1. GDP, CPI and PCE

In an economy, X and Y are two final goods, their unit price and quantity are given in the following table:

Period Qx Px Qy Py
1 80 4 110 7
2 50 6 60 10
  1. Calculate nominal GDP for each period.
gdp1<- 80*4+110*7 
gdp2<- 50*6+60*10

The nominal gdp in year 1 is 1090, and the nominal gdp in year 2 is 900.

  1. Use year 2 as the base year to calculate real GDP for each period.
# base year is 2
#   we have to fix prices in year 2
px<-6
py<-10
real_gdp1.2<- 80*px+110*py
real_gdp2.2<- 50*px+60*py

Under year 2 base, the real gdp in year 1 is 1580, and in year 2 is 900.

  1. Calculate chained-dollar real GDP using year 1 as the base year.
# base year is 1
#   we have to fix prices in year 1
px<-4
py<-7
real_gdp1.1<- 80*px+110*py
real_gdp2.1<- 50*px+60*py


real_growth_base1<-real_gdp2.1/real_gdp1.1
real_growth_base2<-real_gdp2.2/real_gdp1.2
real_chained_growth<- sqrt(real_growth_base1*
                             real_growth_base2)

The chained-dollar real gdp growth is 0.5692137 (i.e. the “authentic” growth).

chained_real_gdp1<-gdp1
chained_real_gdp2<-chained_real_gdp1*real_chained_growth

If we choose year 1 to be the reference or base year, then real gdp in year 1 will be the same as its nominal gdp, which is 1090. Therefore, the chained-dollar real gdp in year 2 will be that value times 0.5692137 which is 620.4428798.

2. GDP based on purchasing power parity (PPP GDP).

  1. Given France GDP denominated in euro and Britain GDP denominated in the British pound, if we convert France GDP to pound denomination based on its exchange rate, will it reflect its final output production size relative to Britain properly? What is the problem of doing so?
Such conversion does not reflect relative production size properly since the exchange rate can change even production sizes do not.
  1. What is PPP GDP? How does it work?
PPP GDP is a measure that can be used to compare final output production size across countries. 

In stead of using nominal exchange rate to convert 
GDP into the same currency. PPP use the relative price ratio of the same basket of goods across countries to determine the conversion ratio. For example, if the basket costs USD10 in the U.S. and costs Euro20 in 
France, then we will use the USD1:Euro2 (i.e. USD10:Euro20) to convert U.S. GDP to France GDP.

3. Present value discount rate

  1. What is present value discount rate? Is a higher present value discount rate good for a saver? How about for a borrower?
If a person's present value discount rate is d, it means that he is able to invest $1 today to generate an income of $1+d tomorrow. Therefore, holding $1+d tomorrow is equivalent to holding $1 today for the person.

The higher the present value discount rate, the higher the return a person's investment can generate. Investment opportunity with a high return is always a good thing regardless of a person's saving or borrowing position.
  1. What defines a person’s present value discount rate?
Since a person's present value discount rate reflects how good a person can manage his investment, many things that affect a person's investment skill will affect this rate, such as a person's education level, familiy background, and even health status.

In the article of the Economist, “Putting it all on red. The rules encourage public-sector pension plans to take more risk”, it says that

Private-sector pension funds in America and elsewhere (and Canadian public funds) regard a pension promise as a kind of debt. So they use corporate-bond yields to discount future liabilities. … American public pension funds are allowed (under rules from the Government Accounting Standards Board) to discount their liabilities by the expected return on their assets.

  1. Why does the choice of discount rates make American public pension funds to take a riskier investment strategy?
Normally riskier investment provides higher expected return, which, when used as a discount rate, can reduce present value of future pension liability. It can improve a pension fund's balance sheet.

4. Intertemporal substitution effect and income Effect

Consider a two-period intertemporal budget constraint: \[(1+i)A_{0}+I_{1}+\frac{I_{2}}{1+i}=C_{1}+\frac{C_{2}}{1+i}+\frac{A_{2}}{1+i}.\] Suppose that only \(C_{1}\) and \(C_{2}\) can be decided, and \(A_0=0,A_2=0\).

  1. In a figure where x-axis represent \(C_{1}\) unit and y-axis is \(C_{2}\) unit, draw the budget constraint line. Please mark the slope and its intercept on both axes.
ans a

ans a

  1. If i increases, draw the new budget constraint line on top of the old one from a.
ans b

ans b

  1. Continue from b. What is a positive income effect? Under what circumstance will there be a positive income effect?

When an impact makes a persion feel wealthier, the action in response to such a feeling is called a positive income effect.

In this question, positive income effect will happen when a persion is a saver, and his reponse will be increasing his consumptions.

  1. What is the intertemporal trade-off between \(C_1\) and \(C_2\)? What is the intertemporal substitution effect of an interest rate increase on \(C_1\)? (Please give a detailed story.)

The intertemporal trade-off: When consumer increase \(C_1\) by one unit today, he sacrifices the opportunity of a saving of one unit, which in turns represents the sacrifice of the opportunity of owning \(1+i\) tomorrow for consumption. Increasing \(C_1\) by one unit means sacrificing \(C_2\) by 1+i units.

When interest rate increases, the intertemporal trade-off as \(C_1\)’s opportunity cose increases. Therefore, \(C_1\) will decrease.

5. Permanent income

We define permanent income level as the level of consumption under perfect consumption smoothing. Suppose a consumer only lives for three periods. He has no initial wealth, and the incomes are \(I_{1}=70,\ I_{2}=100,\ \) and \(I_{3}=50\) for three periods.

  1. Suppose the interest rate \(i=5\%\). What is his permanent income level?
I1<-70
I2<-100
I3<-50
i<-0.05
pv.life.wealth<-I1+I2/(1+i)+I3/((1+i)^2)

total.discount<-1+1/(1+i)+1/((1+i)^2)

c.star<-pv.life.wealth/total.discount

The present discounted value of life time wealth is 210.5895692. The total discount number is 2.8594104, that is \((1+1/(1+i)+1/(1+i)^2)\). Therefore, \(c^*\) is 73.6478985, which is also this household’s permanent income level.

Suppose consumers like perfect consumption smoothing under the current interest rate.

  1. If income today (\(I_1\)) increase temporarily by 20, how much will consumption today (\(C_1\)) increase?
I1<-90
I2<-100
I3<-50
i<-0.05
pv.life.wealth.c<-I1+I2/(1+i)+I3/((1+i)^2)

total.discount.c<-1+1/(1+i)+1/((1+i)^2)

c.star.c<-pv.life.wealth.c/total.discount.c

The presevent discounted value of life time wealth is 230.5895692. The total discount number is 2.8594104. Therefore, \(c^*\) is 80.6423473, which is also this household’s permanent income level.

Consumption today increases by 80.6423473 - 73.6478985 =6.9944489.

  1. If income increase permanently by 20, how much will consumption today (\(C_1\)) increase?

For a permanent inccome increase of 20, consumption will increase by 20 as well.

  1. Why doesn’t perfect consumption smoothing in preference mean perfect consumption smoothing in data observation?

Though consumers prefer a perfectly smoothed consumption pattern, unexpected changes on their income can mess up their initial plan, causing consumption to deviate from their past smoothing plan, which leads to unsmoothed consumption in data observation.