An economy works with a given amount of capital \(K_t\) that is used in production for creating a yield \(Y_t\) which can be either consumed (\(C_t\)) or reinvested to increase the value of capital.
Increment in \(K_t\)
Reduction in \(K_t\)
Production \(Y_t\)
Consumption \(C_t\)
This dynamics of capital can be formally written as
\[
\frac{dK_t}{dt}=Y_t-C_t.
\]
The change in capital \(dK_t\) is expressed with respect to time \(dt\) because we are interested in measuring the rate at which capital evolves. Dividing by \(dt\) allows us to compute capital accumulation per unit of time, independently of the length of the observation period.
In practice, economic variables such as GDP are often measured at discrete intervals—for example, quarterly (every three months). To express changes at an annual rate, the observed variation must be adjusted by the length of the time interval. Since three months correspond to \(dt=\frac{3}{12}=\frac{1}{4}\) of a year, dividing by \(dt\) converts the change into an annualized rate.
For example, suppose capital increases from \(100\) to \(120\) over \(3\) months. The change over that period is \(20\). To compute the annual rate of change, we divide by the fraction of the year represented by three months:
Thus, an increase of \(20\) over one quarter corresponds to an annualized increase of \(80\). This illustrates how dividing by \(dt\) standardizes changes across different time intervals and yields a rate of change per year.
In the dynamics \(dK_t\), the increase in capital is called “investment” (\(I_t\)) and the previous equation can accordingly be written as
\[
I_t=Y_t-C_t.
\]
The right hand side measures saving (i.e. the yield reduced by consumption) and this equation shows that, in equilibrium, the total saving must always coincide with the investment.
Of course, the previous equation can be written in a more common way:
\[
Y_t=C_t+I_t,
\tag{1}\]
i.e., the total revenue can be either consumed or invested.
Note that Eq. ?@eq-Y=C+I is an identity. This means that it is always true because of the accounting principles. Thus, it is just a way to describe the equilibrium between the variables, but it cannot be used for policy purposes. For instance, we cannot be sure that an increment in \(C_t\) will for sure lead to an increment in \(Y_t\). At the same, time, in fact, investment could reduce. This is why in economics we often use the formula “ceteris paribus” (all other things being equal).
1.2 The Government Sector
A government has a total amount of debt \(B_t\) which evolves over time by:
increasing because of the service on debt: if the interest rate is \(r_t\), the total amount of interest is \(B_t r_t\),
increasing because of the public expenditure \(G_t\),
decreasing because of the taxes \(T_t\).
Accordingly, the dynamics of debt can be written as
If we go back to the dynamics of capital \(dK_t\) for private investors, it:
increases because of the interest received by the investors in the public debt,
decreases because of taxes paid to the Government,
decreases because of the amount of money lent to the Government (which is \(\frac{dB_t}{dt}\)).
Increment in \(K_t\)
Reduction in \(K_t\)
Production \(Y_t\)
Consumption \(C_t\)
Debt service \(r_t B_t\)
Tax \(T_t\)
Lent to Gov. \(\frac{dB_t}{dt}\)
Thus, we can write
\[
\underset{\text{Investment}}{\underbrace{\frac{dK_{t}}{dt}}}=\underset{\text{Production}}{\underbrace{Y_{t}}}-\underset{\text{Consumption}}{\underbrace{C_{t}}}+\underset{\text{Interest}}{\underbrace{r_{t}B_{t}}}-\underset{\text{Taxes}}{\underbrace{T_{t}}}-\underset{\text{Lent to Gov.}}{\underbrace{\frac{dB_{t}}{dt}}},
\]
and, if we substitute the value of \(\frac{dB_t}{dt}\) we finally get:
The Government can thus increase the GDP by increasing the public expenditure. Nevertheless, this policy may have some side effects (that we will not check during these lectures).
1.3 The Open Economy
When the domestic production can be sold to foreign countries (exports – \(E_t\)) and foreign products can be bought from abroad, the previous equations must be suitably modified.
The product that are sold abroad are subtracted from what is domestically produced \(Y_t\). Instead, what is bought from abroad (import – \(M_t\)) is added to the total production, i.e. we can consume more that the domestic product.
This evidence shows that when there is a crisis and the private investment decreases, the government can sustain the total demand by increasing its expenditure. This evidence accordingly, shows how the public expenditure is used as a counter cyclical tool.
Finally, we can look at the trade balance: both exports and imports are increasing over time (in absolute value), but their difference has a trend of its own as we see in the following figure where the difference \(E_t-M_t\) is drawn.
The value \(Y_t\) is the Gross Domestic Product at any time, but in economics we are mainly interested in studying the dynamics of this variable. Its dynamics is the change over time and the “relative” change is called “growth rate” \(g_t\):
\[
g_t:=\frac{dY_t}{dt}\cdot\frac{1}{Y_t}.
\]
The value of the GDP can be written as the product between the quantity of goods (and services) produced \(Q_t\) and the average price \(P_t\):
\[
Y_t=Q_t\cdot P_t,
\] and the change over time of this product is (recall the derivative of a product):
\[
\frac{dY_t}{dt}=\frac{dQ_t}{dt}\cdot P_t+Q_t\cdot\frac{dP_t}{dt}
\] or, if we devide by \(Y_t\):
In economics, the components of this equation have a precise meaning:
\(\frac{dY_t}{dt}\frac{1}{Y_t}\): is the relative change of the GDP and is called “nominal growth rate” (\(g_t\))
\(\frac{dQ_t}{dt}\cdot \frac{1}{Q_t}\): is the relative change in the “real” GDP and measure the growth of the quantity of goods and services produced (\(\gamma_t\))
\(\frac{dP_t}{dt}\cdot\frac{1}{P_t}\): is the relative change in the price level and is called “inflation” (\(\pi_t\))
We can find the data about the nominal and real growth of GDP and inflation on the FRED database. The real GDP is labelled GDPC1 and inflation is labelled CPILFESL.
For the case of Italy we can draw a similar graph by using the series on FRED: the real GDP is labelled RGDPNAITA666NRUG and inflation is labelled ITACPIALLMINMEI.
We can see some common interesting points.
The years of the economic boom: Sixties and the first half of Seventies
The years of the two oil shocks (1974 and 1979) that lead to a strong increase in the price of oil around the world; inflation rises all over the world while the real GDP reduces. This phenomenon (high inflation and economic stagnation) is called “stag-flation” and it is quite new
The crisis during the years 1991-1992 which hits the industrialized world
The crisis in 2001 due to the burst of the dot-com bubble: internet is the new frontier of technological change and many people invest in this secton without selecting consistently the new IT firms. When the bubble bursts the economy suffes a slow down. The US Central Bank reduces the interest rate to 1% in order to stimulate the economic growth. With such a small cost for borrowing money, many people (even those who are not so reliable) subscribe mortages for buying houses and a new bubble on the house market begins
The crisis in 2008: those who subscribed mortgages, did that at a floating rate but, while the economy recovers, the Central Bank starts increasing back the interest rate which reaches the level of 5%. At that level, many debtors go bankrupt and the house bubble bursts. Again the Central Bank reduces the interest rate, but this time at the level of 0%
The Greek crisis in 2010: this is EU crisis that does not affect US and it starts because of the Greek problem in paying back its debts
The COVID crisis in 2020: the Central Bank again reduces the interest rate and issues a huge amount of money for sustaining the economy
The war inflation between 2022 and 2023: the Russian invasion of Ukraine determines a new shock on the energy market (Russia is a big producer of oil and gas) and a new wage of inflation hits the world
1.6 Another Way to Compute GDP
The value of production is distributed among those who have participated to it. In particular:
the providers of capital receive an interest
the providers of labor receive wages
the providers of entrepreneurship receive a profit
Let us assume that in an economy there are two firms whose profit & loss are represented in the following scheme.
P&L Firm A
P&L Firm (B)
Costs
Revenues
Costs
Revenues
Purchases
100
Final sales
30
Purch. from (A)
110
Sales
150
Wages
25
Sales to (B)
110
Wages
30
Total
125
Total
140
Total
140
Total
150
Profit
15
Profit
10
The GDP can be computed in three (equivalent) ways:
The sum of the revenues: wages payed by firm A and B (\(55\)) + profits of firm A and firm B (\(25\)), whose total is the GDP \(80\)
The net value of the final sales: the final sales of firm A is \(30\), the final sales of firm B is \(150\), the purchases are \(100\), and the GDP is \(180-100=80\)
The sum of the value added: the value added of firm A is \(140-100=40\) and the value added of firm B is \(150-110=40\), whose total is the GDP \(80\)
The FRED database contains the share of GDP payed as labor income in US: the label of the time series is LABSHPUSA156NRUG.
X=getSymbols('LABSHPUSA156NRUG',src='FRED',return.class='zoo')plot(LABSHPUSA156NRUG,xlab='',ylab='Labor income / US GDP',lwd=2,col='blue')grid()
During the period of the stagflation (1974–1985) the share of labor income on GDP has been constantly reducing. At the end of the Nineties the share increases and, because of the crisis in 2001 the share reduces again to the lowest level. Finally, the COVID crisis implies another reduction and the last abservable value is the lowest of the time series.
1.7 The Three Major Economic Areas
The world GDP is mainly produced by three big continental regions: US, UE, and China. We see how their GDP has been growing over time in the following figure.
We can see that China has become a great superpower only from in the XXI century and this happened exactly because of the international trade. In fact, in 2001 China joins the World Trade Organization (WTO) and start making a lot of money.
We can appreciate the weights of these three countries on the world GDP by looking at the following figure.
plot(GDP[,2:4]/GDP[,1]*100,screens=1,col=c('red','blue','darkorange'),lwd=2,xlab='',ylab='% of World GDP')grid()legend('topright',legend=c('USA','UE','Cina'),col=c('red','blue','darkorange'),lwd=2,bty='n')
1.8 The Relevance of International Trade
An index for measuring the relevance of the international trade in an economy is given by the ratio between the sum of exports and imports (taken with the same sign) and the GDP:
\[
\frac{E_t+M_t}{Y_t},
\]
which is also called “openness index”. The historical values of the index for any country can be found in the World Bank database data.worldbank.org.
library(worldbank)TR=wb_data(indicator="NE.TRD.GNFS.ZS",country=c('US','IT','DE','EU','CN'),lang="en",start_date=NULL,end_date=NULL)TR_zoo=do.call(merge,lapply(split(TR, TR$country_id),function(df) {zoo(x=df$value,order.by=as.numeric(df$date))} ))plot(TR_zoo,screens=1,col=c('orange','black','blue','darkgreen','red'),lwd=2,xlab='',ylab='Trade (% of GDP)')grid()legend('topleft',legend=c('CN','DE','EU','IT','US'),col=c('orange','black','blue','darkgreen','red'),lwd=2,bty='n')
Exports and imports are functions of domestic and foreign GDP. In particular.
When consumers are richer they consume more of both domestic and foreign goods and, accordingly, imports increase; thus, we can write \(M_t \left(\overset{+}{Y_{t}}\right)\).
When foreign consumers are richer, they will consume more of the goods produced in our country. If we call \(Y_t^*\) the foreign GDP, then we can write \(E_t \left(\overset{+}{Y_t^*}\right)\).
We can see such a relationship in the US data about the trade balance and the GPD growth rate.
GDP=fredr(series_id ="GDP",observation_start =as.Date("2010-01-01"),frequency ='q',units ='pc1')GDP=zoo(GDP$value,GDP$date)BP=fredr(series_id ="BOPGSTB",observation_start =as.Date("2010-01-01"),frequency ='q')BP=zoo(BP$value,BP$date)T1=as.Date('2017-01-20')T2=as.Date('2021-01-20')T3=as.Date('2025-01-20')m=min(BP)M=max(BP)# Imposto margini più larghi a destrapar(mar=c(5,5,4,5))# Primo grafico: Bilancia commercialeplot(BP,col='red',lwd=2,xlab='',ylab='Trade Balance (Mln $)')grid()rect(T1,m,T2,M,col=rgb(1,0,0,0.2),border=NA)rect(T3,m,Sys.Date(),M,col=rgb(1,0,0,0.2),border=NA)# Seconda serie sovrappostapar(new=TRUE)plot(GDP,col='blue',lwd=2,xaxt="n",yaxt="n",xlab='',ylab='')# Asse destroaxis(4, col='blue', col.axis='blue')mtext('Var. % GDP Y-o-Y', side=4, line=3, col='blue')# Legendalegend("bottomleft",legend=c("Trade Balance", "Var. % GDP"),col=c("red","blue"),lwd=2,bty="n")
In the figure the red shaded ares show the Trump’s governments. We see that, when the US economy exists from the pandemic crisis, the consumers start increasing their consumption (and the GDP grows very rapidly).
While the GDP grows, we consistently see a deterioration in the trade balance. This happens because both domestic and foreign consumption is increased. While the economic situation stabilizes after the pandemic crisis, also the trade balance stabilizes, but at a deficit that is more important than the pre-pandemic one.
In 2025 Trump becomes again president of the US and he seems to be very worried about the trade balance deficit. Thus, in April 2025 he announces new tariffs to almost all world countries.
Many US firms are worried to be unable to buy foreign goods at the same price as before and they start buying as much as they can to increase their stocks. This is why we see the worst deficit even in the US trade balance exactly after the Trump’s announcement.
After this moment, the trade balance improves, because the US firms have plenty of stocks and they do not need to continue buying new goods. Nevertheless, when the stocks start reducing, the trade balance comes back to the value it had before the Trump policy.
1.9 Trade Balance
The trade balance is not the same for the three great powers. In particular, the US are constantly in deficit, while China and Europe have a positive balance. We can see that in the following figure.
Germany trade balance has improved a lot after the creation of the European Union in 1992. In fact, most the international trade for Germany happens with EU countries and the elimination of the exchange rate risk (since 1999 EU has a unique currency) has improved a lot the opportunity for Germany to sell to other EU countries.
Since its entry in WTO (in 2001), China has improved a lot its trade balance. The trade balance has been increasing at a very high rate untill the sub-prime crisis in 2008, when the other industrialized countries in the world have reduced a lot their GDP and, accordingly, they also imported less goods from China.
US is a chronically importing country and it has a constant trade balance deficit. Nevertheless, after the sub-prime crisis, the weight of this deficit on the US GDP has been almost constant.
1.10 Trump’s Tariffs
When Trump announces new tariffs, the International Trade Administration shows a formula for computing the tariff for each country
where the first two factors are the elasticity of imports with respect to prices (\(\varepsilon\)) while the two last factors contain the semi-elasticity of the price with respect to the tariff (\(\phi\)):
there is no adjustment of the elasticities; in other worlds, the variables \(\varepsilon\) and \(\phi\) are assumed to remain constant through all the process of imposing new tariffs (which is very unrealistic)
imports are assumed to be independent on tariffs; in other worlds, this formula assumes that the other countries in the world will continue buying products from US as if nothing happened.
1.11 The Foreign Trade Multiplier
Eq. ?@eq-Y=C+I+G+E-M is an identity cause it is true for any value of the variables. Now, instead, we assume two behavioral relationships that describe a possible:
consumption is a linear transformation of yield, i.e. \(C_t=c_0+c_1\cdot Y_t\), which means that richer people consume more than poorer people
imports are a linear transformation of yield too, i.e. \(M_t=m_0+m_1\cdot Y_t\), since imports is again a consumption of foreign goods
If we substitute these two hypothesis in Eq. ?@eq-Y=C+I+G+E-M we get
The term \(\frac{1}{1-c_{1}+m_{1}}\) is called “multiplier” and it measures how much the GDP reacts to changes in the investment, public expenditure, or exports.
We can estimate such a multiplier by using a linear regression on the US data that we have already downloaded.
