Introduction to Macroeconomics
                                Answers to First Homework Assignment
                                                 Robert M. Kunst
                                                     April 2012
                1. An open economy produces aniseed (A) and basil (B). All basil is exported. Residents consume aniseed
                   and cinnamon (C). All cinnamon is imported. The following table shows quantities and prices for the
                   three goods for two successive years.
                                                quantities              prices
                                         year   A B C           year  A B C
                                           1   100 50 60          1   5  5   8
                                           2   110 70 50          2   5  7  10
                   Note that, in year 2, households have replaced some expensive imported cinnamon by some additional
                   demand for aniseed, while the world market demands for more basil even at a higher price. Also note
                   that GDP consists of aniseed and basil only. You see this from the production definition, but also
                   from the account-zero identity in an open economy Y = C + I + G + X − IM, with G = I = 0, C
                   corresponding to aniseed plus cinnamon, IM to cinnamon, and X to basil.
                   (a) Determine the GDP deflator for year 2, if it has been set at 1 in the base year 1 and it is assumed
                       to follow the traditional Paasche concept. Also state the rate of inflation for year 2. The answer
                       evolves from direct insertion into the Paasche formula
                                             A A     B B
                                            Q P +Q P        110∗5+70∗7
                                             2  2    2 2  =              ≈1.1556,
                                             A A     B B    110∗5+70∗5
                                            Q P +Q P
                                             2  1    2 1
                       and the rate of inflation in year 2 is thus 15.6%.
                   (b) Determine a consumer price index, which is defined as a Laspeyres index for household consump-
                       tion, in year 2 when it has been set at 100 in the base year 1. Also state the rate of inflation.
                       The answer evolves from direct insertion into the Laspeyres formula
                                             A A    C C
                                           Q P +Q P        100∗5+60∗10
                                             1 2    1  2 =               ≈1.1225,
                                             A A    C C     100∗5+60∗8
                                           Q P +Q P
                                             1 1    1  1
                       thus the Laspeyres index for year 2 will be 112.25, and the measured rate of inflation will be
                       12.2%.
                   (c) Less customary counterparts to (a) and (b): The counterpart to (a) is the Laspeyres index for
                       GDP, i.e.
                                             A A     B B
                                            Q P +Q P        100∗5+50∗7
                                             1  2    1 2  =              ≈1.1333,
                                             A A     B B    100∗5+50∗5
                                            Q P +Q P
                                             1  1    1 1
                       with the implied rate of inflation of 13.3%. The counterpart to (b) is the consumption deflator
                                             A A    C C
                                           Q P +Q P        110∗5+50∗10
                                             2 2    2  2 =               ≈1.1053,
                                           QAPA     C C     110∗5+50∗8
                                                 +Q P
                                             2 1    2  1
                       with the implied rate of inflation of 10.5%.
                       (d) Calculate the nominal GDP for both years, deflate the GDP for year 2 by the GDP deflator
                           from question (a) and compare to the direct (Laspeyres-type) formula for real GDP given in the
                           lecture notes. The two values should coincide. The nominal GDP for both years has already been
                           calculated in the above points (a) and (c). For year 1, we have 750, for year 2, 1040. Deflating
                           the nominal GDP for year 2 by the corresponding deflator yields
                                                                     1040 ≈900,
                                                                     1.1556
                           properly accounting for rounding errors. This, of course, is the same as
                                                          PAQA+PBQB=550+350=900
                                                           1   2    1   2
                    2. The goods market of a closed economy obeys the following equation system:
                                                   C=50+0.9∗(Y −T)             Z =C+I+G,
                      with the given exogenous values I = 100, G = 600, T = 500.
                       (a) Determine the equilibrium values for output at Y = Z, for C, and for disposable income Y −T =
                           YD. Evaluate the fiscal multiplier. The equilibrium output evolves from solving the equation
                                                  Y =50+0.9∗(Y −500)+100+600∴Y =3000,
                           or from insertion into the equilibrium formula. Clearly, C = 2300 and Y   =2500. In this simple
                                                                                                   D
                           model, the fiscal multiplier is determined by 1/(1 − c ) = 10.
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                       (b) This equilibrium is unsatisfactory, as government expenditures exceed revenues and thus public
                           saving is negative. Determine the values for G and T that are required to fulfill the condition
                           G=Tandnevertheless keep output Y at the same level. Evaluate C and Y . Here, the solution
                                                                                                      D
                           procedure changes, Y = 3000 is given and G = T is searched for:
                                                  3000 = 50+0.9∗(3000−T)+100+T ∴T =1500,
                           which then implies Y   =1500 and C = 1400.
                                                D
                        (c) Suppose the government now increases G as well as T by the same amount, say by 100. By how
                           much does Y change if at all? The corresponding value ∆Y/∆G is called the balanced-budget
                           multiplier. We evaluate the equilibrium condition for the new G = T = 1600:
                                                 Y =50+0.9∗(Y −1600)+100+1600∴Y =3100,
                           i.e. GDP increases as much as G, thus the balanced budget multiplier is 1 (one).
                       (d) Now re-consider the original economy in (a), i.e. without assuming G = T, but assume that taxes
                           T depend on output T = 0.2Y. Determine the fiscal multiplier for this modified model. We
                           consider the equilibrium condition
                                               Y =50+0.9∗(Y −0.2Y)+100+G∴Y = 150 + G ,
                                                                                           0.28    0.28
                      which yields the corresponding fiscal multiplier of around 3.57. Of course, this multiplier only applies
                      to fiscal policy related to spending, as taxes cannot be changed here autonomously.
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