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Chapter 7. National Income National Income – effect of changes in government spending, private investment, the marginal propensity to consume and the tax rate. In a closed economy, given the consumption function, the average tax rate, the level of private investment and the level of government spending, calculate the equilibrium level of income and output in the economy. Calculate the expenditure multiplier and show the effect of a) an increase in government spending and b) an increase in private investment. Consumption function: C = 0.8(1 – 0.25)Y + 12 (1) Investment: I = 26 Government spending: G = 14. Solution: The general equation for the consumption function can be represented by C = c(1 – t)Y + k where C is consumption by households, c is the marginal propensity to consume, t is the tax rate, Y is national income and k is a constant (autonomous consumption). Note: Do not confuse consumption “C” (upper case) with the marginal propensity to consume “c” (lower case). The marginal propensity to consume (MPC) is the increase in consumption per unit increase in income. The value of c is between 0 and 1. Any increase in income that is not consumed is savings. The values of c and t determine the slope of the consumption function. In a Closed Economy (no imports or exports), aggregate demand is the sum of household consumption, C, private investment, I, and government spending, G. AD = C + I + G. At equilibrium, aggregate demand equals income and output. Y=C+I+G Y = c(1 – t)Y + k + I + G Y – c(1 – t)Y = k + I + G Y(1 –c + ct) = k + I + G Y= www.wiley.com/college/wiley Page 1 Chapter 7. National Income Given C = 0.8(1 – 0.25)Y + 12 (MPC = 0.8, average tax rate is 25% and autonomous consumption is 12). Y = 0.8(1-0.25)Y + 12 + I + G Y = 0.8 (0.75)Y + 12 + I + G Y = 0.6Y + 12 + I + G 0.4Y = 12 + I + G Y= Y = 30 + 2.5I + 2.5G If private investment, I = 26 and government spending, G = 14, the economy is in equilibrium when Y = 30 + 2.5(26) + 2.5(14) (2) Y = 30 + 65 + 35 Y = 130. Demand equals income and output. Table 7.1 Calculate points for Consumption and Agg. Demand functions Y C = 0.6Y + 12 AD = C + 26 + 14 0 12 40 36 80 60 120 84 160 108 200 132 52 76 100 124 148 172 www.wiley.com/college/wiley Page 2 Chapter 7. National Income 200 180 160 E 140 D 120 e m 100 a n 80 d AD= 0.6Y+12+26+14 60 C=0.6Y+12 40 20 0 0 20 40 60 80 100 120 Income / Output 140 160 180 200 Figure 7.1C. Consumption Function and Aggregate Demand Equilibrium occurs where the 45° line intersects the AD line. Demand is a function of Y, C, c, t, I and G. In economics we often want to see the effect of a change in one variable while holding other variables constant. Given the consumption function, the effect on equilibrium income of a change in government spending, G, while holding private investment, I, constant is given by ∂Y/∂G. This is a partial derivative Y = 30 + 2.5I + 2.5G ∂Y/∂G = 2.5 An increase in government spending results in 2.5 times that increase in national income. If government spending increases by 8, from 14 to 22, the equilibrium level of income increases by 2.5 × 8 = 20. The new equilibrium level of income and output is 150. With government spending increased by 8, the new aggregate demand schedule, AD’ = 0.6Y + 12 + 26 + 22. www.wiley.com/college/wiley Page 3 Chapter 7. National Income Table 7.2 Calculate points for function Y = 0.6Y + 12 + 26 + 14 and Y = 0.6Y + 12 + 26 + 22 Y C = 0.6Y + 12 AD = C + 26 + 14 AD' = C + 26 + 22 0 12 40 36 80 60 120 84 160 108 200 132 52 76 100 124 148 172 60 84 108 132 156 180 200 180 160 140 D e 120 m 100 a n 80 d 60 AD'=0.6Y+12+26+22 AD=0.6Y+12+26+14 C=0.6Y+12 40 20 0 0 20 40 60 80 100 120 140 160 180 200 Income / Output Figure 7.2C. Aggregate Demand – Effect of increase in government spending. An increase in government spending shifts the aggregate demand schedule upwards, parallel to the original aggregate demand schedule. Similarly, an increase in private investment I, with government spending held constant, will result in a 2.5 times increase in national income. Y = 30 + 2.5I + 2.5G ∂Y/∂I = 2.5 www.wiley.com/college/wiley Page 4 Chapter 7. National Income This effect is known as the multiplier. The slope of the consumption function determines the value of the multiplier. The value of the multiplier is given by When c = 0.8 and t = 0.25 the multiplier M = ( ) = = 2.5 Calculate the effect of a change in a) the marginal propensity to consume and b) the tax rate on the value of the multiplier. Solution a) If the tax rate remains constant, the effect of a change in the marginal propensity to consume, c, on the value of the multiplier is given by ∂M/∂c. M = = when t = 25% = ∂M/∂c = ( = = ) ( ( ) ( ( ) ) ) when c = 0.8 = = = 4.69 When the marginal propensity to consume, c, = 0.8, a small change in c will change the value of the multiplier by 4.69 times that change. M= When c = 0.8 and t = 0.25, M = 2.5 (see above) www.wiley.com/college/wiley Page 5 Chapter 7. National Income If c increases to 0.81, M = ( ) = = = 2.547 An increase of 0.01 in the marginal propensity to consume (from 0.8 to 0.81) caused the multiplier to increase by 0.047 (from 2.5 to 2.547), confirming the value of ∂M/∂c above. Solution b) If the marginal propensity to consume remains constant, the effect of a change in the tax rate, t, on the value of the multiplier is given by ∂M/∂t. M = = when c = 0.8 = ∂M/∂t = ∂M/∂t = ( ) ( ( ( ) ) when t = 25% ) = = = -5 When t = 0.25, a small change in t will change the value of the multiplier by 5 times that change, in the opposite direction (indicated by the - sign). M= When c = 0.8 and t = 0.25, M = 2.5 (see above) If t increases to 0.26, M = ( ) = www.wiley.com/college/wiley Page 6 Chapter 7. National Income = = 2.45 An increase of 0.01 in the tax rate (from 0.25 to 0.26) caused the multiplier to decrease by 0.05 (from 2.5 to 2.45), confirming the value of ∂M/∂t above. Summary A change in either government spending or private investment is subject to the multiplier. Any increase (decrease) results in the equilibrium level of income increasing (decreasing) by a multiple of the change. The value of the multiplier depends on the slope of the consumption function. The slope of the consumption function depends on the marginal propensity to consume and on the tax rate. An increase in the MPC increases the slope of the consumption function and increases the value of the multiplier. An increase in the tax rate reduces the slope of the consumption function and reduces the value of the multiplier. www.wiley.com/college/wiley Page 7