© Pilot Publishing Company Ltd Chapter 4 Income Determination I --- Income-expenditure Model

© Pilot Publishing Company Ltd Contents: Assumptions of income-expenditure Model Two-sector economy Three-sector economy Four-sector economy Different kinds of multipliers in different economies Other points to be noticed Paradox of thrift Implications of private saving, public saving & national saving Advanced Material 4.1 : Equality between investment and saving in a two-sector economyAdvanced Material 4.1 : Equality between investment and saving in a two-sector economy

© Pilot Publishing Company Ltd Assumptions of Income- expenditure Model

© Pilot Publishing Company Ltd Assumptions of income-expenditure model (or elementary Keynesian model) The amount of resources and the state of technology remain unchanged, i.e., Y f is a constant. There exists an unemployment of resources. The model is to find out the determinants of equilibrium GNP and the ways to eliminate unemployment. No indirect taxes, subsidies, depreciation or net factor income from abroad, i.e., Y = GDP = GNP. The interest rate and the price level are fixed. So nominal variables = real variables and nom. r = real r.

© Pilot Publishing Company Ltd Two-sector Economy

© Pilot Publishing Company Ltd Two-sector economy Firms Households  Composed of households and firms only  No government sector

© Pilot Publishing Company Ltd Features of households Households provide factor services for income. Factor Services Income

© Pilot Publishing Company Ltd Features of households Consumption Disposable Income Disposable income is either consumed or saved. Saving

© Pilot Publishing Company Ltd C = cY d + C* Disposable Income Consumption function Autonomous Consumption Marginal Propensity to Consume (MPC)  Autonomous consumption (C*) is the consumption at zero disposable income (the minimum amount for subsistence). C* > 0.  Marginal propensity to consume (MPC or c) is the change in consumption resulting from a unit change in disposable income. c < 1.

© Pilot Publishing Company Ltd Graphical Illustration Consumption function C = cY d + C* C YdYd C* c +1 0

© Pilot Publishing Company Ltd S = sY d + S* Disposable Income Saving function Autonomous Saving Marginal Propensity to Save (MPS) Autonomous saving is the saving at zero disposable income. S* = -C*. Why? Marginal propensity to save (MPS or s) is the change in saving resulting from a unit change in disposable income. s = 1 – c. Why?

© Pilot Publishing Company Ltd S = sY d + S* S Y S* = -C* s = (1-c) +1 0 Graphical Illustration Saving function

© Pilot Publishing Company Ltd Features of firms 1. Firms employ factor services to produce goods. 2. Firms also consume final products (fixed investment & inventories) to help production. Factor Services Firms Products

© Pilot Publishing Company Ltd I = i Y + I* National Income Investment function Autonomous Investment Marginal Propensity to Invest (MPI) Autonomous investment is the investment at zero income. I* > 0. Why? Marginal propensity to invest (MPI or i) is the change in investment resulting from a unit change in income. i > 0. Why?

© Pilot Publishing Company Ltd I* I = iY + I* Y r 0 Graphical illustration Investment function +1 i

© Pilot Publishing Company Ltd Equilibrium condition (2-sector economy) Aggregate supply (AS) of final products is GNP or Y.  Without government or taxation  AS = Y = Y d = C + S

© Pilot Publishing Company Ltd Equilibrium condition (2-sector economy) Aggregate demand (AD) for final products is aggregate expenditure (E). AD = E = C + I

© Pilot Publishing Company Ltd AS = AD Y = E C + S = C + I S = I Withdrawal Equilibrium income (or equilibrium GNP) is reached when AS = AD Injection =

© Pilot Publishing Company Ltd  Withdrawal ( 撤出, W)  In a 2 sector economy, saving is the withdrawal.  is the amount of income withdrawn from the circular flow, not being spent on final products.

© Pilot Publishing Company Ltd Injection ( 注入, J)  is the amount of expenditure on final products injected into the circular flow  not financed by income earned from production.  In a 2 sector economy, investment is the injection.

© Pilot Publishing Company Ltd ABC Ltd. Firms Circular flow of a 2-sector economy Investment Saving Injection Withdrawal Consumption Y YdYd E Households When I = S, an equilibrium is achieved.

© Pilot Publishing Company Ltd Z (Y > E) U (Y = E) V (Y < E) Y=E Y E 0 45 o Meaning of a 45 o line Meaning of a 45° line

© Pilot Publishing Company Ltd Aggregate expenditure function Y=E Y E 0 45 o E= C+I C C* I* Add consumption and investment functions vertically I

© Pilot Publishing Company Ltd Adjustment mechanism Income-expenditure approach Injection-withdrawal approach 2 approaches:

© Pilot Publishing Company Ltd Y=E Y E 0 E=C+I YeYe Y1Y1 Y2Y2 Income-expenditure approach  production  production At Y 2, Y < E  Unplanned decrease in inventories At Y 2, Y < E  Unplanned decrease in inventories At Y 1, Y > E  Unplanned increase in inventories At Y 1, Y > E  Unplanned increase in inventories } {

© Pilot Publishing Company Ltd Injection-withdrawal approach At Y 1, J < W  Unintended inventory investment At Y 2, J > W  Unintended inventory disinvestment Y E 0 -C* I* S I YeYe Y1Y1 Y2Y2  production  production { }

© Pilot Publishing Company Ltd Multiplier effect E Y Initial equilibrium ΔEaΔEa E: Y: ΔEiΔEi ΔEi’ΔEi’ Total change in Y = Δ E + c Δ E + cc Δ E + ccc Δ E + … = (1 + c + c 2 + c 3 + …) Δ E = Δ E : Change in Y brought by Δ E a

© Pilot Publishing Company Ltd Multiplier (k)  Mathematical expression: Multiplier effect  is the ratio of the total change in equilibrium income to the initial change in autonomous expenditure (or autonomous withdrawal) that brought it about.

© Pilot Publishing Company Ltd  Equilibrium Y =. (C* + I*) Mathematical derivation of equil. income & multiplier In a 2-sector economy In equilibrium, Y = E = C + I = cY d + C* + I* = cY + C* + I*  E = C + I, where C = cY d + C* and I = I*  Y – cY = C* + I*  (1 – c)Y= C* + I*

© Pilot Publishing Company Ltd When C* or I* changes by Δ E a Mathematical derivation of equil. income & multiplier  Δ Y =. Δ E a 

© Pilot Publishing Company Ltd Q4.1: Calculate the value of multiplier and explain its meaning in each of the following cases. (a) MPC = 1 (b) MPS = 1

© Pilot Publishing Company Ltd Q4.2: If the autonomous expenditure decreases by Δ E, what will be the change in equilibrium income?

© Pilot Publishing Company Ltd Three-sector Economy

© Pilot Publishing Company Ltd HouseholdsFirms Government Three-sector economy

© Pilot Publishing Company Ltd Government  Government’s expenditure is mainly financed by taxation

© Pilot Publishing Company Ltd Government expenditure function (G) G = G*, where G > 0  is fixed by the budget at the beginning of a fiscal year.  G is a constant (G*) independent of any variables.

© Pilot Publishing Company Ltd Tax function Only direct taxes are concerned. There exist three kinds of direct tax systems. No indirect taxes is assumed. Tax SystemFormulaExample Lump-sum tax systemT = T’Poll tax Proportional tax systemT = tY + T*Profits tax Progressive tax systemT = t*Y + T^Salaries tax

© Pilot Publishing Company Ltd  Aggregate supply (AS) of final products is GNP or Y. Equilibrium condition (3-sector economy)  Y d = Y – T = C + S  Y = Y d + T = C + S + T

© Pilot Publishing Company Ltd  Aggregate demand (AD) for final products is E. Equilibrium condition (3-sector economy)  E = C + I + G

© Pilot Publishing Company Ltd  AS = AD  Y = E  C + S + T = C + I + G  S + T = I + G Withdrawal Equilibrium income (or equilibrium GNP) is reached when AS = AD Injection =

© Pilot Publishing Company Ltd ABC Ltd. Households Firms Circular flow of a 3-sector economy Investment Saving Injection Withdrawal Consumption Y E Tax Government expenditure When I + G = S + T, equilibrium is achieved.

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier In a 3-sector economy with a lump-sum tax system In equilibrium, Y = E = C + I + G = cY d + C* + I* + G*  E=C+I+G, where C = cY d + C*; Y d = Y – T’ and I = I* and G = G* Y = c(Y-T’) + C* + I* + G* = cY- cT’+ C* + I* + G*  Equilibrium Y =. (C*+I*+G*- cT’)  (1–c)Y = C* + I*+ G*- cT’

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier When C* or I* or G* changes by Δ E a  Δ Y =. Δ E a  Multiplier =

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier Under a lump-sum tax system Equil. income: Multiplier:  

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier In a 3-sector economy with a proportional tax system In equilibrium, Y = E = C + I + G = cY d + C* + I* + G*  E=C+I+G, where C = cY d + C*; Y d = Y - tY - T*; I = I* and G = G*  (1 - c + ct)Y = C* + I* + G* - cT* Y = c(Y - tY - T*) + C* + I* + G* = cY - ctY - cT* + C* + I* + G*  Equilibrium Y = (C*+I*+G*-cT*)

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier When C* or I* or G* changes by Δ E a  Δ Y = Δ E a  Multiplier =

© Pilot Publishing Company Ltd Under a proportional tax system Equil. income: Multiplier:  The multiplier of proportional tax system is smaller than the multiplier of lump-sum tax system.   As t > 0, (1-c) < (1-c+ct)  

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier With the addition of a proportional transfer payment, where Y d = Y - tY - T* + qY + Q* In equilibrium, Y = E = C + I + G = cY d + C* + I* + G* Equilibrium Y = (C*+I*+G*-cT*-cQ*)  Y = c(Y-tY-T*+qY+Q*) + C* + I* + G*  (1–c+ct-cq) Y = C* + I* + G* -cT* + cQ*

© Pilot Publishing Company Ltd  Δ Y = Δ E a Mathematical derivation of equil. income & multiplier  Multiplier = When C* or I* or G* changes by Δ E a

© Pilot Publishing Company Ltd Under a proportional tax & transfer payment system Equil. income: Multiplier: As q<0, (1-c+ct)<(1-c+ct-cq)   The multiplier with a proportional transfer payment is smaller than the multiplier without it. >

© Pilot Publishing Company Ltd Four-sector Economy

© Pilot Publishing Company Ltd Four-sector economy Foreign Sector HouseholdsFirms Government

© Pilot Publishing Company Ltd Export function: X=X*, where X*>0  Export is determined by foreign economies, not the domestic economy  It is autonomous & is independent of Y

© Pilot Publishing Company Ltd Import function: M= mY +M* Marginal Propensity to Import (MPM) Autonomous Import MPM is the change in the value of imports resulting from a unit change in national income. MPM > 0. M* is the value of imports at zero income. M* > 0.  All economic agents consume imports  Import is positively related to Y

© Pilot Publishing Company Ltd  Aggregate supply (AS) of final products is GNP or Y. Equilibrium condition (4-sector economy)  Y = Y d + T = C + S + T

© Pilot Publishing Company Ltd Equilibrium condition (4-sector economy)  Aggregate demand (AD) for final products is aggregate expenditure (E). E = (C-M C ) + (I-M I ) + (G-M G ) + (X-M X ) = C + I + G + X - (M C +M I +M G +M X ) = C + I + G + X - M

© Pilot Publishing Company Ltd AS = AD Y = E C + S + T = C + I + G + X – M S + T = I + G + X – M S + T + M = I + G + X Equilibrium income (or equilibrium GNP) is reached when Injection = Withdrawal

© Pilot Publishing Company Ltd ABC Ltd. Households Firms Circular flow of a 4-sector economy I - M I Saving Injection Withdrawal C - M C Y E Tax G - M G X - M X When I + G + X – M = S + T, equilibrium is achieved.

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier In a four-sector economy, E = C + I + G + X – M where C = cY d + C*; Y d = Y - tY - T* + qY + Q*; I = I*; G = G*; X = X*; M = mY + M*

© Pilot Publishing Company Ltd  Y = c(Y-tY-T*+qY+Q*)+C*+I*+G*+X*-mY-M*  (1-c+ct-cq+m)Y = C*+I*+G*+X*-M*-cT*+cQ*  Equil.Y = (C*+I*+G*+X*-M*-cT*-cQ*) Mathematical derivation of equil. income & multiplier In equilibrium, Y = E = C+I+G+X-M = cY d +C*+I*+G*+X*-mY-M*

© Pilot Publishing Company Ltd Mathematical derivation of equil. income & multiplier When C* or I* or G* or X* changes by Δ E a Δ Y = Δ E a

© Pilot Publishing Company Ltd Q4.5 Derive the equilibrium income and the multiplier in a four-sector economy if investment is an induced expenditure (I = iY + I*).

© Pilot Publishing Company Ltd Different Kinds of Multipliers in Different Economies

© Pilot Publishing Company Ltd Expenditure multiplier (k) Two-sector economy Three-sector economy Four-sector economy

© Pilot Publishing Company Ltd Comparison of the size of expenditure multipliers (k)  As c>0, i>0, t>0, -q>0, and m>0, k in a 2-sector economy > k in a 3-sector economy > k in a 4-sector economy

© Pilot Publishing Company Ltd Explanation: E Y Initial equilibrium ΔEaΔEa E: Y: ΔEiΔEi Δ E i (2-sector economy) =c Δ E a + i Δ E a : Change in Y brought by Δ E a Δ E i (3-sector economy) =c(1-t+q) Δ E a + i Δ E a Δ E i (4-sector economy) =c(1-t+q) Δ E a + i Δ E a – m Δ E a

© Pilot Publishing Company Ltd Tax multiplier  Tax is a withdrawal  Its multiplier is negative.  When T*:  by $1  Y d :  by $1  C:  by -$c  ∆Y = -c x expenditure multiplier.

© Pilot Publishing Company Ltd Q4.6: Derive the transfer payment multiplier and explain why it is smaller than the expenditure multiplier.

© Pilot Publishing Company Ltd Import multiplier  Import is a withdrawal  Its multiplier is negative.  When M*:  by $1  AE:  by $1  Δ Y = -1 x expenditure multiplier.

© Pilot Publishing Company Ltd Multipliers at full employment  Δ E a > 0  Multiplier = 0  At full employment, as all resources have been used efficiently, real income can no longer be raised (when Δ E a >0, Δ Y=0) but it can be lowered (when Δ E a <0, Δ Y<0).  Δ E a < 0  Multiplier = Unchanged

© Pilot Publishing Company Ltd Other Points to be Noticed

© Pilot Publishing Company Ltd More about aggregate expenditure function E = C + I + G + X - M = c(Y-tY-T*+qY+Q*)+C* +iY+I* +G* +X* - mY- M* E = (c + i - ct + cq-m)  Y + (C*+I*+G*+X*-M*-cT*+ cQ*) Slope of E-function E-intercept

© Pilot Publishing Company Ltd Q4.7: Derive the expenditure multiplier from the income- expenditure diagram. Q4.8: Expenditure multiplier is Δ Y/ Δ E and slope of E-function is Δ E/ Δ Y. Is the expenditure multiplier equal to the inverse of the slope of E-function?

© Pilot Publishing Company Ltd An induced  in E An autonomous  in E E E’ E Y An autonomous change versus an induced change

© Pilot Publishing Company Ltd E Y Y=E E’ (slope > 1) Δ E’> Δ Y ΔYΔY E (Slope < 1) Δ E< Δ Y ΔYΔY 45 o Can’t find the equilibrium Slope of E-function cannot be greater than one The equilibrium

© Pilot Publishing Company Ltd Deflationary AD gap is the amount of expenditure by which the present expenditure falls short of the expenditure achieving full employment. Deflationary income gap is the amount of income by which the equilibrium income falls short of the full employment income. Deflationary Gap

© Pilot Publishing Company Ltd Deflationary income gap 45 o E1E1 } { Deflationary AD gap EfEf E Y Y=E YfYf Y1Y1 Graphical illustration

© Pilot Publishing Company Ltd Inflationary AD gap is the amount of expenditure by which the present expenditure exceeds the expenditure achieving full employment. Inflationary income gap is the amount of income by which the equilibrium income exceeds the full employment income. Inflationary Gap

© Pilot Publishing Company Ltd o E2E2 } Inflationary income gap { Inflationary AD gap EfEf E Y Y=E YfYf Y2Y2 Graphical illustration

© Pilot Publishing Company Ltd Deflationary income gap Inflationary income gap = Deflationary AD gap x multiplier = Inflationary AD gap x multiplier

© Pilot Publishing Company Ltd Paradox of thrift  The puzzle why national income falls (the society gets poorer) when people as a whole save more.

© Pilot Publishing Company Ltd SP’SP’ Ye’Ye’ } Unintended inventory investment 0 SPSP IPIP S,I Y YeYe Note: If investment is an autonomous expenditure, the results are  Y , C  & S unchanged (= I) Saving is detrimental when  Saving  but the unspent income does not re-enter the circular flow  Saving  but the unspent income does not re-enter the circular flow  Firms cut production  Income (Y) 

© Pilot Publishing Company Ltd SP’SP’ 0 SPSP IPIP S,I Y Ye’Ye’ YeYe } Unintended inventory investment Saving is detrimental Note: If investment is an induced expenditure, the results are  Y , C  & S  (= I  )

© Pilot Publishing Company Ltd SP’SP’ Ye=Ye’Ye=Ye’ IP’IP’ Saving is beneficial when  Saving  and the unspent income can re-enter the circular flow as investment Then Y is unchanged. In addition, as I , productivity . 0 SPSP IPIP S,I Y Note: If investment is an autonomous expenditure, the results are  Y unchanged, C , S  & I 

© Pilot Publishing Company Ltd SP’SP’ 0 SPSP IPIP S,I Y Ye=Ye’Ye=Ye’ I P ’ Saving is beneficial Note: If investment is an induced expenditure, the results are  Y unchanged, C , S  & I 

© Pilot Publishing Company Ltd Implications of private saving, public saving & national saving Private saving (S P or S, 私人儲蓄 ) is the saving of households, i.e., S P = Y d – C = Y – T – C. Public saving (S G, 公共儲蓄 ) is the saving of the government, also called fiscal surplus. S G = T – G. National saving (S N, 國民儲蓄 ) is the saving of the economy as a whole. S N = S P + S G. Definition:

© Pilot Publishing Company Ltd Implications of private saving In equilibrium, total withdrawal = total injection. S P + T + M = I + G + X  S P = I + (G – T) + (X – M) ………………. (1) In equilibrium, AS = AD. Resources not consumed by households (private saving) must be consumed by other economic agents – by firms as investment (I), and/or by the government creating fiscal deficit (G - T), and/or by the foreign sector as net exports (X - M).

© Pilot Publishing Company Ltd Implications of public saving In equilibrium, total withdrawal = total injection. S P + T + M = I + G + X  Fiscal surplus = S G = T – G = (I – S P ) + (X – M) …….. (2)  Fiscal deficit = -S G = G – T = (S P – I) + (M – X) …….. (3) Equation (2): In equilibrium, AS = AD. Resources not consumed by the government (public saving) must be consumed by other economic agents – by private sector (I – S P ), and/or by the foreign sector (X - M). Equation (3): If there exists fiscal deficit, the resources have to be supplied by the private sector and/or the foreign sector, through the issuance of internal debt (S P - I), and/or external debt (which enables the economy to have net imports, M – X).

© Pilot Publishing Company Ltd Implications of natioinal saving By definition, S N = S P + S G = [I + (G-T) + (X-M)] + (T-G)  S N = I + (X – M) …….. (4) In equilibrium, CA + KA = 0  CA = X – M = -KA. From equation (4), S N – I = X – M = CA = -KA …….. (5) Equation (4): In equilibrium, AS = AD. Resources not consumed by households and the government (national saving) must be consumed by other economic agents – by firms as investment (I), and/or by the foreign sector as net exports (X - M). Equation (5): In equilibrium, AS = AD. Resources not consumed by our economy (S N - I) must be consumed by foreign economies as net exports (X - M) and illustrated by our current account surplus. To have external balance, the capital account must have deficit (CA = -KA).

© Pilot Publishing Company Ltd Advanced Material 4.1 Equality between investment and saving in a two-sector economy  Meaning of equality between ex-ante investment & ex-ante saving  In a two-sector economy, equality between ex-ante (or planned or desired) investment and ex-ante saving is the equilibrium condition.

© Pilot Publishing Company Ltd With unintended inventory investment  Y  0 SPSP IPIP S,I Y YeYe Y1Y1 IPIP SpSp (Sp>Ip) Derivation

© Pilot Publishing Company Ltd SpSp 0 SPSP IPIP S,I Y YeYe Y2Y2 IPIP (Sp<Ip) With unintended inventory disinvestment  Y 

© Pilot Publishing Company Ltd SPSP IPIP S,I Y YeYe (Sp=Ip) No unintended change in inventories  Y e Equilibrium condition of a 2-sector economy

© Pilot Publishing Company Ltd  Meaning of equality between ex-post investment and ex-post saving  In a two-sector economy, equality between ex-post (or actual or realized or observed) investment and ex- post saving is an identity.  As they must always be equal, the equality is a tautology without any economic meaning or implication. Equality between investment and saving in a two-sector economy

© Pilot Publishing Company Ltd SPSP S,I Y YeYe Y1Y1 IPIP [S p (= S a ) >I p ] but I a = I p + I u (>0) = S a [S p (= S a ) >I p ] but I a = I p + I u (>0) = S a Derivation Y2Y2 IPIP I u (>0) IPIP I u (<0) IPIP [S p (= S a ) < I p ] but I a = I p + I u (<0) = S a [S p (= S a ) < I p ] but I a = I p + I u (<0) = S a [S p (= S a ) = I p ] I a = I p + I u (=0) = S a [S p (= S a ) = I p ] I a = I p + I u (=0) = S a

© Pilot Publishing Company Ltd Implications 1. In a 4-sector economy, equality between planned total withdrawal and planned total injection is the equilibrium condition. 2. In a 4-sector economy, equality between actual total withdrawal and actual total injection is an identity and is meaningless in economics.

© Pilot Publishing Company Ltd  Planned investment (I p ) is the planned change in fixed capital & inventories. Different terms related to investment  Unplanned investment (I u ) is the unplanned change in inventories, which is the amount not purchased by any economic agents.  Realized investment is the amount of actual investment (= I p + I u ).  Unrealized investment is the amount of actual investment falling short of the amount of planned investment (= I u < 0).

© Pilot Publishing Company Ltd Correcting Misconceptions: 1. The multiplier of an autonomous decrease in expenditure is negative. 2. The import function is represented by the linear equation: M = mY d + M*. 3. Multipliers must be positive. 4. An increase in aggregate expenditure would shift the E-curve upward in a parallel manner.

© Pilot Publishing Company Ltd Equilibrium income is the same as full employment income. 6. Equality between total injection & total withdrawal is the equilibrium condition of goods market. 7. An increase in saving (a withdrawal) is detrimental to an economy. Correcting Misconceptions: