Econ303 What are derivatives 1The derivative of a univariate function f(x) evaluated at x, denoted either by f0(x) ordf(x)dx,is defined as the follows.df(x)dx= lim→0f(x + ) − f(x)1. Given function f(x) = 10 + 3x, if x = 10, f (x) = . If x increases by 1,f(x) increases by .The rate of change is defined by the ratio between change of values of function andchange 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 and rate of change is .The derivatives of f(x) at x = 10 is .The slope of the straight line represented by function f(x) is .2. Given function f(x) = x2, if x = 10, f(x) = . If x increases by 1, f(x)increases by . 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?9Econ303 Find derivatives 11. f(x) =1x2. f(x) =√x + ln ax3. f(x) = 4x0.34. f(x) = e3x2−2x10Econ303 Measure GDP 11. A wheat and bread only economyThe 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 $3.50per loaf.Determine the GDP.(Additional thoughts) In the US, a large fraction of transactions among banks takesplace over Fedwire, which is an electronic payments system operated by the FederalReserve System. During 2008, on an average day, 521, 000 payments were made overFedwire, with a total value of $2.7 trillion. To put this in context, annual GDP in 2008was $14.3 trillion, so on average total daily transactions over Fedwire were about 19%of total annual GDP. How could these be?11Econ303 Measure GDP 22. 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 2million tons are exported, the rest is sold to domestic households.• pays $40 million in wagesDetermine GDP using three approaches.(Additional thoughts) Determine the GNP if coal company is owned by foreigners.12Econ303 Price index and inflation 1Consider the data for a simple toy economy given below.Price 2000 2001 Quantity 2000 2001oranges $15 $17 oranges 100 120apples $10 $20 apples 150 1201. 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.13Econ303 Chain-weighted real GDP 1Consider the following data from a simple toy economy as shown.Year 2000 Year 2001 Year 2002Price of car $15,000 $17,500 $20,000Price of apple $10 $20 $25Number of cars produced 100 120 130Pounds of apples produced 10,000 12,000 15,000Calculate the Chain-weighted real GDP, where 2001 is the base year.1. 2000-2001 index using 2000 price2. 2000-2001 index using 2001 price3. 2000-2001 Fisher index4. 2001-2002 index using 2001 price5. 2001-2002 index using 2002 price14Econ303 Chain-weighted real GDP 26. 2001-2002 Fisher index7. real GDP in 20018. real GDP in 20009. real GDP in 200215Econ303 Cost of living adjustment 1We have following data on US federal minimum wage.year minimum wage CPI1964 $1.15 31.32010 $7.25 220.31. $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?16Econ303 Some algebra with growth rate 1We have the following data on China.Year 1960 1990GDP (billion $) 378.3 1,500.9Population (billion) 0.667 1.1341. 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 p opulation.4. Compute the annual average growth rate of GDP per capita.17Econ303 Solve for unknowns 11. 1.5 + 3x = 6 + 1.5x2.5x= 63. 7x0.3= 5x4.√3 + 5x = 25. 1.05x= 10 · 1.02x18Econ303 Power of growth 1Two countries, Fast and Slow, have real GDP per capita $1,000 and $5,000 in 2000, respec-tively.1. In country Fast, GDP per capita increases on average 8% per year. By 2014, what willits GDP per capita be?2. In country Slow, its GDP per capita in 2010 was $6,095. What was its annual averagegrowth rate during this time period?3. If the growth rate in each country stays the same in the future, how many years after2000 will country Fast catch up with country Slow?19Econ303 Production and Marginal product 1Consider a production function Y = F (K, L) = 100K0.3L0.7.1. If 100 units of capital and 2 units of labor are used in production, how many units ofoutput can be produced?2. If capital doubles while labor stays the same, how many units of output can be pro-duced? 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 pro-duced? 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 outputcan be produced? Is output more or less than doubled?20Econ303 Production and Marginal product 25. Give a general expression for MPKand MPL.6. Evaluate the MPKand MPLfor the following combination of inputs.(K, L) MPKMPL(100, 2)(200, 2)(100, 4)(200, 4)7. Currently, 200 units of capital and 2 units of labor are used in production. If you areconsidering to add one more worker, what would be your estimate of his/her contribu-tion towards output?(Further thoughts) If the TFP rose from 100 to 120, what would happen to MPKand MPL?21Econ303 Steady state of Malthusian world 1Country Maltopia had production functionY = 250D13N23Y is measured by pounds of food and D acres of land. The US Department of Agricultureestimates that in the US on average each person eats 5 pounds of food per day. Assume thatthe subsistence level of consumption is bit less than half of that, which means 700 poundsper year.1. Compute the steady state population density in country Maltopia.2. Many years passed. The