗ t+1

1 + rt+1

1 + r∗ t+2

1 + rt+2

1 + r∗ t+3

1 + rt+s

ǫt+s+1.finally,

ǫ = ǫ0+ ǫr· (r − r∗), 0 < ǫr< 1

3.4.2 The Balance of Trade Function Form

The above discussion allows for the specification of a simple function form relating the trade balance withits determinants as:

T B = T B0+ T Bex· (Y∗− T∗) − T Bim· (Y − T ) + T Bǫǫwith

Tiêu đề	Basic macroeconomics bmaka qualification course for fa and ba students at the vietnamese german university
Tác giả	Dr. Le Van Ha
Trường học	Vietnamese - German University
Chuyên ngành	Basic Macroeconomics
Thể loại	lecture manual
Năm xuất bản	2018
Thành phố	Binh Duong

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Số trang	68
Dung lượng	1,29 MB