Chapter 22: Adding Government and Trade to the Simple Macro Model Answers to Study Exercises Question 1 a) Public saving is equal to the government budget surplus, which is equal to T G. If G is $155 billion for any level of national income, the surplus is straightforward to compute. The completed table is shown below. National Income (Y) Net Tax Revenues (T) Public Saving (T G) 100 45 110 200 70 85 300 95 60 400 120 35 500 145 10 600 170 15 700 195 40 800 220 65 b) See the diagram below. The public saving function is upward sloping because as national income rises the government collects more tax revenue and spends less on transfers. Thus net tax revenue (T) rises for a given amount of government purchases (G). T G therefore rises as national income rises.
2 c) The net tax rate is the amount by which net tax revenues rise in response to a rise in national income. It is the slope of the public saving function. In this case, each $100 billion increase in real national income leads to a $25 billion increase in net tax revenues. Thus the net tax rate is 0.25, which is the slope of the T G function in the figure above (note that scales on the two axes are different). d) The increase in G by $15 billion means that the amount of public saving falls by $15 billion at each level of national income. Thus the public saving function shifts down by $15 billion. Question 2 a) Net exports are equal to exports minus imports. If exports are equal to $300 billion for any level of national income, then net exports are easily computed. The completed table is shown below. National Income (Y) Imports (IM) 100 85 215 200 120 180 300 155 145 400 190 110 500 225 75 600 260 40 700 295 5 800 330 30 Net Exports (X IM) b) See the figure below. The net export function is downward sloping because increases in real national income lead to increases in imports (IM) but leave exports (X) unchanged. Thus X IM falls as real national income rises. c) The marginal propensity to import is the change in imports that results from a $1 change in national income. From the information provided in the table above, for each $100 billion increase in national income, imports increase by $35 billion. Thus the marginal propensity to import is 0.35. This is the (absolute value of the) slope of the net export function.
3 d) If one of Canada s major trading partners experiences a recession, firms and households in that country will demand fewer Canadian goods and services. This results in a reduction in Canada s exports, X. Net exports will decline at any level of Canadian national income. Thus the net export function will shift down. For example, a reduction in Canadian exports of $20 billion would shift the net export function vertically downward by $20 billion. Question 3 a) The decrease in G shifts AE down and hence, other things being equal, leads to a reduction in equilibrium national income. b) The increase in Canadian wheat exports shifts AE up and hence, other things being equal, leads to an increase in equilibrium national income. c) The decrease in I shifts AE down and hence, other things being equal, leads to a reduction in equilibrium national income. d) The (permanent) reduction in income-tax rates raises disposable income, and hence desired consumption expenditure, in relation to national income. As a result, AE shifts up (and also becomes steeper). Other things being equal, equilibrium national income rises. (In the extreme case where Ricardian Equivalence holds exactly (see Chapter 31), individuals realize that today s reduction in tax rates must be offset by an increase in future tax rates, and thus they are not made wealthier by the tax reduction. In this case, desired consumption will not rise and there will be no effect on equilibrium national income.) e) The import restriction reduces U.S. imports from Canada. If the reduction in Canadian lumber exports is not replaced by exports to other countries, the AE function in Canada shifts down. Other things being equal, Canadian equilibrium national income falls. f) The Asian slump leads to a reduction in demand for Canadian exports, especially British Columbia forestry products. The reduction in exports means a downward shift in the AE function and a reduction in equilibrium national income. g) The depreciation of the Canadian dollar (which in this chapter we treat as an exogenous event) leads to a rise in Canadian exports and a fall in Canadian imports. The net export function shifts up and becomes flatter. The AE function shifts upward (and flattens out) and the equilibrium level of national income rises. Question 4 a) Desired investment (I) is 50, as shown by the height of the line marked I in the figure. b) Government purchases (G) is 50, shown by the difference between the I and G+I lines in the figure. c) The vertical intercept of the AE function is 150. This is all autonomous expenditure, including consumption, investment, government, and exports. We know that government and investment sum to 100. We are told that autonomous exports are 25. Thus the level of autonomous consumption must be 25.
d) Total autonomous expenditure is 150, the vertical intercept of the AE function. 4 e) A decrease in G by $25 billion will shift the AE function down (in a parallel fashion) by $25 billion. Equilibrium national income will fall by $25 billion times the simple multiplier. f) An increase in the net tax rate shifts the AE function down and also makes it flatter. Equilibrium national income will fall. g) An increase in desired investment by $50 billion will shift the AE function up (in a parallel fashion) by $50 billion. Equilibrium national income will rise by $50 billion times the simple multiplier. h) A reduction in the marginal propensity to import will shift the AE function upwards and also make it steeper. Equilibrium national income will rise. Question 5 a) C = 15 + (.75)Y D b) T = (0.2)Y c) Y D = Y T Y D = Y (.2)Y Y D = (.8)Y d) C = 15 + (0.75)Y D C = 15 + (.75)(.8)Y C = 15 + (.6)Y e) The marginal propensity to spend out of national income, z, is the slope of the AE function. The AE function for Sunrise Island is AE = 15 + (.6)Y + 20 + 30 AE = 65 + (.6)Y In this model, z = 0.6. f) The simple multiplier for Sunrise Island is equal to 1/(1-z) = 1/(1-.6) = 1/.4 = 2.5. Question 6 a) Recall that the marginal propensity to spend out of national income is equal to z = MPC(1 t) m The values of z for the various hypothetical economies are: Economy A: z = 0.75 (1 0.2) 0.15 = 0.45 Multiplier = 1.82 Economy B: z = 0.75 (1 0.2) 0.30 = 0.30 Multiplier = 1.42 Economy C: z = 0.75 (1 0.4) 0.30 = 0.15 Multiplier = 1.17 Economy D: z = 0.90 (1 0.4) 0.30 = 0.24 Multiplier = 1.32 Recall that the simple multiplier is equal to 1/(1 z).
5 b) Comparing Economies A and B, we see that the marginal propensity to spend out of national income is higher in the economy with the lower value of m. A lower marginal propensity to import means that each $1 increase in national income leads to a smaller increase in expenditure on imports, and thus a larger increase in expenditure on the output of domestic producers. Thus, the multiplier will be higher when m is smaller. c) Comparing Economies B and C, we see that the economy with the lower income-tax rate has the higher marginal propensity to spend out of national income. Other things equal (like MPC and m), a lower tax rate means that each $1 increase in national income leads to a larger increase in disposable income and thus a larger increase in consumption expenditures. Thus the multiplier is larger when t is smaller. d) Comparing Economies C and D, we see that the economy with the higher MPC has the higher marginal propensity to spend out of national income. Other things equal (like t and m), a higher MPC means that each $1 increase in national income leads to a larger increase in consumption expenditures. Thus the multiplier is larger when MPC is higher. Question 7 a) The National Asset Formation function is downward sloping only because the net export function is downward sloping. (Recall that desired investment is autonomous with respect to national income.) The net export function is downward sloping because an increase in national income leads to an increase in imports (but no change in domestic exports). Thus, as national income rises, I and X are constant but IM rises. Therefore (I + X IM) falls. The slope of the National Asset Formation function, in absolute value, is the marginal propensity to import. b) National saving is equal to private saving (S) plus government saving (T G). Government purchases are autonomous with respect to national income, but both S and T rise with national income. Thus, as national income rises, (S + T G) also rises. The slope of the National Saving function is the marginal propensity to save plus the net tax rate. c) An increase in the level of government purchases (G) would lead to a downward shift in the National Saving function. The equilibrium level of national income would rise. (In the 45-degree line diagram, the AE function would shift up and equilibrium national income would increase.) d) An increase in desired investment by domestic firms would lead to an upward shift in the National Asset Formation function. Equilibrium national income would rise. (In the 45-degree line diagram, the AE function would shift up and equilibrium national income would increase.) e) A reduction in Canadian income-tax rates would reduce the net tax rate and lead to a flattening of the National Saving function (it would rotate downward). Equilibrium national income would increase. (In the 45-degree line diagram, the AE function would rotate up, becoming steeper, and equilibrium national income would increase.)
6 Question 8 a) The consumption function is: C = a + by D Y D is the difference between national income and total tax revenues: Y D = Y ty = Y(1 t) Putting this expression for Y D into the consumption function, we get the relationship between consumption and national income: b) The AE function is: C = a + by(1 t ) AE = a + by(1 t) + I 0 + G 0 + (X 0 my) We can collect all of the autonomous terms together, and collect all of the terms in Y together, to simplify the AE function as: AE = [a + I 0 + G 0 + X 0 ] + [b(1 t) m]y c) The equilibrium condition is Y = AE. Imposing this condition, and using A to be the sum of all the autonomous terms, we get Y = A + [b(1 t) m]y d) The equilibrium value of national income is the value that solves the above equation. Call this value Y E. The solution is Y E = A/(1 z) Where z = [b(1 t) m] is the marginal propensity to spend out of national income. e) If the level of autonomous spending increases by A, then the equilibrium level of national income rises by 1/(1 z) times A. 1/(1 z) is the simple multiplier.
7 Question 9 a) The AE function is: AE = 50 + (.7)Y D + 75 + 100 + (50 -.15Y) AE = 225 + (.7)(Y -.2Y) + (50 -.15Y) AE = 225 + (.7)(.8)Y + (50 -.15Y) AE = 275 + (.41)Y See the figure below. Total autonomous expenditure is 275, which is the vertical intercept of the AE function. b) The slope of the AE function is 0.41, which is the marginal propensity to spend out of national income, z. c) To compute the equilibrium level of national income, use the equilibrium condition: Y = AE Y = 275 + (.41)Y Y(1 -.41) = 275 Y E = 275/(.59) = 466.1 d) If X rises from 50 to 100, the AE function shifts up by 50. Equilibrium national income rises by 50 times the simple multiplier. Since z = 0.41, the simple multiplier is 1/(1-.41) = 1/.59 = 1.69. So equilibrium national income rises by 50 1.69 = 84.5 to 550.6. e) The simple multiplier is 1/(1-z) = 1/(1-.41) = 1/.59 = 1.69.