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EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Problem set 2 Solution 1) Consider an economy with only two firms, Firm A that produces apples, and Firm J that produces apple juice. The transactions carried by the two firms in the economy in 2009 are written below: Firm A Wages paid to employees £20000 Profits £15000 Rent for Land £10000 Revenues from selling apples to the £15000 public Revenues from selling apples to Firm J £30000 Wages paid to employees Sales revenues Apples purchased from Firm A Juices boxes purchased from abroad Firm J £10000 £50000 £30000 £10000 a) Calculate the GDP in this economy in 2009 using the production, the expenditure and the income method. Show that all methods give the same answer. b) Provide some intuition to the fact that the Real GDP may not be a good measure of overall welfare of a given economy. Solution a) Product approach: in this case the GDP is the defined as the sum of the value of all goods and services produced minus the value of all intermediate goods used in production From the data we have the value of production of Firm A is £15000 + £30000 (its revenues). That represents the value of the final goods produced and sold by Firm A. Firm A does not use any intermediate good. Firm J has a value given by: £50000. Firm J uses intermediate goods for a value of £30000+£10000=£40000. Therefore, according to the product approach the GDP is given by: £45000 + £50000 - £40000 = £55000 Expenditure approach: the GDP is defined as the total spending on all final goods and services produced in the economy in a given period of time. We know that Total Expenditure = C + I + G + NX In our economy, I = 0 (notice that while we have Land in the table we do not have the purchased value of it but instead only the rental income during a given period of time), G = 0 and NX = -£10000. We have that total consumption is: £15000 (what the public consumes from buying the goods from Firm A) + £50000 (consumption of juices) - £10000 (imports) = £55000 Income approach: the GDP is defined as the sum of all income received by economic agents contributing to production Here we have: £20000 of wages paid by firm A, £15000 of profits distributed by Firm A, £10000 of rent paid to the owner of land by Firm A. Then we have £10000 of wages paid by Firm J. Summing up all those values we have: £20000 + £15000 + £10000 + £10000 = £55000. b) We know that the GDP measures equivalently the total output produced, the total income and the total expenditure. Are those things equivalent to a measure of welfare? When we talk about welfare we think about something like the well-being of an individual or of a group of individuals. We need to notice that the GDP measures the value that arises mainly from market transactions (some elements of the public expenditure that enter in the GDP may be valued at the cost since a market may not exist, for example, national defense expenditure). Then we can ask if there are elements that may affect the well being of a society that do not arise from market transactions. The answer is yes. For example, the GDP does not measure leisure since we do not have a market for leisure; nevertheless, leisure tends to provide utility to individuals. Moreover, the GDP also does not take into account home produced goods and the quality of these goods. Family care by a mother is normally considered superior to hiring a stranger to care for your child. Although this service is considered superior for most people they are not included in GDP. There are also market transactions that affect the well-being of societies but are not recorded in the DGP calculations. For example, all the transactions related to the underground economy (black markets and illegal markets). People can obtain goods and services (and this affects their well-being) from black markets and in many countries the underground economy can be as large as the legal economy. Finally GDP does not take into account income distribution. If a country has a very high GDP per capita, but all the income is taken by few individuals, the majority of the population is likely to have a very low level of welfare. For example according to the IMF, in 2009, the GDP per capita in Qatar was $78260 while the GDP per capita in UK in the same year was $34388. This would imply that the welfare of an average citizen in Qatar is more than twice the welfare of an average citizen in UK. Obviously, a precise measure of welfare is quite difficult to be calculated, while calculating the GDP is normally quite simple. Therefore, the GDP can be considered an approximation to the level of welfare (like the consumer surplus is an approximation of welfare for the single consumer in a market). 2) (Income distribution) Consider an economy where total output is produced according to the following aggregate production function: Y = K α Lβ Where Y denotes output, K is the level of capital and L is the level of labour. The terms α and β are positive constants. Assume that the input markets (for capital and labour) and the output market are competitive. Show that when α + β > 1 then the Product Exhaustion Theorem does not hold. Provide an intuition for your finding. Solution The Product Exhaustion Theorem tells that if there are constant returns to scale and all markets are competitive, then the input factors are paid according to their marginal productivity and those payments exhaust the value of output. Here we have the case of a Cobb-Douglas production function that under the assumption that α + β > 1 displays Increasing Returns to Scale. Then we already know that the Product Exhaustion Theorem will not work. To prove it, we need to use the Euler’s Theorem for Homogenous equation. First we define what a homogeneous function is. A function f ( x1 , x 2 ,..., x n ) (where we consider a function of n variables just to be general enough, but this will be true also for a function of a single variable) is homogeneous of degree k if it’s true that: f (rx1 , rx 2 ,..., rx n ) = r k f ( x1 , x 2 ,..., x n ) Where r>0 is a number (positive) and k is a constant. Once we know what a homogeneous function is we can apply the Euler’s Theorem. The Euler’s theorem says that if a function f ( x1 , x 2 ,..., x n ) is homogenous of degree k, then we can write: ∂f ∂f ∂f kf ( x1 , x 2 ,... , x n ) = x1 + x2 + ...+ x n ∂ x1 ∂x2 ∂xn Meaning: the original function (scaled by k) can be written in terms of its partial derivatives. We already know that the Cobb-Douglas production function is homogenous of degree α + β . To prove that you just need to check that: F ( rK , rL) = (rK ) ( rL) β = r α K α r β Lβ = ( r )α + β K α Lβ α = r (α + β ) F ( K , L ) Using the previous notation we have here k = (α + β ) Using the Euler’s theorem we have then: ∂F ( K , L) ∂F ( K , L) (α + β ) F ( K , L) = K+ L ∂K ∂L That expression is equivalent to: ∂F ( K , L) ∂F ( K , L) (α + β )Y = K+ L ∂K ∂L ∂F ( K , L) ∂F ( K , L) Where is the marginal productivity of capital while is the ∂K ∂L marginal productivity of labour. ∂F ( K , L) r If all markets are competitive then it must be true that = , and ∂K P ∂F ( K , L) w = , where r is the rental price of capital and w is the price of labour (the ∂L P wage). To see this, suppose that there is a single representative firm in the economy. The profit of this representative firm is: π = PY − wL − rK ⇒ PF ( K , L) − wL − rK If markets are competitive then the firm takes the prices (P, w and r) as given. The firm maximises profits by choosing the level of inputs (L and K) to be used to produce the output Y. By maximising the profit function with respect to K and L and by setting those derivatives equal to zero we have: ∂π ∂F ( K , L) =P −r =0 ∂K ∂K ∂π ∂F ( K , L) =P −w=0 ∂L ∂L ∂F ( K , L) r ∂F ( K , L) w = , = , meaning that if ∂K P ∂L P markets are competitive a firm chooses the level of capital and the level of labour such that the marginal productivity of each input is equalised to the real price of that input. Multiply by P each side of the following equation: Those two equations imply that: (α + β )Y = ∂F ( K , L) ∂F ( K , L) K+ L ∂K ∂L So we have: ∂F ( K , L) ∂F ( K , L) K+P L 1) ∂L ∂K ∂F ( K , L) r ∂F ( K , L) ∂F ( K , L) Using the fact that = ⇒r=P and w = P into P ∂K ∂K ∂L equation 1) we have: (α + β ) PY = rK + wL PY is the value of total product, or the nominal GDP. If α + β > 1 , then we have that (α + β ) PY > PY , meaning that: PY < rK + wL When there are increasing returns to scale and each input is paid according to its marginal productivity, then the payments to the inputs (rK is the payment received by the owner of capital, while wL is the payment received by the workers) more than exhaust the total value of output. This means that IF inputs are paid at their marginal productivity value (P multiplied the marginal productivity) then the value generated is MORE than the value that will be distributed (PY). How can it be? A possible explanation is that with increasing returns to scale there is some sort of positive externality that will create extra value but nobody (workers and capitalists) can appropriate this extra value. This also implies that since what is distributed is only PY, at least one input must be paid LESS than its marginal product. This implies that the respective market CANNOT be competitive, that is the contrary of what we have assumed. This result also explains why in economic models the use of constant returns to scale is normally preferred. This is because by assuming constant returns to scale for the aggregate production function the problem of income distribution is not an issue anymore. (α + β ) PY = P 3) (Twin deficit) From the national accounting identity derive the relationship between Aggregate Saving, Investment and the Current Account. Using that relationship explain why a fiscal deficit can be associated with a current account deficit. Solution The national accounting identity is: Y = C + I + G + NX To derive the link between aggregate saving, investment and current account we need to define what is aggregate saving. Let’s call it S Define S = S Pr + S Pub , where S Pr = Y + NFP − T − C is private saving. In particular Y + NFP is the Gross National Product (i.e. GDP plus Net Factor Payments from abroad), T are taxes and C is aggregate consumption. In practice aggregate private saving is just aggregate disposable income (i.e. income after taxes) minus aggregate consumption. S Pub = T − G is public saving which is just tax revenues minus government expenditure. Using the definitions of S Pr and S Pub into the definition of S we get: S = Y + NFP − T − C + T − G = Y + NFP − C − G Using the national accounting identity to substitute Y into that equation: S = C + I + G + NX + NFP − C − G = I + NX + NFP NX + NFP is the current account and therefore we can write: S = I + CA Aggregate domestic saving equals domestic investment plus the current account. To see how a fiscal deficit can be related to a current account deficit we can do the following. Write the previous equation as: S Pr + S Pub = I + CA Now assume that S Pub = 0 which means that the government is running a balanced budget ( T = G ) . Moreover assume that S Pr is equal to I . Then the above equation implies that CA = 0 so the current account is balanced. Now assume that the government starts running a fiscal deficit, that is S Pub < 0 . If S Pr is still equal to I , i.e. they have not changed because of the fiscal deficit, then for the equation above to hold it must be that CA < 0 . Therefore it is possible that a fiscal deficit can be associated (in this case it is the cause) of a current account deficit. This is known as the Twin Deficit Hypothesis. The typical case is the US that in the last 30 years have experienced high fiscal deficits and high current account deficit. One possible explanation is that the current account deficits were the consequences of fiscal deficits. However for this hypothesis to work (i.e. that fiscal deficits are the cause of current account deficits) we have to assume that private saving is equal to domestic investment and they remain constant after the fiscal deficit has been created. Those assumptions may not hold true in reality (and in general they are not true).