APPENIX Should the Private Sector Provide Public Capital? Santanu Chatterjee epartment of Economics Terry College of Business University of eorgia
Appendix A The appendix describes the optimization problem and the equilibrium dynamics for the private agent under the three regimes of public capital provision A Private Provision with a overnment Subsidy The Hamiltonian function for the private agent in this regime can be expressed as γ βt βt N I H = ( C e + λ e N r N C ( s ( τ Ψ Ω + γ T βt βt + qe [ I δ ] + qe [ δ ] (A / where = A a( + b( + η ( ecalling the stationary variables z = / c = C / n = N / and y = / the above optimization exercise leads to the following equilibrium dynamics (where we have set = in equilibrium: ( s q ( s h z q = δ δ z h { } ( q ( ( ( s h (Aa q s z n N q = r( n z + c y δ n N n + + h (Ab h ( c C r n z β q = δ c C ( γ h + ( τ ba ( y ( q q + + δ = r( n q q hq z (Ac (Ad d + ( τ ( ( / ( b η A y z q s ( + + q q q s h q ( δ = r n z (Ae
The fraction of output devoted to public investment g ( = is given by g ( t ( q s z = ( s h y (Af ifferentiating (Af with respect to time we can describe the dynamic evolution of g ( t : q z g ( t = + g ( t { y ( y z } (Ag ( y / z ( s h z The steady-state equilibrium is attained when z = n = c = q = q = 0 and is characterized by sustained balanced growth: N C N C φ = = = = Further from (Ag we see that these conditions imply that g = 0 Imposing these conditions in (Aa-(Ag along with the government s balanced budget condition (7 leads to the steady-state conditions (3a-(3f in the text The linearized dynamics corresponding to the steady-state equilibrium (3a-(3f can be described as ( Χ Χ (A3 Χ = where Χ = ( z n c q q Χ = ( z n c q q the 5x5 coefficient matrix of the linearized system functional form for the upward-sloping supply curve of debt: i i ( The linearized matrix can be described as follows: and a 5 ij i j = represents 5x5 For the analytics we assume the following i i αn ( + z r n z = r + e α 0 i = (A4 The determinant of the coefficient matrix of (A3 can be shown to be positive under the condition that ( rn z > φ Since (Aa-Ae is a fifth-order system a positive determinant implies that there could be 3 or 5 positive (unstable roots In order to yield a saddlepoint-stable solution we require that there be three unstable roots to match the three jump variables ( c q q Our numerical simulations yield saddlepoint-stable behavior for all plausible ranges of parameters with three positive (unstable and two negative (stable roots 3
( a = 0; a = 0; a = 0; a = z h ; a = z s a 3 4 5 ( q ( s + α( n e y = + αn ( z ( + z ( s h z a h = + = αn αne r φ a 3 a 4 q n = zq a5 = h ( s h αn αn αcne α( + τ y e c 3 3 33 34 h a 35 a = a = a = 0 a = c = 0 ( γ ( γ αn αq n e y a4 = ba ( + ( τ ( y y a z = = 0 αn α( + τ y qe 4 a 43 a44 = ( + τ y r φ a 45 = 0 αn αq n e y y z y a5 ( b η A ( ( τ = + y ( + z z z ( z a a a a αn α( + τ y e q 5 = 53 = 0 54 = 0 55 = ( τ y r φ + where + y y = ( ba z z A Private Provision with overnment egulation The Hamiltonian for the private agent in this regime is given by γ βt βt N I H = ( C e + λ e N r N C Ψ Ω + γ T βt βt βt + qe [ I δ ] + qe [ δ ] + ve [ g ] (A The equilibrium dynamics for this regime can be expressed as z q v q = δ δ (Aa z h h 4
{ } q v z n N q = r( n z + c y δ n N n + + h h h ( q ( ( c C r n z β q = δ c C ( γ h (Ab (Ac + ( + v g ba ( y ( q q + + δ q q hq ( = r n z (Ad ( ( η ( + + v g b A y / z q v + + δ r( n z q q q h q where q = q λ q = q λ v = v λ = (Ae The evolution of the resource cost of regulation ( v can be derived from the first-order condition for investment in public capital ( : y + h g q = v (A3 z Setting z = n = c = q = q = 0 in (Aa-(Ae and noting (A3 leads to the steady-state conditions (5a-(5f in the text The linearized dynamics corresponding to this steady state can be described as ( (A4 Χ = Χ Χ where Χ = ( z n c q q Χ = ( z n c q q the 5x5 coefficient matrix of the linearized system where y y b = g b = 0 b3 = 0 b4 = z h b5 0 = z z and b 5 ij i j = represents 5x5 ( q v ( ( αn + z α n e y y y b = + + g ( q v h z z z 5
αn αne b = + r φ b 3 = b q n = = 0 4 b5 h αn αn αc n e αe c b3 = b 3 = b33 = 0 b34 = c h b 35 = 0 ( γ ( γ αn αq n e y y y y b4 = ba ( + ( y h ( g ( ( v g + + z z z z b = = 0 b44 = r φ αn αqe 4 b 43 + 45 = ( b bg A y αn αq n e y y z y y b5 ( b η A = ( v g ( h + + g ( z z z + ( z z g y z y ( q v z z ( z αn αe q y b5 = b53 = 0 b54 = 0 b55 = r ( b η A g z + + δ where + y y = ( ba z z A3 irect overnment Provision The relevant Hamiltonian for the private agent s optimization problem is given by γ βt βt H = ( C e + λ e N r( N / T N C Ψ ( I / + ( τ γ βt + q e [ I δ ] (A3 where ( ( = A b ( + b / The equilibrium dynamics for this regime can be expressed as 6
z y q g δ = δ z z h (A3a ( q n N y n N n h h ( = r( n z + c + + ( g ( g y q δ h (A3b ( c C r n z β q = δ c C ( γ h + ( τ ba ( y ( q + + δ = r n q q q hq ( z (A3c (A3d Setting z = n = c = q = 0 in (Aa-(A3d leads to the steady-state conditions (6a-(6d in the text The linearized dynamics corresponding to this steady state can be described as ( (A33 Χ = Χ Χ where Χ = ( z n c q Χ = ( z n c q 4x4 coefficient matrix of the linearized system where y y c = g c = 0 c3 = 0 c4 = z h z z and c 4 ij i j = represents the 4x4 αn α( n e y h y y y c = ( g ( g + z z z z αn αn e c = + r φ c 3 = c 4 q n = h αn αn αc n e ( + τ y αe c 3 = 3 = 33 = 34 = c h c c c 0 c ( γ ( γ 7
αn αq n e y c4 = ba ( + ( τ ( y y c z c44 = ( + τ y r φ = = 0 αn α( + τ y qe 4 c 43 where + y y = ( ba z z Appendix B The Provision of Public Capital and Optimal Fiscal Policy An important objective for a government in a decentralized economy might be to maximize economic welfare by ensuring the provision of an optimal amount of public investment either directly or through private providers This requires the design of an optimal tax and subsidy policy that will replicate the (first-best social optimum in a centrally planned economy To solve the problem of time inconsistency we introduce lump sum taxes into the framework and examine how the particular mode of providing public capital can influence the design of optimal fiscal policy B The First-Best Equilibrium The following set of equations describes the evolution of a centrally planned economy where the social planner s resource allocation decision internalizes both the production and the borrowing externalities (all dynamic variables for the centrally planned economy are denoted by a superscript * : z q q = δ δ (Ba z h h { } ( ( q z n q q = r( n z + c + + y n n h h h δ (Bb c c { } ( * ( ( r n z + n + z r β q = δ ( γ h (Bc + ( ( ( ba y q q r n n δ r n z r + + + = + q q hq + z q + z ( ( (Bd 8
+ ( b A ( y / z { q } r ( q n n δ r n z r + + + = + q q hq + z q + z ( ( (Be The central planner corrects for three sources of externalities First from (Bc we see that in performing his optimization the planner takes into account the marginal cost of borrowing from international capital markets This is also reflected in the second term on the right hand sides of (Bd and (Be Second the third term on the left hand sides of (Bd and (Be reflect the fact that the accumulation of both types of capital enhances the economy s debt-servicing capacity by reducing its debt-capital ratio This raises the return from each type of capital by reducing the cost of borrowing Finally the central planner internalizes the social benefits of public capital formation and the interaction of the shadow prices of the two types of capital in making his allocation decisions Therefore irrespective of the mode of provision of public capital replicating the first-best solution in a decentralized economy requires a fiscal policy that can correct these three sources of distortions B Optimal Fiscal Policy under Private Provision In an economy with private provision the government can use three policy instruments to achieve the first-best optimum: a tax on borrowing τ N a tax on output τ and a tied subsidy to the private agent s for investment in public capital Introducing a lump sum tax T enables us to rewrite the budget constraint (7 in the following manner ( / τ τn ( T s r N N T Ω = + + (B A first-best tax and subsidy policy is defined as the set of policy instruments ( ˆ d ˆ d ˆ d N s τ τ that not only replicates the steady-state equilibrium in a centrally planned economy but also its dynamic adjustment path Comparing the steady-state versions of (3a-(3e with (Ba-(Be we get the following set of equations (note that to denote optimal levels we have dropped the superscripts and * for the dynamic variables: ( ˆ ( ( γ ( { ( } ( ( γ + τn r n z β r n z + n + z r β = ( ˆ τ ba ( y ba ( y ( + + n r = + + z q (Ba (Bb 9
( ˆ τ ( b η + { q ( s } ( y A + z s h { q } + y n = ( b A + + r z h + z ( (Bc Equations (Ba-(Bc can be solved for the optimal values of the three policy instruments: ˆ τ N ( n z ( n r = + z r n z (B3a ( n z = + z ba y n r ˆ τ + (B3b (B3c {( } ( sˆ = q +Σ 4Σ q +Σ where η ( η ( n z + n r y n Σ= h b A r ( n + z + z ba y z + z The optimal fiscal policy package described in (B3a-(B3c corrects for both the external effects of public capital accumulation as well as capital market imperfections The optimal tax on foreign borrowing given in (B3a corrects the borrowing externality by taxing foreign debt at a rate equal to the elasticity of foreign debt with respect to the debt-capital ratio The optimal tax on income given in (B3b turns out to be a subsidy reflecting a reward to the private agent for accumulating private capital which improves the economy s aggregate debt-servicing capability As a result it is tied to the reduction in the marginal cost of borrowing due to investment in private capital valued by its marginal product Conversely if the private agent were a net creditor to the rest of the world then the appropriate policy would be to impose a tax on income The optimal subsidy for public investment given in (B3c corrects for three distortions First it enables the private agent to internalize the social benefits from the aggregate stock of public capital Second it rewards the agent for accumulating public capital which improves the economy s debt-servicing capability Third it takes into account the effect of the accumulation of private capital on the shadow price of public capital q 0
Finally we should note that the optimal tax and subsidy policy is constant over time By substituting (6a-(6c in (4 it can be easily verified that an economy with private provision of public capital can not only replicate the steady-state equilibrium in a centrally planned economy but also its dynamic adjustment path B3 Optimal Fiscal Policy under overnment Provision In the presence of a lump sum tax T government provision of public capital can be written as the flow budget constraint in an economy with direct ( / τ τn ( T Ω = + r N N T + (B3 Then a first-best taxation policy is defined as the set of policy instruments ( ˆ ˆ N the equilibrium for a centrally planned economy τ τ that replicate To characterize the first-best taxation policy we compare the steady-state versions of (6a and (6d with (Ba and (Bd This gives the following set of optimal tax rates: ˆ τ N ( n z ( n r = + z r n z (B3a ( n z = + z ba y n r ˆ τ v g + (B3b Setting the two tax rates in accordance with (B3a and (B3b will enable the decentralized economy to replicate the steady-state equilibrium of a centrally planned economy The optimal tax on foreign borrowing given by (B3a is exactly similar to that in the economy with private provision However from (B3b we see that the income tax rate is slightly different from the one in specified in (Bb This is because direct government provision of public capital in a decentralized economy implies that the allocation of output to public investment g is arbitrary and therefore may be above below or equal to its social optimum Hence the first component of the income tax rate specified in (8b corrects for this deviation For example when g of allocating an extra unit of output to public investment is positive ie < g (the social optimum the shadow value v > 0 As a result income should be subsidized to encourage private capital accumulation which by increasing the flow of output The expression in (B3b is derived under the assumption that the central planner sets g = g comparison with direct government provision in a decentralized economy arbitrarily to enable a
also increases the stock of public capital The second component of the income tax is similar to (B3b and reflects the reward to the private agent for accumulating private capital which improves the economy s aggregate debt-servicing capability There is one important caveat to the first-best policy described in (B3a and (B3b If the government maintains τ ˆ = τ during the transition to the steady-state equilibrium the adjustment path followed by the decentralized economy will fail to replicate that of its centrally planned counterpart Therefore ˆ τ is the first-best tax rate only in the steady state but not in transition To see this let us denote the linearized matrix for the central planner s equilibrium dynamics in (Ba- (Be by polynomial Then the eigenvalues in the centrally planned economy are the unique solutions to the F ( µ et µ I = 0 (B33 where I is a 5x5 identity matrix The equilibrium dynamics will be characterized by two stable (negative eigenvalues and for the centrally planned economy we will denote these by µ < 0 and µ < 0 Then from (B33 it must be the case that F ( µ F( µ Substituting for ˆ τ and τ in the linearized matrix N ˆ corresponding to (B33 and (B34: where ( i i J = = (B34 0 ( µ J( µ for the decentralized economy we get a result = = (B35 0 µ = are the stable eigenvalues and J( is a polynomial corresponding to F( in the decentralized model It can be verified that 3 ( 0 F µ i = (B36 i It then follows from (B35 and (B36 that µ µ and µ µ Therefore if ˆ τ N and ˆ τ are fixed over time the decentralized economy will converge to the first-best steady-state equilibrium at a rate that is non-optimal relative to that of the centrally planned economy This happens because when the 3 ue to the analytical complexity of our model (B36 was verified numerically For a more complete proof in a related but simpler model see Turnovsky (997
government directly provides public capital the private agent takes it as exogenously given and therefore does not internalize the effect of its private investment decisions on the shadow price of public capital during transition This externality is not accounted for by a time-invariant income tax rate To correct the above problem let us modify the tax on private income to ( t ˆ z ( t z n ( t τ = τ + θ + θ n (B37 where θ and θ are constants to be determined The income tax rate in (B37 is time-varying and tracks the dynamic evolution of the economy as the stocks of public and private capital and debt change over time As Turnovsky (997 points out a time-varying tax rate enables the private agent to track the dynamic adjustment of the shadow price of public capital Moreover θ and θ are only relevant along the transition path and do not affect the steady-state equilibrium However the appropriate determination of the constants θ and θ is crucial for the first-best tax policy to replicate the dynamic adjustment of a centrally planned economy To ensure this the government must set θ and θ such that J J ( F( µ θ = µ = 0 (B38a ( F( µ θ = µ = 0 (B38b When θ and θ are chosen in accordance with (B38a and (B38b the speed of adjustment in the decentralized economy will replicate that of the centrally planned economy Moreover θ and θ are only relevant along the transition path and do not affect the steady-state equilibrium As z ( t z ( ( and n t n τ t will converge to its long-run optimal rate ˆ τ 3
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