9 Econ303 What are derivatives 1 The derivative of a univariate function f x evaluated at x denoted either by f 0 x or df x dx is defined as the follows df x f x f x lim 0 dx 1 Given function f x 10 3x if x 10 f x f x increases by If x increases by 1 The rate of change is defined by the ratio between change of values of function and change of values of argument Calculate the rate of change in this case If another variable y f x the rate of change measures the impact of x on y locally i e around x 10 If x increases by 0 1 f x increases by The derivatives of f x at x 10 is and rate of change is The slope of the straight line represented by function f x is 2 Given function f x x2 if x 10 f x increases by If x increases by 1 f x Calculate the rate of change in this case If x increases by 0 1 f x increases by and rate of change is Give a general expression for the derivatives of f x The derivatives of f x at x 10 is 3 If we use derivatives of f x to estimate the rate of change what is its limitations 10 Econ303 1 f x 2 f x Find derivatives 1 x x ln ax 3 f x 4x0 3 2 2x 4 f x e3x 1 11 Econ303 Measure GDP 1 1 A wheat and bread only economy The wheat producer grows 30 million bushels of wheat out of which 25 million sold to bread producer at 3 per bushel 5 million sold to himself as seeds The bread producer made 100 million loaves of bread and sold to consumers at 3 50 per loaf Determine the GDP Additional thoughts In the US a large fraction of transactions among banks takes place over Fedwire which is an electronic payments system operated by the Federal Reserve System During 2008 on an average day 521 000 payments were made over Fedwire with a total value of 2 7 trillion To put this in context annual GDP in 2008 was 14 3 trillion so on average total daily transactions over Fedwire were about 19 of total annual GDP How could these be 12 Econ303 Measure GDP 2 2 Three approaches for a coal steel economy a Coal producer produces 15 million tons and sells it for 5 per ton pays 50 million in wages b Steel producer uses 25 million tons of coal out of which 10 million tons are imported produces 10 million tons of steels and sells it for 20 per ton out of which 2 million tons are exported the rest is sold to domestic households pays 40 million in wages Determine GDP using three approaches Additional thoughts Determine the GNP if coal company is owned by foreigners 13 Econ303 Price index and inflation 1 Consider the data for a simple toy economy given below Price 2000 2001 oranges 15 17 apples 10 20 Quantity oranges apples 2000 2001 100 120 150 120 1 Using year 2000 price to calculate the real GDP Find the GDP deflators in each year 2 Year 2000 is the base year Calculate the CPI in each year 3 Calculate the inflation rate in 2001 using each type of price index 14 Econ303 Chain weighted real GDP 1 Consider the following data from a simple toy economy as shown Year 2000 Year 2001 Price of car 15 000 17 500 10 20 Price of apple Number of cars produced 100 120 Pounds of apples produced 10 000 12 000 Year 2002 20 000 25 130 15 000 Calculate the Chain weighted real GDP where 2001 is the base year 1 2000 2001 index using 2000 price 2 2000 2001 index using 2001 price 3 2000 2001 Fisher index 4 2001 2002 index using 2001 price 5 2001 2002 index using 2002 price 15 Econ303 6 2001 2002 Fisher index 7 real GDP in 2001 8 real GDP in 2000 9 real GDP in 2002 Chain weighted real GDP 2 16 Econ303 Cost of living adjustment We have following data on US federal minimum wage year minimum wage CPI 1964 1 15 31 3 2010 7 25 220 3 1 31 3 in 1964 has the same purchasing power as to how many dollars in 2010 2 1 15 in 1964 has the same purchase power as how many dollars in 2010 3 Has the living standard of a minimum wage earner increased 1 17 Econ303 Some algebra with growth rate We have the following data on China Year 1960 GDP billion 378 3 Population billion 0 667 1990 1 500 9 1 134 1 Compute GDP per capita in 1960 and 1990 2 Compute the annual average growth rate of GDP 3 Compute the annual average growth rate of population 4 Compute the annual average growth rate of GDP per capita 1 18 Econ303 1 1 5 3x 6 1 5x 2 5 x 6 3 7x0 3 5x 4 3 5x 2 5 1 05x 10 1 02x Solve for unknowns 1 19 Econ303 Power of growth 1 Two countries Fast and Slow have real GDP per capita 1 000 and 5 000 in 2000 respectively 1 In country Fast GDP per capita increases on average 8 per year By 2014 what will its GDP per capita be 2 In country Slow its GDP per capita in 2010 was 6 095 What was its annual average growth rate during this time period 3 If the growth rate in each country stays the same in the future how many years after 2000 will country Fast catch up with country Slow 20 Econ303 Production and Marginal product 1 Consider a production function Y F K L 100K 0 3 L0 7 1 If 100 units of capital and 2 units of labor are used in production how many units of output can be produced 2 If capital doubles while labor stays the same how many units of output can be produced By how much has the output increased Is output more or less than doubled 3 If labor doubles while capital stays the same how many units of output can be produced By how much has the output increased Is output more or less than doubled 4 If both capital and labor input double at the same time how many units of output can be produced Is output more or less than doubled 21 Econ303 Production and Marginal product 2 5 Give a general expression for M PK and M PL 6 Evaluate the M PK and M PL for the following combination of inputs K L 100 2 200 2 100 4 200 4 M PK M PL 7 Currently 200 units of capital and 2 units of labor are used in production If you are considering to add one more worker what would be your estimate of his her contribution towards output Further thoughts If the TFP rose from 100 to 120 what …