Macro Economics Assignment 1, Assignments of Macroeconomics

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University of Western Ontario

Prof. Simona Cociuba

Fall 2017

Assignment 1 Econ 2220A: Solutions

Due on Monday, September 25, 2017

1 Growth Rates

Question 1. (1 point) Consider the following economic facts: (i) Per capita real GDP in Krakozhia in year 2002 was 12 ; 000 Krakozhian dollars. (ii) Per capita real GDP in Krakozhia in year 2008 was 15 ; 000 Krakozhian dollars. If Krakozhiaís per capita real GDP grew at the CONSTANT ANNUAL growth rate g between year 2002 and year 2008 ; what is g? Show your derivations! Answer 1: y 2002 = 12; 000 and y 2008 = 15; 000 : Given constant annual growth, we have:

y 2008 = y 2007 (1 + g) = ::: = y 2002 (1 + g)^6

1 + g =

y 2008 y 2002

^16

^16

g = 3 : 8 percent

Question 2. (1 point) Suppose that per capita real GDP doubled from year t = 0 to year t = 1. What was the growth rate of per capita real GDP between period 0 and 1? Give your result in percent. Show your derivations! Answer 2: The growth rate was 100 percent.  y 1 y 0

100 = (2 1)
100 = 100 percent

Question 3. (1 point) Let yt denote GDP at time t; where subscript t is an integer that indicates the year. Suppose that the value of GDP at "the beginning of time" is y 0 = $500: If GDP grows at the constant rate of 2% per year, how many years will it take until GDP equals $788: 4496? Show your derivations! Answer 4: The answer is 23. This is obtained from solving the equation below:

y 0 (1 + 0:02)T^ = yT 500 (1 + 0:02)T^ = 788 : 4496

Taking natural logarithm on both sides, we get:

ln (500) + T ln (1:02) = ln (788:4496) , T =

ln (788:4496) ln (500) ln (1:02)

GDP according to the income approach After-tax wage income 14 ; 500 = 7; 000 + 5; 500 + 3; 500 1 ; 500 After-tax proÖts 9 ; 250 = 2; 950 + 6; 300 Taxes 3 ; 500 = 800 + 1; 200 + 1; 500 GDP 27 ; 250

GDP according to the product approach Value-added by plantation 10 ; 750 = 2; 500
2 + 500
2 + 1; 500
2 :5 + 500
2 Value-added by restaurant 13 ; 000 = 18; 000 2 ; 500
2 Value-added by government 3 ; 500 GDP 27 ; 250

GDP according to the expenditure approach Consumption 19 ; 000 = 500
2 + 18; 000 Investment 1 ; 000 = 500
2 Government expenditures 3 ; 500(= 800 + 1; 200 + 1; 500) Net exports (Exports - Imports) 3 ; 750 = 1; 500
2 : 5 GDP 27 ; 250

(5b). Write down a general expression for the accounting identity which deÖnes national savings. How much is national savings in this economy? Write down a general expression for the accounting identity which relates national savings to the current account. Is the current account in surplus or deÖcit in this economy? How large is the surplus/deÖcit? National savings equals

S = Sprivate^ + Sgovernment^ = Y + N F P C G Sprivate^ = Y d^ C = Y| + (^) {zN F P } GN P

+ T R T C

Sgovernment^ = T T R G

where N F P : net factor payments from the rest of the world, Y d^ : disposable income, T R : transfers, T : taxes. Note that N F P is zero in this economy. National savings in this economy equal Y C G = 27; 250 19 ; 000 3 ; 500 = 4; 750 : Write down national savings as function of current account:

S = Sprivate^ + Sgovernment^ = Y + N F P C G use Y = C + I + G + N X ) S = I + N X| +{z N F P } current account

= I + CA

The current account is in surplus because goods are exported. CA = 3; 750 :

Question 6. (1 point) Consider an economy with two goods. The quantities sold of each good in years 1 ; 2 and 3 , and the corresponding prices are given in the table below. Compute nominal GDP. Compute Öxed-weight real GDP and chained-weight real GDP using year 2 as the base year. Compute the GDP price deáator and the ináation rate. Fill in your answers in the table below. Show all your derivations!

Year 1 Year 2 Year 3 Prices of good 1 6 7 8 Prices of good 2 5 4 3 Quantities of good 1 42 43 44 Quantities of good 2 30 35 40

Nominal GDP 402 441 472

Fixed-weight real GDP (base=year 2) 414 441 468

Chain-weight real GDP (base=year 2) 411 : 7 441 465 : 8

Implicit GDP price deáator (uses fwRGDP) 97 : 10 100 100 : 85 Ináation rate (uses fwRGDP) n/a 2 :99% 0 :85%

Implicit GDP price deáator (uses cwRGDP) 97 : 64 100 101 : 33 Ináation rate (uses cwRGDP) n/a 2 :42% 1 :33%

Nominal GDP

Nominal GDP 1 = 6
42 + 5
30 = 402 Nominal GDP 2 = 7
43 + 4
35 = 441 Nominal GDP 3 = 8
44 + 3
40 = 472

Fixed-weight real GDP in base year 2.

f wRGDP 1 = 42
7 + 30
4 = 414 f wRGDP 3 = 44
7 + 40
4 = 468

Chained-weight real GDP in base year 2 is computed as follows:

cwRGDPt = cwRGDPt 1

s P pt 1
qt P pt 1
qt 1

 P

pt
qt P pt
qt 1

| {z } Fisher Index (t 1 ;t)

The Fisher index measures the ratio of real GDP in two consecutive years by taking a geometric average of the Laspeyres and Paasche indices. (The Laspeyres and Paasche indices also measure the ratio of real GDP in two consecutive years using di§erent prices as base

3 Measurement: Price Indices

Question 7. (1 point) Use the information in Table 1 to answer (a) ; (b) and (c) below.

Table 1. Consumerís purchases of two goods

Prices in year 1 pgood1 1 = 5 pgood2 1 = 6 Quantities in year 1 q 1 good1 = 20 qgood2 1 = 10

Prices in year 2 pgood1 2 = 7 pgood2 2 = 6 Quantities in year 2 q 2 good1 = 15 qgood2 2 = 12

(a) : Give a deÖnition of the consumer price index.

Current year CP I =

Cost of base year quantity at current prices Cost of base year quantity at base year prices

(b) : Use the information in Table 1 to calculate the CPI in year 1 and year 2 : Use Year 2 as base year. If we take year 2 as the base year:

CP I 1 base year 2 =

pgood1 1 qgood1 2 + pgood2 1 qgood2 2 pgood1 2 qgood1 2 + pgood2 2 qgood2 2

CP I 2 base year 2 = 100

(c) : What is the CPI ináation rate from year 1 to year 2? The ináation rate is equal to the growth rate in the consumer price index.

CP I 2 CP I 1 CP I 1

If year 2 is base year, we get: 10083 :^8305 : 05
100 = 20:41%.

4 Math Refresher

Question 8. (2 points) (a) : Consider the following functions. The function h is a function of a single variable and is deÖned on the space of positive real numbers and takes values in the space of positive real numbers h : R+! R+: The functions f and g are deÖned on R^2 + and take values in R+: Find the partial derivative of these functions with respect to the variable x: (Hint: For f and g; treat the variable y as a constant when taking the derivative).

h (x) = ln (x) derivative is = (^1) x

f (x; y) = x

1 (^2) y 1 (^3) derivative is =^12 x^ 1 (^2) y 1 3

g (x; y) = ln (xy^2 + 5) derivative is = (^) xy (^21) +5
y^2 = y

2 xy^2 +

(b) : Solve for x that satisÖes: log (x + 1) log (x + 4) = log

x

Use the properties of the log function

log

x + 1 x + 4

= log

x

x + 1 x + 4

x

) x^2 + x = x + 4 ) x =  2

but x needs to be > 0 so that log

x

doesnít tend to - 1 ; so x = 2

(c) : Solve for x and y that satisfy the following system of equations: 8 < :

(^3) y 1+x

0 :3 + (1 + x)
0 :2 = y

Equation 1 becomes

3 y 1 + x

3 y 1 + x

) 1 :4 + 1: 4 x = 3 y ) y = 1: 6 1 : 4 x

Equation 2 becomes: 0 :3 + 0:2 + x
0 :2 = y ) 0 :5 + 0: 2 x = y So, we have

y = 1 : 6 1 : 4 x = 0:5 + 0: 2 x ) 1 :1 = 1: 6 x ) x =

) y = 1: 6 1 : 4
0 :6875 = 0: 6375