HT7 Macroecons (Debt and Fiscal Policy)
Week's outline
Lecture I: Normative theories of public debt
o Ricardian equivalence
o Sustainability and dynamics of government debt
o Intertemporal tax smoothing
o Keynesian stabilisation policy & Maastricht norms
Lecture II: Positive theories of public debt
o Two-period recap of tax smoothing with endogenous public spending & precautionary
buffers
o present in-office bias
o weak Minister of Finance: common-pool problem and debt bias
o partisan politics and strategic debt management
o signal political ability
o other debt distortions
o delayed stabilisation: war of attrition/game of chicken
Lecture III: Cost of debt, default and other issues
o Cost of debt and model of default risk and default
o Tackling debt bias and Fiscal Councils
o The balance sheet approach to government liabilities & assets and economic crises
o Background: UK government debt 1694-2016
Lecture 19: Normative theories of public debt
Outline
Flow and intertemporal government budget constraints: solvency and no-Ponzi games
Review of Ricardian equivalence result
Reasons why Ricardian equivalence might fail in practice
Correcting for growth and inflation
Sustainability of government debt
Maastricht convergence norms
Review of intertemporal tax smoothing: minimising present discounted value (PV) of tax
distortions
Keynesian stabilisation policy
Government budget constraint
Consumer and government flow budget constraints
Flow budget constraints: describe the flow/ rate of change of a stock in a time period
For consumers/ households the flow budget constraints are
o d(B+E)/dt = r(B+E) + Y – T – C
o with initial bond and equity holdings B(0) and E(0), where bond and equity represent
asset holdings of consumers/ households
o RHS = private sector savings = return on assets + income – tax – consumption
, o The change in asset level (LHS) = private sector savings (RHS); Since savings are used to
buy assets (eg. Lending to hold cash to the next period is equivalent to a bond)
For the government the flow budget constraint is
o dB/dt = rB + G – T
o with initial government debt B(0) given
o G is the primary public/ government spending (excludes interest payments)
o G – T: primary deficit (excludes interest payments)
o RHS = government deficit = interest payments + primary public spending – taxes
o The increase in government debt (LHS) = current period government deficit
o Here, we ignore seigniorage revenues
Integrating these flow budget constraints from the distant future back to the present together
with the no-Ponzi-games conditions gives the present-value budget (PVB) constraints
o Integrating these 2 flow budget constraint equations produces the consumer and
government PVB constraints (derivation not needed)
Present value budget (PVB) constraints
Consumers present-value budget (PVB) constraint is
o 𝑃𝑉𝑡 [𝐶] ≤ financial assets + human wealth = 𝐴𝑡 + 𝑃𝑉𝑡[𝑌−𝑇]
o Where 𝐴𝑡 = 𝐸𝑡 + 𝐵𝑡 is the financial assets value
o Where 𝑃𝑉𝑡[𝑌−𝑇] is the human wealth (PV of disposable income)
o PV of consumption (what consumers intend to spend) is constrained by and must be less
than their wealth (financial assets + human wealth)
Here the present value of disposable income is defined as
o
o Disposable income = Income – Taxes = Y – T
o PV obtained by continuous integration of future flows (hence ‘e’) at discount rate r, for
cashflows [Y(s) - T(s)] in each time period s
Given initial period t, and s goes from current period t to infinity
(s – t) gives duration to discount by for cashflows in each time period s
Government’s PVB constraint states that
o 𝑃𝑉𝑡 [𝑻𝒕] ≥ net public liabilities = 𝑩𝒕 +𝑷𝑽𝒕[𝑮]
o Net public liabilities = existing government debt + PV of commitments and plans of
future government spending
o For the government to be solvent, PV of all future taxes (government revenue) must be
larger than net public liabilities
Alternative way to think about Government’s PVB constraint
o Rearrangement: 𝑃𝑉𝑡 [𝑻𝒕 – 𝑮] ≥ 𝑩𝒕.
o PV of government surpluses ≥ existing government debt
o Intuitively: must payoff current debt with future surpluses
Consumers’ PVB constraint thus becomes
o 𝑃𝑉𝑡[𝐶] ≤ 𝐸𝑡 + 𝑃𝑉𝑡[𝑌−𝐺]
o Substitute government PVB constraint (assume it binds) into consumer PVB constraint
𝑃𝑉𝑡 [𝐶] ≤ 𝐴𝑡 + 𝑃𝑉𝑡[𝑌−𝑇]
, 𝑃𝑉𝑡 [𝐶] ≤ 𝐴𝑡 + 𝑃𝑉𝑡[𝑌] − 𝑃𝑉𝑡[𝑇]
𝑃𝑉𝑡 [𝐶] ≤ 𝐴𝑡 + 𝑃𝑉𝑡[𝑌] − 𝑩𝒕 − 𝑷𝑽𝒕[𝑮] (substitution occurs here)
𝑃𝑉𝑡 [𝐶] ≤ (𝐴𝑡 − 𝑩𝒕) + 𝑃𝑉𝑡[𝑌−𝑮]
o There is no tax variable, suggesting that the path of taxes has no effect on consumption.
Ricardian equivalence
Consumption growth = r - 𝜌
In week 6, you derived the discrete-time Euler equation as
o 𝒖′(𝑪𝒕) = 𝜷(𝟏+𝒓) 𝑬[𝒖′(𝑪𝒕+𝟏)]
With logarithmic utility and abstracting from uncertainty, this becomes:
o
Let 𝜷 = 1/(1+𝜌), then:
o
o If 𝜌 is small or time interval is short (when time interval is short, you naturally discount
less, so smaller 𝜌)
o Smaller 𝜌 (larger 𝜷), discount future utilities less
Thus, the growth rate of consumption (LHS) = interest rate − the rate of time preference (RHS)
Hence, postpone consumption and save more if r high and 𝜌 low (high growth rate of
consumption)
o Intuitively: prefer to save when r is high (earnings from saving is high) and 𝜌 is low (you
are patient)
Consumption as a function of wealth
Assume logarithmic utility, so that utility of consumers is
o
o where 𝜌 > 0 denotes pure rate of time preference
o Own thoughts: PV of utility = continuous discount of utilities from all future
consumption at each point in time s, from current period t to infinity, given initial
consumption level Ct.
Notice use of large U on LHS (not single period utility)
o Note: for total utility, discount by 𝜌; for present value, discount by r
The continuous-time Euler equation is
o
o Equivalent to the discrete version derived above: the growth rate of consumption (LHS)
=r−𝜌
Hence,
, o
o First equation: PV of consumption = continuous integration of all future consumption C s
at each point in time s.
o Second equation: obtain Cs in each period by considering initial C t at period t and
accounting for growth rate of consumption (r − 𝜌) using continuous compounding
o Third equation: this is similar to a growing perpetuity, where PV = initial cashflow/
(discount rate – growth rate) = C t/[r − (r − 𝜌)] = Ct/𝜌
Ricardian equivalence
Optimal aggregate consumption is a fraction of total wealth: financial wealth (𝐴𝑡 ) and human
wealth (𝑃𝑉𝑡[𝑌−𝑇])
o
o 𝑪𝒕 = 𝝆𝑷𝑽𝒕[𝑪𝒕]: from 𝑷𝑽𝒕[𝑪𝒕] = Ct/𝜌 (above)
o The rest: from binding PBV constraints
Hence, timing of taxes does not affect the household budget constraint (middle expression) and
does not affect consumption – it does not matter whether public spending is financed by current
taxes or by debt (i.e., future taxes)
If rational households anticipate that a tax cut must be followed by a tax hike, private
consumption is unaffected
o PV of future tax hike = PV of current tax cut; so 𝑃𝑉𝑡[𝑌−𝑇] is unaffected
o Ricardian equivalence: intertemporal pattern of taxes does not affect consumption
o Intuitively, this makes sense since the household’s wealth level and thus consumption is
unaffected by timing of taxes
But if they anticipate that a tax cut must be followed by future cut in public spending, they
increase consumption
o To compensate for previous tax cuts, right wing governments tend to cut future public
spending, while left wing governments tend to hike taxes
o Own thoughts:
We have thus far assumed a fixed path of government spending, meaning
constant PV(G), which under the government PVB constraint, means constant
PV(T). So intertemporal shifts in tax occurs while PV(T) is constant, so
consumption is unaffected, as seen in the consumer PVB constraint
However, another way to satisfy the government PVB constraint is to increase/
decrease PV[T] and PV[G] on both sides. For example, a current tax cut
(decreased PV[T]) paid for by future cut in public spending (decreased PV[G])
increases PV[C] as seen in the consumer PVB constraint. This leads to higher
consumption level 𝑪𝒕 = 𝝆𝑷𝑽𝒕[𝑪𝒕]
For reference:
Government PVB: 𝑃𝑉𝑡 [𝑻𝒕] = 𝑩𝒕 +𝑷𝑽𝒕[𝑮]
Consumer PVB: 𝑃𝑉𝑡 [𝐶] = 𝐴𝑡 + 𝑃𝑉𝑡[𝑌−𝑇]
Week's outline
Lecture I: Normative theories of public debt
o Ricardian equivalence
o Sustainability and dynamics of government debt
o Intertemporal tax smoothing
o Keynesian stabilisation policy & Maastricht norms
Lecture II: Positive theories of public debt
o Two-period recap of tax smoothing with endogenous public spending & precautionary
buffers
o present in-office bias
o weak Minister of Finance: common-pool problem and debt bias
o partisan politics and strategic debt management
o signal political ability
o other debt distortions
o delayed stabilisation: war of attrition/game of chicken
Lecture III: Cost of debt, default and other issues
o Cost of debt and model of default risk and default
o Tackling debt bias and Fiscal Councils
o The balance sheet approach to government liabilities & assets and economic crises
o Background: UK government debt 1694-2016
Lecture 19: Normative theories of public debt
Outline
Flow and intertemporal government budget constraints: solvency and no-Ponzi games
Review of Ricardian equivalence result
Reasons why Ricardian equivalence might fail in practice
Correcting for growth and inflation
Sustainability of government debt
Maastricht convergence norms
Review of intertemporal tax smoothing: minimising present discounted value (PV) of tax
distortions
Keynesian stabilisation policy
Government budget constraint
Consumer and government flow budget constraints
Flow budget constraints: describe the flow/ rate of change of a stock in a time period
For consumers/ households the flow budget constraints are
o d(B+E)/dt = r(B+E) + Y – T – C
o with initial bond and equity holdings B(0) and E(0), where bond and equity represent
asset holdings of consumers/ households
o RHS = private sector savings = return on assets + income – tax – consumption
, o The change in asset level (LHS) = private sector savings (RHS); Since savings are used to
buy assets (eg. Lending to hold cash to the next period is equivalent to a bond)
For the government the flow budget constraint is
o dB/dt = rB + G – T
o with initial government debt B(0) given
o G is the primary public/ government spending (excludes interest payments)
o G – T: primary deficit (excludes interest payments)
o RHS = government deficit = interest payments + primary public spending – taxes
o The increase in government debt (LHS) = current period government deficit
o Here, we ignore seigniorage revenues
Integrating these flow budget constraints from the distant future back to the present together
with the no-Ponzi-games conditions gives the present-value budget (PVB) constraints
o Integrating these 2 flow budget constraint equations produces the consumer and
government PVB constraints (derivation not needed)
Present value budget (PVB) constraints
Consumers present-value budget (PVB) constraint is
o 𝑃𝑉𝑡 [𝐶] ≤ financial assets + human wealth = 𝐴𝑡 + 𝑃𝑉𝑡[𝑌−𝑇]
o Where 𝐴𝑡 = 𝐸𝑡 + 𝐵𝑡 is the financial assets value
o Where 𝑃𝑉𝑡[𝑌−𝑇] is the human wealth (PV of disposable income)
o PV of consumption (what consumers intend to spend) is constrained by and must be less
than their wealth (financial assets + human wealth)
Here the present value of disposable income is defined as
o
o Disposable income = Income – Taxes = Y – T
o PV obtained by continuous integration of future flows (hence ‘e’) at discount rate r, for
cashflows [Y(s) - T(s)] in each time period s
Given initial period t, and s goes from current period t to infinity
(s – t) gives duration to discount by for cashflows in each time period s
Government’s PVB constraint states that
o 𝑃𝑉𝑡 [𝑻𝒕] ≥ net public liabilities = 𝑩𝒕 +𝑷𝑽𝒕[𝑮]
o Net public liabilities = existing government debt + PV of commitments and plans of
future government spending
o For the government to be solvent, PV of all future taxes (government revenue) must be
larger than net public liabilities
Alternative way to think about Government’s PVB constraint
o Rearrangement: 𝑃𝑉𝑡 [𝑻𝒕 – 𝑮] ≥ 𝑩𝒕.
o PV of government surpluses ≥ existing government debt
o Intuitively: must payoff current debt with future surpluses
Consumers’ PVB constraint thus becomes
o 𝑃𝑉𝑡[𝐶] ≤ 𝐸𝑡 + 𝑃𝑉𝑡[𝑌−𝐺]
o Substitute government PVB constraint (assume it binds) into consumer PVB constraint
𝑃𝑉𝑡 [𝐶] ≤ 𝐴𝑡 + 𝑃𝑉𝑡[𝑌−𝑇]
, 𝑃𝑉𝑡 [𝐶] ≤ 𝐴𝑡 + 𝑃𝑉𝑡[𝑌] − 𝑃𝑉𝑡[𝑇]
𝑃𝑉𝑡 [𝐶] ≤ 𝐴𝑡 + 𝑃𝑉𝑡[𝑌] − 𝑩𝒕 − 𝑷𝑽𝒕[𝑮] (substitution occurs here)
𝑃𝑉𝑡 [𝐶] ≤ (𝐴𝑡 − 𝑩𝒕) + 𝑃𝑉𝑡[𝑌−𝑮]
o There is no tax variable, suggesting that the path of taxes has no effect on consumption.
Ricardian equivalence
Consumption growth = r - 𝜌
In week 6, you derived the discrete-time Euler equation as
o 𝒖′(𝑪𝒕) = 𝜷(𝟏+𝒓) 𝑬[𝒖′(𝑪𝒕+𝟏)]
With logarithmic utility and abstracting from uncertainty, this becomes:
o
Let 𝜷 = 1/(1+𝜌), then:
o
o If 𝜌 is small or time interval is short (when time interval is short, you naturally discount
less, so smaller 𝜌)
o Smaller 𝜌 (larger 𝜷), discount future utilities less
Thus, the growth rate of consumption (LHS) = interest rate − the rate of time preference (RHS)
Hence, postpone consumption and save more if r high and 𝜌 low (high growth rate of
consumption)
o Intuitively: prefer to save when r is high (earnings from saving is high) and 𝜌 is low (you
are patient)
Consumption as a function of wealth
Assume logarithmic utility, so that utility of consumers is
o
o where 𝜌 > 0 denotes pure rate of time preference
o Own thoughts: PV of utility = continuous discount of utilities from all future
consumption at each point in time s, from current period t to infinity, given initial
consumption level Ct.
Notice use of large U on LHS (not single period utility)
o Note: for total utility, discount by 𝜌; for present value, discount by r
The continuous-time Euler equation is
o
o Equivalent to the discrete version derived above: the growth rate of consumption (LHS)
=r−𝜌
Hence,
, o
o First equation: PV of consumption = continuous integration of all future consumption C s
at each point in time s.
o Second equation: obtain Cs in each period by considering initial C t at period t and
accounting for growth rate of consumption (r − 𝜌) using continuous compounding
o Third equation: this is similar to a growing perpetuity, where PV = initial cashflow/
(discount rate – growth rate) = C t/[r − (r − 𝜌)] = Ct/𝜌
Ricardian equivalence
Optimal aggregate consumption is a fraction of total wealth: financial wealth (𝐴𝑡 ) and human
wealth (𝑃𝑉𝑡[𝑌−𝑇])
o
o 𝑪𝒕 = 𝝆𝑷𝑽𝒕[𝑪𝒕]: from 𝑷𝑽𝒕[𝑪𝒕] = Ct/𝜌 (above)
o The rest: from binding PBV constraints
Hence, timing of taxes does not affect the household budget constraint (middle expression) and
does not affect consumption – it does not matter whether public spending is financed by current
taxes or by debt (i.e., future taxes)
If rational households anticipate that a tax cut must be followed by a tax hike, private
consumption is unaffected
o PV of future tax hike = PV of current tax cut; so 𝑃𝑉𝑡[𝑌−𝑇] is unaffected
o Ricardian equivalence: intertemporal pattern of taxes does not affect consumption
o Intuitively, this makes sense since the household’s wealth level and thus consumption is
unaffected by timing of taxes
But if they anticipate that a tax cut must be followed by future cut in public spending, they
increase consumption
o To compensate for previous tax cuts, right wing governments tend to cut future public
spending, while left wing governments tend to hike taxes
o Own thoughts:
We have thus far assumed a fixed path of government spending, meaning
constant PV(G), which under the government PVB constraint, means constant
PV(T). So intertemporal shifts in tax occurs while PV(T) is constant, so
consumption is unaffected, as seen in the consumer PVB constraint
However, another way to satisfy the government PVB constraint is to increase/
decrease PV[T] and PV[G] on both sides. For example, a current tax cut
(decreased PV[T]) paid for by future cut in public spending (decreased PV[G])
increases PV[C] as seen in the consumer PVB constraint. This leads to higher
consumption level 𝑪𝒕 = 𝝆𝑷𝑽𝒕[𝑪𝒕]
For reference:
Government PVB: 𝑃𝑉𝑡 [𝑻𝒕] = 𝑩𝒕 +𝑷𝑽𝒕[𝑮]
Consumer PVB: 𝑃𝑉𝑡 [𝐶] = 𝐴𝑡 + 𝑃𝑉𝑡[𝑌−𝑇]
