Description of dynamical processes Jacek Szanduła

Lecture plan Average changes. Aggregate indices. Fixed base and chain base measures. Changes and individual indices. Average changes. Aggregate indices. Lowe index. Laspeyres price and quantity indices. Paasche price and quantity indices. Other price indices: Fisher, Walsh, Marshall–Edgeworth,Törnqvist. Deflator. Coping with inflation. Jacek Szanduła

Fixed base (jednopodstawowy, o stałej bazie) and chain base (łańcuchowy) measures Fixed base – a particular period t0 is chosen arbitrarily and measures of dynamics are calculated with the use of values of the current period t and the base period t0. Chain base – measures of dynamics are calculated with the use of values of the current period t and the previous period t – 1. Jacek Szanduła

Changes and individual indices Fixed base Chain base Absolute change Percentage change Index Jacek Szanduła

Some relations between measures of dynamics Jacek Szanduła

Changes and indices – example Average monthly wages in PLN (X). t0 = 2002 Year X 2002 2133  2003 2201 68 3.2 1.032 2004 2290 157 89 7.4 4.0 1.074 1.04 2005 2380 247 90 11.6 3.9 1.116 1.039 2006 2477 344 97 16.1 4.1 1.161 1.041 2007 2691 558 214 26.2 8.6 1.262 1.086 2008 2944 811 253 38.0 9.4 1.38 1.094 2009 3103 970 159 45.5 5.4 1.455 1.054 2010 3225 1092 122 51.2 1.512 2011 3400 1267 175 59.4 1.594 Jacek Szanduła

Average changes Average Rate of Change (zmiana średniookresowa) Compound (Annual) Growth Rate (średniookresowe tempo zmian) Jacek Szanduła

Aggregate indices An aggregate index measures relative changes in values ​​of several variables taken as a combination. Example: Monthly expenditures Month May June Food Price Quantity Bread 2 30 2.50 25 Butter 5 10 6 12 Tomato 4 Ham 20 3 Cheese 2.5 22 How to measure growth of prices using single index? Jacek Szanduła

Lowe price index – Lowe price index – price of ith good in the tth moment – quantity of ith good in the tth moment Jacek Szanduła

Laspeyres price index Month May June Food Month May June Food SUM 270 Quantity Bread 2 30 2.50 25 Butter 5 10 6 12 Tomato 4 Ham 20 3 Cheese 2.5 22 Month May June Food Price Quantity Bread 2 30 2.50 25 2  30 = 60 2.5  30 = 75 Butter 5 10 6 12 5  10 = 50 6  10 = 60 Tomato 4 4  10 = 40 Ham 20 3 20  3 = 60 Cheese 2.5 22 20  2.5 = 50 22  2.5 = 55 SUM 270 290 Jacek Szanduła

Laspeyres quantity index Month May June Food Price Quantity Bread 2 30 2.50 25 Butter 5 10 6 12 Tomato 4 Ham 20 3 Cheese 2.5 22 Month May June Food Price Quantity Bread 2 30 2.50 25 2  30 = 60 2  25 = 50 Butter 5 10 6 12 5  10 = 50 5  12 = 60 Tomato 4 Ham 20 3 20  3 = 60 Cheese 2.5 22 20  2.5 = 50 20  2 = 40 SUM 270 Jacek Szanduła

Paasche price index Month May June Food Month May June Food SUM 296.5 Quantity Bread 2 30 2.50 25 Butter 5 10 6 12 Tomato 4 Ham 20 3 Cheese 2.5 22 Month May June Food Price Quantity Bread 2 30 2.50 25 2.5  25 = 72.5 2  25 = 50 Butter 5 10 6 12 6  12 = 72 5  12 = 60 Tomato 4 4  12 = 48 Ham 20 3 20  3 = 60 Cheese 2.5 22 22  2 = 44 20  2 = 40 SUM 296.5 270 Jacek Szanduła

Paasche quantity index Month May June Food Price Quantity Bread 2 30 2.50 25 Butter 5 10 6 12 Tomato 4 Ham 20 3 Cheese 2.5 22 Month May June Food Price Quantity Bread 2 30 2.50 25 2.5  25 = 72.5 2.5  30 = 75 Butter 5 10 6 12 6  12 = 72 6  10 = 60 Tomato 4 4  12 = 48 4  10 = 40 Ham 20 3 20  3 = 60 Cheese 2.5 22 22  2 = 44 22  2.5 = 55 SUM 296.5 290 Jacek Szanduła

Fisher price index Jacek Szanduła

Other price indices Walsh Marshall–Edgeworth Törnqvist (share of ith good in all expenditures at the tth moment) Jacek Szanduła

Törnqvist price index – example Month May June Food Price Quantity Bread 2 30 2.50 25 60 72.5 1.25 0.222 0.245 0.233 1.053 Butter 5 10 6 12 50 72 1.2 0.185 0.243 0.214 1.040 Tomato 4 48 0.8 0.162 0.174 0.962 Ham 20 3 1 0.202 0.212 Cheese 2.5 22 44 1.1 0.148 0.167 1.016 SUM 270 296.5  Jacek Szanduła

Deflator Deflator (deflator) – value used to convert current prices into inflation-adjusted prices. One uses a deflator when one seeks to determine whether or not prices are rising in real terms. Jacek Szanduła

Coping with inflation Year X CPIt/t-1 Y Z X – average monthly wages in PLN. t0 = 2002 CPIt/t-1 – Consumer Price Index year to previous year Y – real average monthly wages in PLN (base year 2002) Z – real average monthly wages in PLN (base year 2011) Year X CPIt/t-1 Y Z 2002 2133 1.019 2715 2003 2201 1.008 2201 : 1.008 = 2184 2779 2004 2290 1.035 2290 : 1.008 : 1.035 = 2195 2793 2005 2380 1.021 2234 2844 2006 2477 1.01 2302 2930 2007 2691 1.025 2440 3106 2008 2944 1.042 2562 3261 2009 3103 2609 3103  1.026  1.043 = 3321 2010 3225 1.026 2643 3225  1.043 = 3364 2011 3400 1.043 2672 Jacek Szanduła

Exercise 1 Year Profit [mln PLN] CPIt/t-1 2007 1.9 1.025 2008 2.1 1.042 2009 2 1.035 2010 1.026 2011 1.043 Present the company's profit in prices of 2009. Jacek Szanduła

Exercise 2 Calculate indices: Laspeyres, Paasche, Fisher. Year 2010 2011 Good Price Quantity Bread 2.50 200 2.00 Bus tickets 500 400 Beer 100 120 Calculate indices: Laspeyres, Paasche, Fisher. Jacek Szanduła