Public Economics Ben Heijdra Chapter 8: Taxation and ... · Public Economics: Chapter 8 1 Public...

Public Economics: Chapter 8 1

Public EconomicsBen Heijdra

Chapter 8: Taxation and Economic Growth

Public Economics – Chapter 8 Version 1.1 – November 2004Ben J. Heijdra


Aims of this chapter

• What is the effect of taxation on macroeconomic growth in general equilibrium?

• Exogenous growth models

– Solow-Swan model: ad hoc savings function

– Ramsey model: dynamic optimization under perfect foresight

– Extended Ramsey model: endogenous labour supply

– Mention: two-sector models

• Endogenous growth models

– capital fundamentalist models

– human capital and growth

– R&D and growth

• Conclusions



Exogenous Growth Models

• Background literature: H & vdP (2002, ch. 14)

• Exogenous growth:

– long-run growth rate is exogenously determined

– growth during transition may depend on policy parameters such as tax rates

• First wave of growth theory (1955-1970) sees capital accumulation as the engine of

growth

• Second wave of growth theory (1985-) identifies additional sources of growth:

human capital, knowledge, new products

• Selective overview of exogenous growth models and the effects of taxation



Solow-Swan Model

• Key originators: Robert Solow (1956) and Trevor Swan (1956)

• React to earlier “Keynesian” growth theory by Harrod and Domar [no capital-labour

substitutability]

• Overview of a basic Solow-Swan model



• CRTS production function:

Y (t) = F [K(t), A (t) L(t)︸︷︷︸]≡N(t)

– Y (t) is aggregate output

– K (t) is the aggregate stock of capital [machines, buildings, PCs, and the like]

– L (t) is employment [number of workers]

– A (t) is the exogenous index of labour-augmenting productivity

– N (t) is employment measured in efficiency units [workers get more productive

as time evolves]



• Technological assumptions:

– CRTS: F [λK(t), λN(t)] = λF [K(t), N(t)] for λ > 0

– positive but diminishing marginal products: FK , FN > 0, FKK , FNN < 0, and

FKN > 0

– Inada conditions: nice curvature properties around the origin (with K or N equal

to zero) and in the limit (with K or N approaching infinity):

limK→0

FK = limN→0

FN = +∞lim

K→∞FK = lim

N→∞FN = 0



• Keynesian savings function:

S(t) = sY (t), 0 < s < 1

– s is the constant exogenous propensity to save

– S (t) is total saving

• Closed economy: output exhausted by household consumption C(t) and investment

I(t)

Y (t) = C(t) + I(t)

= C (t) + S (t)

– assume government consumption is zero (G (t) = 0)

– I (t) is gross investment



• Aggregate gross investment:

I(t) = δK(t) + K(t)

– δK(t) is replacement investment

– δ is the constant depreciation rate (δ > 0)

– K (t) is the stock of capital so K (t) ≡ dK(t)dt

is net investment

• Forcing equations [changes in exogenous variables]:

– labour supply is exogenous and the population grows as a whole at a constant

exponential rate nL:

L(t)

L(t)= nL ⇔ L(t) = L(0)enLt

where L (0) is the base-year population



– labour-augmenting technology is exogenous and grows at the constant

exponential rate nA:

A(t)

A(t)= nA ⇔ A(t) = A(0)enAt

where A(0) is the base-year technology level.

• Summary of the model:

I (t) = S (t) (M1)

S(t) = sY (t) (M2)

I(t) = δK(t) + K(t) (M3)

Y (t) = F [K (t) , N (t)] (M4)

– Endogenous: I (t), S (t), K (t), and Y (t)

– Exogenous: N (t). Parameters: s and δ



• Fundamental differential equation for capital per effective labour:

k(t) = sf(k(t))− (δ + nL + nA)k(t) (FDE)

– k (t) ≡ K (t) /N (t)

– y(t) ≡ Y (t)/N(t)

– intensive-form production function:

y (t) = f(k(t)) ≡ F [K(t)/N(t), 1]

with properties:

f ′(k(t)) ≡ FK [k(t), 1]

f (k (t))− k (t) f ′(k(t)) = FN [k(t), 1]

f ′′(k(t)) ≡ N(t)FKK [k(t), 1]



Proof : write Y = NF [K/N, 1] and differentiate:

(FK ≡)∂Y

∂K= NFK

[K

N, 1

]1N

= FK

[K

N, 1

](1)

Similarly, since Y = Nf(k), we also have that:

∂Y

∂K= Nf ′(k)

1N

= f ′(k) (2)

Combining (1) and (2) we find f ′(k) = FK

[KN , 1

]. We also find from (2) that:

(FKK ≡)∂2Y

∂K2= f ′′(k)

1N

For labour we find:

(FN ≡)∂Y

∂N= F

[K

N, 1

]+ NFK [.]

−K

N2

= f(k)− f ′(k)k (3)

Q.E.D.



• In Figure 8.1 we show the phase diagram for k(t)

– the straight line (δ + nL + nA)k(t) represents the amount of investment

required to replace worn-out capital and to endow each efficiency unit of labour

with the same amount of capital [recall that the stock of efficiency units of labour

grows at rate nL + nA]

– for a constant savings rate, s, the per capita saving curve has the same shape as

the intensive-form production function [Inada conditions: f(k(t)) is vertical at the

origin, is concave, and flattens out as more and more capital per worker is

accumulated]

– there is a unique stable equilibrium at E0 [at point A: sf (k) > (δ + nL + nA) k

so that k > 0 and vice versa at point B]



• In the Balanced Growth Path the capital-effective-labour ratio is k∗ and we can

deduce the following effects:

– k = 0 so that:

k∗ =sf (k∗)

δ + nL + nA

y∗ = f (k∗)

– k∗ ≡ (KN

)∗and y∗ ≡ (

YN

)∗are constant whilst N grows exponentially at rate

nL + nA so: (K

K

)∗

=

(Y

Y

)∗

=

(N

N

)∗

= nL + nA



– but S = I = sY and s is constant so [also along the BGP] we have:(

S

S

)∗

=

(I

I

)∗

=

(Y

Y

)∗

= nL + nA

– output per worker is Y/L and it grows at the following rate along the BGP:

(Y

Y

)∗

−(

L

L

)∗

= nA



< = =<

!

!

sf(k(t))

k(t)

(* + nL + nA)k(t)

E0

A

sf(k(t))

(*+nL+nA)k(t)

k*

f(k(t))

!

B

Figure 8.1: The Solow-Swan Model



• Major conclusions on the Solow-Swan model:

– the long-run growth rate in the economy is fully explained by exogenous factors

(viz. the rate of population growth and the rate of labour-augmenting

technological change)

– the only place where taxes can have an effect in the model is via the savings rate

s. Suppose that s depends negatively on the capital tax tC . An increase in tC will

then have the effects as illustrated in Figure 8.2 .

¦ initially in steady-state E0

¦ savings rate falls from s0 to s1

¦ immediate effect: jump from E0 to A [k predetermined at impact]



¦ at point A: actual investment falls short of required investment so k < 0

¦ gradual move to new steady-state at E1

¦ growth less than in BGP during transition (see Figure 8.3 for transitional

dynamics)

¦ long-run growth rate unaffected by the capital tax

– model is somewhat unsatisfactory because the one of the key variables, viz. the

savings rate, is exogenous. Must provide a theoretical foundation for savings

behaviour: the Ramsey model



!

!

s1 f(k(t))

k(t)

(* + nL + nA)k(t)

E0

A

sf(k(t))(*+nL+nA)k(t)

!

s0 f(k(t))

E1

k*0k*

1

Figure 8.2: Fall in the Savings Rate in the Solow-Swan Model



[ln K (t)]0*

ln N(t)

ln K(t)ln N(t)

t

[ln K (t)]1*

!

!

t = 0

E0

E1

t 6 4

Figure 8.3: Transitional Dynamics



Ramsey Model

• We now return to the dynamic optimization approach of Chapter 5

• Abstract from exogenous sources of economic growth:

– no population growth (nL = 0)

– no labour-augmenting technological progress (nA = 0)

• Representative household:

– infinitely lived

– perfect foresight

– exogenous labour supply (say L (t) = 1)



• objective function:

Λ (0) ≡∫ ∞

0

U [C (t)] e−ρtdt (LU)

– U [·] is the felicity function (or instantaneous utility): U ′ [·] > 0 > U ′′ [·]– C (t) is (the flow of) consumption of the household

– ρ is the pure rate of time preference [ρ > 0]

– Λ (0) is life-time utility

• portfolio investment opportunities of the household:

– purchase shares in existing firms

– purchase (short-period) government bonds

– no risk so they are perfect substitutes in the portfolio



• budget identity:

B (t) + s (t) E (t) + C (t) = (1− tL) W (t) + (1− tR) r (t) B (t) + Z (t) (BI)

– B (t) is the stock of government debt [B ≡ dBdt

]

– s (t) is the price of shares [s ≡ dsdt

]

– E (t) is the outstanding stock of equities [E ≡ dEdt

]

– tL is the tax on wage income

– W (t) is the wage rate

– tR is the tax on interest income

– r (t) is the interest rate on government bonds

– Z (t) is the lump-sum transfer received from the government

– Note : compared to earlier model we have tG = 0 (no capital gains tax) and

D (t) = 0 (firm pays no dividends)



• remarks:

– initial conditions: E(0) and B (0) predetermined at time t = 0

– household chooses paths for C (t), B (t), and E (t) in order to maximize (LU)

subject to (BI) and some transversality conditions

– non-standard optimal control problem which can be solved by transforming it [see

also Chapter 5 ]



• the first-order conditions for the household’s optimization problem are:

U ′ [C] = λ (1)

(1− tR) r =s

s(2)

λ

λ= [ρ− (1− tR) r] (3)

where λ is the co-state variable associated with aggregate financial wealth.

– (1) is the (implicit) Frisch demand for consumption

– (2) is the no-arbitrage equation between government bonds and equities

– (3) describes the optimal time path for λ



Digression: Solving the Household Problem

• all assets are perfect substitutes in the household portfolio so that we can define

total assets A (t) as:

A (t) = B (t) + s (t) E (t)

• it follows that:

A (t) = B (t) + s (t) E (t) + s (t) E (t)

• rewriting (BI) we get [by adding s (t) E (t) to both sides]:

B (t) + s (t) E (t) + s (t)E (t)︸︷︷︸≡A(t)

= (1− tL)W (t) + (1− tR) r (t)B (t)

+Z (t)− C (t) (A)



• by adding and deducting (1− tR) r (t) s (t) E (t) to the RHS of (A) we get:

A (t) = (1− tL) W (t) + (1− tR) r (t)

B (t) + s (t) E (t)︸︷︷︸

≡A(t)

+

[s (t)

s (t)− (1− tR) r (t)

]s (t) E (t) + Z (t)− C (t) (B)

• rewrite (B):

A (t) = (1− tR) r (t) A (t) + (1− tL) W (t) + Z (t)− C (t)

+

[s (t)

s (t)− (1− tR) r (t)

]s (t) E (t) (C)

• we now have a single aggregate state variable (A (t)) whose dynamic evolution

must be determined.



• current-value Hamiltonian [dropping time subscripts]:

H ≡ U [C] + λ

((1− tR) rA + (1− tL) W + Z − C

+

[s

s− (1− tR) r

]sE

)

+µ [A−B − sE]

• control variables: C , E, B; state variable: A; co-state variable: λ; Lagrange

multiplier: µ

• first-order conditions [Kuhn-Tucker conditions]:

∂H∂C

= U ′ [C]− λ ≤ 0, C ≥ 0, C∂H∂C

= 0 (F1)



∂H∂E

= λs

[s

s− (1− tR) r

]− µs ≤ 0,

E ≥ 0, E∂H∂E

= 0 (F2)

∂H∂B

= −µ ≤ 0, B ≥ 0, B∂H∂B

= 0 (F3)

λ− ρλ = −∂H∂A

= −λ (1− tR) r − µ (F4)

– assume consumption is essential [i.e. U ′ [0] = −∞] so that C > 0 and it

follows from (F1) that U ′ [C] = λ

– if government bonds are held by the household (B > 0), then it follows from (F3)

that µ = 0

– if shares are also held (E > 0) then it follows from (F2) that ss− (1− tR) r = 0



• hence, in the interior solution we get from (F1)-(F4):

U ′ [C] = λ

(1− tR) r =s

s

λ

λ= [ρ− (1− tR) r]

which coincides with our results. Q.E.D.



Representative Firm

• key simplifying features:

– abstract from corporate debt

– no dividends paid

– no adjustment costs of firm investment

• gross profit is:

Π (t) ≡ F [K (t) , L (t)]−W (t) L (t)

– K (t) is the capital stock

– L (t) is labour demand

– Π (t) is gross profit [tax base of the corporate income tax]

• retained earnings are equal to after-tax profit:

(1− tC) Π (t) = RE (t) (A)



– corporate profit is taxed at rate tC

– RE (t) is retained earnings

– D (t) = 0 no dividend payments

• capital accumulation identity:

K (t) = I (t)− δK (t)

– K (t) ≡ dK(t)dt

– I (t) is gross investment

– δK (t) is replacement investment, δ is the depreciation rate

• financing constraint of the firm:

RE (t) + s (t) E (t) = I (t) (B)



• combining (A) and (B) [assuming RE (t) > 0] yields:

s (t) E (t) = I (t)− (1− tC) Π (t) (C)

– if I > (1− tC) Π then the firm issues new shares

– if I < (1− tC) Π then the firm buys back its own shares

• the value of outstanding shares is:

V (t) = s (t) E (t) (D)

• we can now derive the differential equation for V (t):

V (t) = (1− tR) rV (t)− [(1− tC) Π (t)− I (t)] (E)

Derivation :

– from (D) we get:

V (t) = s (t) E (t) + s (t) E (t) (F)



– from the household no-arbitrage equation we recall:

(1− tR) r =s (t) E (t)

s (t) E (t)⇐⇒

(1− tR) rV (t) = s (t) E (t) (G)

– combining (C), (F), and (G) we get (E). Q.E.D.

• the only sensible (no-bubble) solution for V (0) is obtained in the usual manner by

solving (E) forward in time and imposing a terminal condition:

V (0) =

∫ ∞

0

[(1− tC) Π (t)− I (t)] exp

[−

∫ t

0

θ (τ) dτ

]dt (OF)

where the cost of capital is:

θ (τ) ≡ r (τ) [1− tR (τ)] (COC)



and we have used:

limt→∞

V (t) exp

[−

∫ t

0

θ (τ) dτ

]= 0 (TC)

– (TC) is the terminal condition ensuring that the value of the firm remains finite

[thus we focus on the fundamental value of the firm]

– (COC) shows that the firm’s cost of capital equals the after-tax interest rate on

government bonds

– (OF) is the firm’s objective function. It must be maximized by appropriate choice

of K (t), I (t), and L (t) subject to the accumulation identity,

K (t) = I (t)− δK (t), the profit function,

Π (t) ≡ F [K (t) , L (t)]−W (t) L (t), and taking as given K (0).

• The optimization problem is solved with the method of optimal control [see H & vdP



pp. 700-702]. The current-value Hamiltonian is:

H ≡ (1− tC) [F [K (t) , L (t)]−W (t) L (t)]− I (t) + q (t) [I (t)− δK (t)]

– q (t) is the co-state variable

– K (t) is the state variable

– L (t) and I (t) are the control variables

• first-order conditions [interior solution]:

∂H∂L (t)

= (1− tC) [FL [·]−W (t)] = 0 (A)

∂H∂I (t)

= q (t)− 1 = 0 (B)

q (t)− r (t) (1− tR (t)) q (t) = − ∂H∂K (t)

= − [(1− tC)FK [·]− δq (t)] (C)

K (t) =∂H

∂q (t)(D)



• these conditions can be simplified to:

W (t) = FL [K (t) , L (t)] (LD)

r (t) (1− tR (t)) + δ = (1− tC)FK [K (t) , L (t)] (KD)

K (t) = I (t)− δK (t) (CA)

– (LD) is the standard labour demand equation

– (KD) is the demand for capital

– (CA) is the dynamic expression for the capital stock



• Summary of the Full Ramsey Model

U ′ [C (t)] = λ (t) (M1)

λ (t)

λ (t)= ρ + δ − (1− tC) f ′ (k (t)) (M2)

k (t) = f (k (t))− C (t)− δk (t) (M3)

W (t) = f (k (t))− k (t) f ′ (k (t)) (M4)

– (M1) is the Frisch demand for consumption [C is negatively related to the marginal utility

of wealth]

– (M2) is the household Euler equation, taking into account the demand for capital [rental

rate on capital expression]

– (M3) is the goods market clearing condition [abstracting from government consumption]

– (M4) is the usual expression for the gross wage rate

– Endogenous: C , λ, k, and W ; exogenous: tC ; parameters: ρ and δ.



K(t)

8(t)

K(t) = 0.

8(t) = 0.

KKR

!

C(t) 0

!

!

E0E0

KGRCGR CKR

!

AA

SP

8(t) = UN[C(t)]

Figure 8.4: The Ramsey Growth Model



• Figure 8.4 presents the phase diagram for the Ramsey model in (λ, k) space

– the λ = 0 line is obtained from (M2) and defines a unique capital-labour ratio,

kKR [KR = “Keynes-Ramsey”]:

f ′(kKR

)=

ρ + δ

1− tC

– for points to the left (right) of the λ = 0 line, k is too low (too high), f ′ (k) is too

high (too low), and λ decreases (increases) over time. See the vertical arrows.

– the k = 0 line is obtained by setting k = 0 in (M3) and substituting C (t) into

(M1):

k (t) = 0 ⇐⇒ U ′ [f (k (t))− δk (t)] = λ (t)

– the slope of the k = 0 line is thus:(

dλ

dk

)

k=0

= U ′′ [·] [f ′ (k (t))− δ]



so that [since U ′′ < 0] we have:

(dλ

dk

)

k=0

<

=

>

0 ⇐⇒ f ′ (k (t))

>

=

<

δ

– for points above (below) the k = 0 line, λ is too high (too low), C is too low (too

high), and the capital stock increases (decreases) over time. See the horizontal

arrows.

– there is a downward sloping saddle path, SP, through the unique equilibrium E0

• Growth properties:

– capital-labour ratio constant in the long run, k = kKR

– all variables grow at the same rate: zero in this case as nL = nA = 0



• We can now re-examine the effects of the capital tax tC . Figure 8.5 illustrates the

impact, transitional, and long-run effects of an unanticipated and permanent

increase in the corporate tax tC

– initially in steady-state E0

– corporate tax rises from tC0 to tC1: the λ = 0 line shifts to the left [no effect on

k = 0 line]:

(λ = 0 line) f ′(kKR

)=

ρ + δ

1− tCso:

f ′′(kKR

)dkKR =

ρ + δ

(1− tC)2dtC

– immediate effect: jump from E0 to A [k predetermined at impact]

– at point A: actual investment falls short of required investment so k < 0 and

ρ + δ > (1− tC) f ′ (k (t)) so that λ > 0



– gradual move in north-westerly direction to the new steady-state at E1

– growth less than in BGP during transition [i.e. negative growth]

– long-run growth rate unaffected by the capital tax [remains zero]

• Tax incidence: an interesting implication of the model is that tax incidence differs in

the short-run and the long-run

– in the short run, k is fixed so it follows from (M4) that the wage rate is also

unchanged. From the rental expression (KD) we find that the after-tax reward to

capital owners, (1− tC) f ′ (k) falls. Hence, in the impact period capital bears

the full burden of the corporate tax.



– in the long run, k is crowded out, and the after-tax reward to capital owners is

restored to (1− tC) f ′ (k) = ρ + δ. Since k falls, it follows from (M4) that

wages fall:

dW (∞)

dtC= [f ′ (k)− f ′ (k)− kf ′′ (k)]

dk (∞)

dtC

= −kf ′′ (k)dk (∞)

dtC< 0

Hence, labour bears the full burden of the corporate tax in the long run!



K(t)

8(t) [8(t) = 0]0

.

!

C(t) 0KKR

!

!

E0E0

KGRCGR

!AA

SP

8(t) = UN[C(t)]

[8(t) = 0]1

.

E1E1

!

BB

!!

!

K(t) = 0.

1KKR

Figure 8.5: A Rise in the Corporate Tax in the Ramsey Model



Extended Ramsey Model

• Weakness of standard Ramsey model [for tax policy analysis] is the assumption that

labour supply is exogenous. This limits the kinds of distortions that can be

distinguished with the model

• We now extend the Ramsey model by distinguishing an endogenous labour supply

decision by the representative household

• For background on the extended Ramsey model: H & vdP (2002, pp. 478-483) and

Judd (JPE 1987)



• Objective function representative household is modified to:

Λ (0) ≡∫ ∞

0

U [C (t) , 1− L (t)] e−ρtdt (LU)

– L (t) is labour supply (1− L (t) is leisure)

– U (·) is a strictly concave function in C and 1− L, i.e. UC > 0, U1−L > 0,

UCC < 0, U1−L,1−L < 0, and:

UCCU1−L,1−L − (UC,1−L)2 > 0

Note: indifference curves bulge toward the origin.

• The budget identity of the household is modified to:

B (t) + s (t) E (t) + C (t) = (1− tL)W (t)L (t) + (1− tR) r (t)B (t) + Z (t) (BI)

– wage income is now WL (rather than W , as before)



• remarks:

– initial conditions: E(0) and B (0) predetermined at time t = 0

– household chooses paths for [real decisions] C (t), and L (t), and for [portfolio

decisions] B (t) and E (t) in order to maximize (LU) subject to (BI) and some

transversality conditions

– non-standard optimal control problem which can again be solved by transforming

it [see above]



• the first-order conditions for the household’s optimization problem are:

UC [C (t) , 1− L (t)] = λ (t) (1)

U1−L [C (t) , 1− L (t)] = λ (t) (1− tL) W (t) (2)

(1− tR) r (t) =s (t)

s (t)(3)

λ (t)

λ (t)= [ρ− (1− tR) r (t)] (4)

where λ is the co-state variable associated with aggregate financial wealth.

– (1)-(2) implicitly define the Frisch demands for consumption and leisure

– (3) is the no-arbitrage equation between government bonds and equities

– (4) describes the optimal time path for λ

– Note : only (2) is new compared to the standard Ramsey model



• The rest of the model is unchanged.....

– from the firm side we still have:

W (t) = FL [K (t) , L (t)] (5)

r (t) (1− tR (t)) + δ = (1− tC) FK [K (t) , L (t)] (6)

K (t) = I (t)− δK (t) (7)

– and the loose ends are:

Y (t) = C (t) + I (t) (8)

Z (t) = tC [F [K (t) , L (t)]−W (t) L (t)] + tLW (t) L (t) (9)



Digression: Solving the Household Problem

• (not in class)

• all assets are perfect substitutes in the household portfolio so that we can define

total assets A (t) as:

A (t) = B (t) + s (t) E (t)

• it follows that:

A (t) = B (t) + s (t) E (t) + s (t) E (t)

• rewriting (BI) we get [by adding s (t) E (t) to both sides]:

B (t) + s (t) E (t) + s (t)E (t)︸︷︷︸≡A(t)

= (1− tL) W (t)L (t) + (1− tR) r (t)B (t)

+Z (t)− C (t) (A)



• by adding and deducting (1− tR) r (t) s (t) E (t) to the RHS of (A) we get:

A (t) = (1− tL) W (t) L (t) + (1− tR) r (t)

B (t) + s (t) E (t)︸︷︷︸

≡A(t)

+

[s (t)

s (t)− (1− tR) r (t)

]s (t) E (t) + Z (t)− C (t) (B)

• rewrite (B) by adding and deducting (1− tL) W (t):

A (t) = (1− tR) r (t) A (t) + (1− tL) W (t) + Z (t)− C (t)

−W (t) [1− L (t)] +

[s (t)

s (t)− (1− tR) r (t)

]s (t) E (t) (C)

• we now have a single aggregate state variable (A (t)) whose dynamic evolution

must be determined.



• current-value Hamiltonian [dropping time subscripts]:

H ≡ U [C, 1− L] + λ

((1− tR) rA + (1− tL) W + Z − C

−W (1− L) +

[s

s− (1− tR) r

]sE

)

+µ [A−B − sE]

• control variables: C , 1− L, E, B; state variable: A; co-state variable: λ; Lagrange

multiplier: µ



• interesting first-order conditions [interior solution]:

∂H∂C

= UC [C, 1− L]− λ = 0 (F1)

∂H∂ (1− L)

= U1−L [C, 1− L]− λ (1− tL) W = 0 (F2)

∂H∂E

= λs

[s

s− (1− tR) r

]= 0 (F3)

λ− ρλ = −∂H∂A

= −λ (1− tR) r (F4)

which coincides with our results. Q.E.D.



Compact Summary of the Extended Ramsey Model

• using the various model expressions, the model can be written in very compact form

as follows:

C (t) = c [λ (t) , K (t) , tL] (M1)

L (t) = l [λ (t) , K (t) , tL] (M2)

K (t) = L (t) f

(K (t)

L (t)

)− C (t)− δK (t) (M3)

λ (t)

λ (t)= ρ + δ − (1− tC) f ′

(K (t)

L (t)

)(M4)

– (M1) and (M2) are the expressions for, respectively, consumption and labour

supply, conditional on the state variables (λ and K) and the tax rate. They are

obtained by noting that (1), (2) and (5) define implicit functions (c (·) and l (·)) for



these variables:

UC [C (t) , 1− L (t)] = λ (t)

U1−L [C (t) , 1− L (t)] = λ (t) (1− tL) FL [K (t) , L (t)]

– (M3) is the capital accumulation expression obtained by combining (7) and (8)

and noting that Y = Lf (K/L) where f (K/L) ≡ F[

KL, 1

]

– (M4) is the usual expression for the marginal utility of wealth which is obtained by

combining (4) and (6) and noting that FK (·) = f ′ (·)– Endogenous: C , L, λ, and K ; exogenous: tC and tL; parameters: ρ and δ.



• growth properties:

– in the steady state, λ = K = 0, and (M4) defines a unique capital-labour ratio

(k∗). Using k∗, (M1)-(M3) then yield solutions for λ, C , and L.

– Hence, there is no long-run growth in the model...

– ....and thus taxes do not affect long-run growth either.

• tax incidence analysis:

– Judd (JPE 1987) uses the most general version of the model for tax policy

analysis [difficult because many different elasticities affect the different functions].

– here we study a Mickey Mouse version of the model in which all useful

substitution elasticities are set equal to unity



Tax Incidence in the Unit-Elastic Ramsey Model

• We now use the following specific form for the felicity function:

U [C, 1− L] = log[Cα (1− L)1−α]

, 0 < α < 1 (FF)

– intertemporal substitution elasticity is equal to unity [logarithmic felicity]

– intratemporal substitution elasticity between consumption and leisure is also unity

[Cobb-Douglas sub-felicity]

• We use the following specific production function:

F [K, L] ≡ KεL1−ε, 0 < ε < 1 (PF)

– intratemporal substitution elasticity between capital and labour is unity

[Cobb-Douglas technology]



• Using (FF) and (PF) we find that the model simplifies a lot:

C (t) =α

λ (t)(A)

1− α

1− L (t)= λ (t) (1− tL) (1− ε)

(K (t)

L (t)

)ε

(B)

K (t) = L (t)

(K (t)

L (t)

)ε

− C (t)− δK (t) (C)

λ (t)

λ (t)= ρ + δ − (1− tC) ε

(K (t)

L (t)

)ε−1

(D)



• Since C only depends on λ, we can condense the model even more by eliminating

λ:

(1− α) C (t)

α [1− L (t)]= (1− tL) (1− ε)

(K (t)

L (t)

)ε

(M1)

K (t) = L (t)

(K (t)

L (t)

)ε

− C (t)− δK (t) (M2)

C (t)

C (t)= (1− tC) ε

(K (t)

L (t)

)ε−1

− (ρ + δ) (M3)

– (M1) is the labour market equilibrium (LME) condition, expressing combinations

of C , L, and K for which labour supply (LHS) equals labour demand (RHS)

– (M2) is the capital accumulation expression

– (M3) is the household Euler equation



C(t)

K(t)

!

K(t) = 0.

KKKGR

CGR

0! !!

KC

!

!

!CC

C(t) = 0.

E0

ASP0

Figure 8.6: Phase Diagram of the Extended Ramsey Model



• In Figure 8.6 the phase diagram of the model is presented.

– The LME condition defines an implicit function relating equilibrium employment to

consumption and the capital stock:

L (t) = l

[C (t) K (t)−ε

1− tL

](EE)

with l′ [·] < 0 [see below]. We find the partial effects:

∂L (t)

∂C (t)= l′ [·] K (t)−ε

1− tL< 0

∂L (t)

∂K (t)= −εl′ [·] C (t) K (t)−(1+ε)

1− tL> 0

∂L (t)

∂tL= l′ [·] K (t)−ε

(1− tL)2 < 0



– The C = 0 line is the combination of C and K such that the capital labour ratio

is constant. The equilibrium capital-labour ratio is obtained from (M3):

k∗ ≡(

K

L

)∗=

[ρ + δ

(1− tC) ε

]1/(ε−1)

(ECLR)

Using (ECLR) in (M1) we find that C/ (1− L) is constant along the C = 0 line.

Since K/L is also constant, it follows that the C = 0 line is a straight downward

sloping line.

– Consumption dynamics can be found by noting that (M3) can be written as:

C (t)

C (t)= (1− tC) ε

[(K (t)

L (t)

)ε−1

− (k∗)ε−1

]

It follows that ∂[C/C

]/∂C < 0 [since ∂L/∂C < 0]: see the vertical arrows

in Figure 8.6.



– The K = 0 line is the combination of C and K such that the capital stock is

constant. By using (M2) and (EE) and setting K = 0 we find:

C (t) =

(l

[C (t) K (t)−ε

1− tL

])1−ε

K (t)ε − δK (t)

It is straightforward [but tedious] to show that the K = 0 line is as drawn in

Figure 8.6 [see H&vdP (2002, pp. 530-533) for details]

– Capital dynamics is obtained by noting from (M2) and (EE):

K (t) =

(l

[C (t) K (t)−ε

1− tL

])1−ε

K (t)ε − C (t)− δK (t)

It follows that ∂K/∂C < 0 [since ∂L/∂C < 0]: see the horizontal arrows in

Figure 8.6.

– There is a unique saddle-point stable equilibrium at point E0



• In Figure 8.7 we illustrate the effects of an increase in the corporate tax rate, tC

– the K = 0 line is unaffected

– the C = 0 line rotates counter-clockwise around point KC [since L = 1 and

C = 0 in that point]

– at impact K is predetermined and the economy jumps from E0 to A.

– over time, the economy moves gradually from A to E1

– in the long run both capital and consumption are lower

• Student exercise: determine the impact, transitional, and long-run effects of the

corporate tax increase on:

– the wage rate; the reward to capital owners

– output

– who bears the burden of the tax?



C(t)

K(t)

!

K(t) = 0.

KK0! !

KC

!

!

!

E0

A

SP1

!E1

!

[C(t) = 0]0

.

[C(t) = 0]1

.

Figure 8.7: A Rise in the Corporate Tax in the Extended Ramsey Model



• Closing remarks on the extended Ramsey model

– the general [non unit-elastic] case can easily be handled by performing local

policy analysis, i.e. by using the log-linearization approach used elsewhere in

these lectures [see Judd (1987) for details]

– model is still one of exogenous growth, i.e. per definition tax rates cannot affect

the long-run growth rate in the economy! It is time to move on to the endogenous

growth literature.



Digression: Deriving the Unit-Elastic Model

• (not in class)

• using (FF) and (PF) we find:

– that (1), (2) and (5) reduce to:

α

C= λ

1− α

1− L= λ (1− tL) W

W = (1− ε)

(K (t)

L (t)

)ε



– whilst (4) and (6) reduce to:

λ

λ= ρ− r (1− tR)

r (1− tR) = (1− tC) ε

(K

L

)ε−1

− δ

– combining (7) and (8) yields:

K = Y − C − δK

• derivation of the equilibrium employment equation (EE):

– first we write (M1) as:

[φ (L) ≡] [1− L] L−ε = ω0

(CK−ε

1− tL

)

where ω0 ≡ (1−α)α(1−ε)

> 0 is a constant. We find that φ (L) is positive and



downward sloping in the feasible range L ∈ [0, 1]:

φ′ (L) = −L−(1+ε) [L + ε (1− L)] < 0

φ′′ (L) = εL−(2+ε) [(1− ε) L + 1 + ε] > 0

– it follows that φ (·) can be inverted, i.e. that L = φ−1[ω0

(CK−ε

1−tL

)]≡ l [·].

The derivative of l [·] with respect to its argument is then obtained in the usual

manner [Implicit Function Theorem]:

φ′ (L) dL = ω0d

(CK−ε

1− tL

)=⇒

l′ [·] ≡ dL

d(

CK−ε

1−tL

) =ω0

φ′ (L)< 0



Endogenous Growth Models

• Background literature: H & vdP (2002, pp. 448-473)

• Endogenous growth:

– long-run growth rate is endogenously determined

– growth during transition and in the long run may depend on policy parameters

such as tax rates

• In the first wave of growth theory (1955-1970):

– endogenous growth was known but....

– not taken seriously.

– Why? Capital accumulation is the engine of growth and there are diminishing

returns to capital in the long run [labour gets scarce] which chokes off the growth

process.



• In the second wave of growth theory (1985-):

– there are some who deny the existence of diminishing returns to capital in the

long run [capital fundamentalists]

– ...others who see human capital accumulation as an additional engine of growth

[human capital theory]

– ...others who see endogenous technological change as the key engine of growth

[endogenous technology theory]

• Selective overview of endogenous growth models and the effects of taxation



sf(k(t))/k(t)

k(t)

* + nLB

k(0)

gk(4)!

!

C

!!A

!

Figure 8.8: The Capital-Fundamentalist Model



Capital Fundamentalist Models

• Type 1, recognized by Solow (1956): easy substitutability between labour and

capital. Labour never an effective constraint to growth as firms continue to substitute

toward capital. See book and Figure 8.8 .

• Type 2: for various reasons there may be constant returns to scale to capital.

Examples:

– Barro (1990): productive government spending

– Rebelo (1991): there is a “core” of capital goods that is produced under CRTS

using only accumulable factors

– Arrow (1962)-Romer (1986): learning by doing with spillovers

• Here we show a simple example of the basic Rebelo (1991) model

• Two production sectors:



– capital good sector produces investment goods using only existing capital under

CRTS:

I (t) = AIKI (t)

¦ I (t) is output in the capital sector (I (t) ≥ 0)

¦ AI is the general technology index in that sector

¦ KI (t) is capital use in the sector

– consumption good sector produces consumption goods, using existing capital

and land under CRTS:

C (t) = AC [KC (t)]ε T 1−ε

¦ C (t) is output in the consumption goods sector

¦ AC is the general technology index in that sector

¦ KC (t) is capital use in the sector

¦ T is land



• two types of capital:

– reproducible capital which can be accumulated over time and used in the two

sectors:

K (t) = KI (t) + KC (t) (CME)

K (t) = I (t)− δK (t) (CA)

Note : K can also be seen as a composite of physical and human capital [skilled

labour]

– non-reproducible capital which is available in same quantity in each period, e.g.

land T

• Market structure:

– competitive firms in both sectors: representative firm

– firms rent factors from households



– perfect mobility of capital between sectors

• Representative household:

– infinitely lived

– perfect foresight

• objective function:

Λ (0) ≡∫ ∞

0

[C (t)1−1/σ − 1

1− 1/σ

]e−ρtdt (LU)

– the felicity function is of the CRRA type: σ is the intertemporal substitution

elasticity (σ > 0)

– C (t) is (the flow of) consumption of the household





• household directly decides on the capital accumulation decision [chooses I (t)]

• budget identity:

C (t) + pI (t) I (t) = RT (t) T + RK (t) K (t) + Z (t) (BI)

– RT (t) is the rental rate on land

– RK (t) is the rental rate on capital

– Z (t) is lump-sum transfer received from the government

– pI (t) is the relative price of the investment good [pI ≡ PI/PC and we set

PC = 1]

• remarks:

– initial conditions: K (0) is predetermined at time t = 0

– household chooses paths for C (t), I (t), and K (t) in order to maximize (LU)

subject to (BI) and (CA) (and some transversality conditions)



• the current-value Hamiltonian for this problem is [provided I (t) > 0]:

H ≡ C (t)1−1/σ − 11− 1/σ

+ λ (t)[RT (t) T + RK (t) K (t) + Z (t)− C (t)

pI (t)− δK (t)

]

– control variable: C (t)

– state variable: K (t)

– co-state variable: λ (t)

– the first-order conditions for the household’s optimum are:

C (t)−1/σ =λ (t)

pI (t)(A)

λ (t) =

[ρ + δ − RK (t)

pI (t)

]λ (t) (B)



– Note : if the constraint I (t) ≥ 0 becomes binding the household simply chooses

to set I (t) = 0 and to consume its net income.

• Representative firm in the capital good sector:

– profit function [in terms of consumption goods]:

ΠI (t) ≡ (1− tI) pIAIKI (t)−RK (t) KI (t)

where tI is an output tax

– competitive behaviour leads to dΠI/dKI = 0 or:

RK (t) = (1− tI) pI (t) AI (C)

ΠI (t) = 0



• Representative firm in the consumption good sector:

– profit function [in terms of consumption goods]:

ΠC (t) ≡ (1− tC) ACKC (t)ε T 1−ε −RK (t) KC (t)−RT (t) T

– competitive behaviour leads to dΠC/dKC = dΠC/dT = 0 or:

RK (t) = ε (1− tC) AC

(KC (t)

T

)ε−1

(D)

RT (t) = (1− ε) (1− tC) AC

(KC (t)

T

)ε

(E)

ΠC (t) = 0



• Drawing things together we find the following key expressions:

C (t)

C (t)= σ

[pI (t)

pI (t)− λ (t)

λ (t)

](M1)

λ (t)

λ (t)= ρ + δ − (1− tI) AI (M2)

pI (t)

pI (t)= (ε− 1)

(φ (t)

φ (t)+

K (t)

K (t)

)(M3)

C (t)

C (t)= ε

(φ (t)

φ (t)+

K (t)

K (t)

)(M4)

– (M1) is obtained by differentiating (A) with respect to time

– (M2) is the combination of (B) and (C)



– (M3) is obtained as follows. First, we combine (C) and (D) to obtain:

pI (t) = ε

(1− tC1− tI

)(AC

AI

)(KC (t)

T

)ε−1

(F)

Next we differentiate (F) with respect to time [assuming time-invariant taxes,

tC = tI = 0]:

pI (t)

pI (t)= (ε− 1)

KC (t)

KC (t)(G)

Finally, we define φ (t) ≡ KC(t)K(t)

as the fraction of capital employed in the

consumption good sector, note that:

KC (t)

KC (t)=

φ (t)

φ (t)+

K (t)

K (t)(H)

and rewrite (G) as in (M3)



– (M4) is obtained by differentiating the production function,

C (t) = AC [KC (t)]ε T 1−ε, with respect to time, and noting (H)

• The endogenous growth rate of consumption can now be determined:

– by combining (M3) and (M4) we find:

pI (t)

pI (t)=

ε− 1

ε

C (t)

C (t)(I)

– by using (I) and (M2) into (M1) we find:

C (t)

C (t)= σ

[ε− 1

ε

C (t)

C (t)− (ρ + δ) + (1− tI) AI

]

or:

gC ≡ C (t)

C (t)=

εσ [(1− tI) AI − (ρ + δ)]

σ + ε (1− σ)(GRC)



• Key insights from (GRC):

– growth rate, gC , is time invariant, i.e. there is no transitional dynamics at all!

– the tax rate on the investment good sector decreases the rate of growth. [Note

the stark contrast in this respect with the exogenous growth literature]. Increasing

tI is equivalent to decreasing AI and this directly affects the engine of growth in

this economy

– the tax rate on the consumption good sector does not affect the rate of growth.

[just as in the exogenous growth literature]. Increasing tC is equivalent to

decreasing AC and this only affects the level of consumption [but not its rate of

growth] in this economy. tC is like a lump-sum tax.



• Note 1 : determination of φ (t).

– using (M4) and K (t) = AIKI (t)− δK (t) we find:

K (t)

K (t)= AI [1− φ (t)]− δ

gC = ε

(φ (t)

φ (t)+

K (t)

K (t)

)

– combining we derive the differential equation in φ (t):

φ (t)

φ (t)=

gC

ε+ δ + AI [φ (t)− 1]



– this is an unstable differential equation [see Figure 8.9 ] for which the only

economically sensible solution is that φ (t) adjusts at all times to ensure that

φ (t) = 0, i.e.

1− φ∗ =gC + εδ

εAI

– Note : an increase in tI leads to a decrease in gC and a downward shift in the

capital allocation line (say from CAL0 to CAL1 in Figure 8.9). The economy jumps

from E0 to E1 and φ∗ increases, i.e. a larger proportion of capital is used in the

consumption good sector as a result of the shock.



N(t)0 ! !

+

!

E0

N(t).

N(t)

E1

CAL0 CAL1

N*0

N*1

Figure 8.9: Intersectoral Allocation of Capital



• Note 2 : growth rate of net aggregate output Y (t).

– define aggregate net output [in terms of the consumption good] as:

Y (t) = C (t) + pI (t) [I (t)− δK (t)]

– differentiate with respect to time:

Y (t)

Y (t)=

C (t)

Y (t)

C (t)

C (t)+

pI (t) I (t)

Y (t)

[I (t)

I (t)+

pI (t)

pI (t)

]

−δpI (t) K (t)

Y (t)

[K (t)

K (t)+

pI (t)

pI (t)

]



– since φ (t) = φ∗ is constant we know that I (t) /K (t) is constant. By also

using (M3) and (M4) we derive:

gY ≡ Y (t)

Y (t)=

C (t)

Y (t)εK (t)

K (t)+

pI (t) I (t)

Y (t)

[K (t)

K (t)+ (ε− 1)

K (t)

K (t)

]

−δpI (t) K (t)

Y (t)

[K (t)

K (t)+ (ε− 1)

K (t)

K (t)

]

= εK (t)

K (t)= gC

Hence, net output grows at the same rate as consumption. The savings rate is

constant.



• Closing remarks on the capital fundamentalist model

– Clear where the engine of growth is in the model described above:

I (t) = AIKI (t). Endogenous growth almost a direct assumption (rather than

a result) of the model

– In some variations of the capital fundamentalist model there is transitional

dynamics [adding realism to the story]

– Rebelo (1991 JPE) has shown that the model can be generalized by

disaggregating capital into physical and human capital. Provided there is a core

of capital goods that is produced under CRTS using no non-reproducible factors

[directly or indirectly], the story remains valid.



Human Capital and Economic Growth

• Uzawa (1965)-Lucas (1988) place human capital at the forefront of the economic

growth process

• Lucas model is the “learning or doing” model

• We follow the version developed by Rebelo (1991 JPE)

• Lifetime utility of the representative household is the same as before:

Λ (0) ≡∫ ∞

0

[C (t)1−1/σ − 1

1− 1/σ

]e−ρtdt (LU)

– the felicity function is of the CRRA type: σ is the intertemporal substitution

elasticity (σ > 0)

– C (t) is (the flow of) consumption of the household





• Consumption and investment goods produced in the same sector. Technology is:

Y (t) = AY [φ (t) K (t)]1−γ [LY (t) H (t)]γ , 0 < γ < 1

– Y (t) is aggregate output in the production sector [Y (t) = C (t) + I (t)]

– K (t) is the aggregate capital stock

– H (t) is the aggregate stock of human capital

– φ (t) ≡ KY (t) /K (t) so φ (t) K (t) = KY (t) is the amount of capital used

in the production sector

– LY (t) is the number of hours worked in the production sector by an individual

with H (t) units of human capital

– AY is an index of technology



• Human capital is embodied in the worker and appreciates at rate δ [the same rate as

for physical capital]. Human capital accumulation proceeds according to the

following technology:

H (t) = AH [(1− φ (t)) K (t)]1−β [(L− LY (t)

)H (t)

]β − δH (t) (HCA)

– 0 < β < 1

– H (t) is net investment in human capital

– (1− φ (t)) K (t) is the amount of capital used for human capital accumulation

– L is the (fixed) labour supply so(L− LY (t)

)H (t) is the amount of human

capital used for human capital accumulation

– δH (t) is the depreciation on existing human capital



• The household directly decides on:

– the physical capital accumulation decision [choice of I (t)]: K (t) is rented to

firms at rental rate RK (t)

– the human capital accumulation decision [choice of H (t)]: LY (t) H (t) is

rented to firms at rental rate RH (t)

– the consumption decision [choice of C (t)]

• Budget identity:

C (t) + I (t) = φ (t) RK (t) K (t) + RH (t) LY (t) H (t) + Z (t) (BI)

– RK (t) is the rental rate on physical capital

– φ (t) K (t) is the amount of capital rented out to the production firms

– RH (t) is the rental rate on human capital

– Z (t) is lump-sum transfer received from the government



• remarks:

– initial conditions: K (0) and H (0) both predetermined at time t = 0

– capital accumulation equation:

K (t) = I (t)− δK (t) (CA)

– household chooses paths for C (t), I (t), and H (t) in order to maximize (LU)

subject to (BI) and (CA) (and some transversality conditions)



• the current-value Hamiltonian for this problem is [provided I (t) > 0]:

H ≡ C (t)1−1/σ − 11− 1/σ

+λ (t)[φ (t)RK (t)K (t) + RH (t) LY (t)H (t) + Z (t)− C (t)− δK (t)

]

+µ (t)[AH [(1− φ (t)) K (t)]1−β [(

L− LY (t))H (t)

]β − δH (t)]

– control variables: C (t), LY (t), and φ (t)

– state variables: K (t) and H (t)

– co-state variable: λ (t) and µ (t)



– the (interesting) first-order conditions for the household’s optimum are:

C (t)−1/σ = λ (t) (A)

λ (t)RH (t) = βµ (t)AH

[(1− φ (t))K (t)(L− LY (t)

)H (t)

]1−β

(B)

λ (t) RK (t) = (1− β)µ (t)AH

[(1− φ (t))K (t)(L− LY (t)

)H (t)

]−β

(C)

λ (t)− ρλ (t) = − [φ (t)RK (t)− δ

]λ (t)

−µ (t) (1− β) (1− φ (t)) AH

[(1− φ (t)) K (t)(L− LY (t)

)H (t)

]−β

(D)

µ (t)− ρµ (t) = −RH (t)LY (t) λ (t)

−µ (t)

β

(L− LY (t)

)AH

[(1− φ (t)) K (t)(L− LY (t)

)H (t)

]1−β

− δ

(E)



• Representative firm the production sector:

– profit function [in real terms]:

Π(t) ≡ (1− tY )Y (t)−RK (t) KI (t)−RH (t) LY (t)H (t)

= (1− tY )AY [φ (t)K (t)]1−γ [N (t)]γ −RK (t)φ (t) K (t)−RH (t)N (t)

where tY is an output tax and N (t) ≡ LY (t) H (t) labour measured in

efficiency units

– competitive behaviour leads to:

RK (t) = (1− γ) (1− tY ) AY

[φ (t) K (t)

LY (t) H (t)

]−γ

(F)

RH (t) = γ (1− tY ) AY

[φ (t) K (t)

LY (t) H (t)

]1−γ

(G)

Π (t) = 0

• We can combine the various expressions



– [static] optimal allocation of human capital (obtained from (B) and (G)):

RH (t) = βµ (t)

λ (t)AH

[(1− φ (t)) K (t)(L− LY (t)

)H (t)

]1−β

= γ (1− tY ) AY

[φ (t) K (t)

LY (t) H (t)

]1−γ

(SC1)

– [static] optimal allocation of physical capital (obtained from (B) and (G)):

RK (t) = (1− β)µ (t)

λ (t)AH

[(1− φ (t)) K (t)(L− LY (t)

)H (t)

]−β

= (1− γ) (1− tY ) AY

[φ (t) K (t)

LY (t) H (t)

]−γ

(SC2)



– combining (SC1) and (SC2) we can eliminate µ (t) /λ (t) to get:

β

1− β

(1− φ (t))(L− LY (t)

) =γ

1− γ

φ (t)

LY (t)(A1)

– [dynamic] evolution of the shadow price of physical capital (obtained from (D) and

(C)):

λ (t)

λ (t)= ρ + δ −RK (t) (DC1)



– [dynamic] evolution of the shadow price of human capital (obtained from (E) and

(B)):µ (t)

µ (t)= ρ + δ − λ (t)

µ (t)LRH (t) (DC2)

– In the steady state, K (t) /H (t), φ (t), and LY (t) are constant and it follows

from (SC1) and (SC2) that µ (t) /λ (t) is also constant, i.e.:

µ (t)

µ (t)=

λ (t)

λ (t)(SSC)

– combining (DC1), (DC2) and (SSC) we find:

RK (t) =λ (t)

µ (t)LRH (t)

or (by using (B) and (F)):



(1− γ) (1− tY ) AY

[φ (t) K (t)

LY (t) H (t)

]−γ

= LβAH

[(1− φ (t))K (t)(L− LY (t)

)H (t)

]1−β

(A2)

• We now have two expressions, (A1) and (A2), which can be solved for the

capital-labour intensities in the two activities. We find after some manipulations that

the steady-state capital labour ratio in the production sector is:

(φK

LY H

)∗=

[(1− γ) (1− tY ) AY

βLAH

(β (1− γ)

γ (1− β)

)1−β]1/(1−β+γ)

(A3)

• Using this expression in (SC2) we find:

(RK

)∗= (1− γ) (1− tY ) AY

[(φK

LY H

)∗]−γ

= Ψ [(1− tY ) AY ]θ(AHL

)1−θ(A4)



where Ψ and θ are defined as:

Ψ ≡[(1− γ)(1−β)(1−γ) (1− β)γ(1−β) ββγγγ(1−β)

]1/(1−β+γ)

> 0

θ ≡ 1− β

1− β + γ, 0 < θ < 1

• It follows from the Euler equation (A) and (DC1) that:

gC ≡(

C

C

)∗

= −σ

(λ

λ

)∗

= σ[(

RK)∗ − δ − ρ

]

= σ[Ψ [(1− tY ) AY ]θ

(AHL

)1−θ − δ − ρ]

(GRC)

where we have used (A4) in the final step.

– gC is the long-run endogenous growth rate in the economy [I (t), K (t), and

H (t) all grow at this same rate in the steady state]Public Economics – Chapter 8 Version 1.1 – November 2004

Ben J. Heijdra


– net output is Y ≡ C + I − δK and gY = gC

– there is transitional dynamics [too tedious to discuss here. See Bond, Wang, and

Yip (1996 JET )]



– Student exercise: compute the effect of the tax rate, tY , on the steady-state

growth rate. Explain the intuition behind the results.

• Closing remarks on the human capital model

– engine of growth: CRTS in K and H in the production of goods and new human

capital [the “core property” once again]

– scale effect : large countries grow faster than small countries do (L larger for the

former). This is easily falsified.



Research & Development and Economic Growth

• Key idea: purposeful conduct of R&D activities is the source of growth.

– In the absence of human and physical capital, households can nevertheless save

by accumulating patents.

– Patents are blueprint for the production of “slightly unique” products.

– The patent holder has a little bit of monopoly power which can be exploited.

– Hence, in this literature we leave the competitive framework and enter the realm

of monopolistic competition. [Schumpeterian models of “creative destruction” can

be built along the lines of the present model]

• Three productive sectors

– final goods sector

– R&D sector

– intermediate goods sectorPublic Economics – Chapter 8 Version 1.1 – November 2004

Ben J. Heijdra


• Final goods sector

– CTRS, perfectly competitive, external effect “returns to specialization”

– produces a homogenous good using differentiated inputs in the production

process.

– Technology:

Y (t) ≡ N(t)η

[N(t)−1

∫ N(t)

0

Xj(t)1/µdj

]µ

, µ > 1, 1 ≤ η ≤ 2

¦ Xj is intermediate input j

¦ N is the existing number of varieties

¦ µ and η are parameters. if η > 1 there are returns to specialization as in

Adam Smith’s famous pin factory. If intermediate inputs are more finely

differentiated then firms can use a more roundabout production process [η < 2

is a mild and reasonable assumption used later]



– Pricing decision; price equals marginal cost:

PY (t) ≡ N(t)−η

[N(t)µ/(1−µ)

∫ N(t)

0

Pj(t)1/(1−µ)dj

]1−µ

– Derived demand for input j:

Xj(t)

Y (t)= N(t)(η−µ)/(µ−1)

(Pj(t)

PY (t)

)µ/(1−µ)

, j ∈ [0, N(t)]

so µ/(1− µ) is the demand elasticity.

• R&D sector

– CRTS, perfectly competitive, external effect “standing on the shoulder of giants”

– produces blueprints for new intermediate inputs, using labour as an input.

– Technology:

N(t) = (1/kR)N(t)LR(t)



Labour engaged in the R&D sector becomes more productive as more patents

already exist. Today’s engineers “stand on the shoulders of giants.”

– Pricing decision:

PN(t) =kR (1− sR) W (t)

N(t)

where sR is a wage subsidy in the R&D sector.

• Intermediate goods sector

– many small monopolistically competitive firms: each firm [patent holder] uses

labour to produce its own slightly unique variety of the intermediate input.

– Technology:

Xj(t) = (1/kX)Lj(t)

constant marginal production costs.



– Pricing decision:

Pj(t) = µW (t)kX ,

where µ is the gross monopoly markup.

• Households

– lifetime utility function:

Λ(0) =

∫ ∞

0

[C(t)1−1/σ − 1

1− 1/σ

]e−ρtdt

– household budget identity:

PY (t)C(t) + PN(t)N(t) = W (t)L + N(t)Π(t) + Z (t)

where Z (t) is lump-sum transfers from the government [or tax, if Z (t) < 0]



– optimality conditions:

C(t)

C(t)= σ [r(t)− ρ]

r(t) =Π(t) + PN(t)

PN(t)

where r(t) is the rate of return on blueprints.

• Growth aspects: there is no transitional dynamics [no capital]. The growth rates are:

γN =σ(µ− 1)(L/kR)− σρ (1− sR)

σ (µ− 1) + [σ (2− η) + η − 1] (1− sR)> 0

γC = γY = (η − 1)γN

– The innovation rate increases with the monopoly markup (µ), the R&D subsidy

(sR), and the size of the labour force (L), and decreases with the rate of time

preference (ρ).



– Consumption and aggregate output grow only if the returns to specialization are

operative (so that η > 1)

– The R&D subsidy affects growth .....

• Closing remarks on the R&D model

– engine of growth: N(t) = (1/kR)N(t)LR(t). Linear in N

– Efficiency aspects: now no longer obvious that market equilibrium is efficient as

there are both external effects and non-competitive behaviour. The

quick-and-dirty intuition would seem to suggest that there is too little innovation

[under-investment in R&D] because the innovator does not capture all the

beneficial effects of his act. It turns out that:

¦ if η = µ [knife-edge case]: q&d intuition is OK!

¦ if η ≈ 1 [weak specialization effect]: there may be too much innovation!



– Problematic aspect : the growth rate depends on the scale of the economy (L in

this case). Hence, large countries should grow faster than small countries. This is

not observed in reality. Jones removes the scale effect by replacing the R&D

technology by:

N(t) = (1/kR)LR(t)N(t)φ1[LR(t)

]φ2−1,

where LR is average R&D labour per R&D firm.

¦ We had φ1 = 1 but now assume 0 < φ1 < 1 [the giants don’t grow forever]

¦ We had φ2 = 1 but now assume 0 < φ2 ≤ 1 [duplication externality:

individual R&D firms think the production function is linear, but in actuality it

features diminishing returns to labour]



¦ Assuming that the population grows at an exponential rate, nL and setting

sR = 0 the growth rates are now [see book for details]:

γN =φ2nL

1− φ1

γy = γY − nL = (µ− 1)γN

γc = γC − nL = γy

¦ We reach the striking conclusion that by eliminating the scale effect we are

back in the realm of exogenous growth and the Solow model!!!



Digression: Deriving the Growth Rate in the R & D Model

• following the steps outlined in H&vdP (2002, pp. 465-466) we find:

Π(t)

PN(t)= (µ− 1)

(kXN(t)X(t)

kR (1− sR)

)(D1)

PN(t)

PN(t)= (η − 2)

(N(t)

N(t)

)(D2)

C(t) = N(t)η−1N(t)X(t) (D3)



• assuming a time-invariant subsidy (sR = 0) we obtain:

γC(t) = σ

[(µ− 1

kR (1− sR)

)LX(t) + (η − 2)γN(t)− ρ

](D4)

γC(t) = (η − 1)γN(t) +LX(t)

LX (t)(D5)

γN(t) =L− LX(t)

kR

(D6)

• using (D4)-(D6) we derive the following differential equation:

LX(t)

LX (t)=

σ (µ− 1)

kR (1− sR)LX(t)− σρ

+ [σ(2− η) + (η − 1)]

(LX (t)− L

kR

)(D7)



– it follows that:

∂

∂LX (t)

[LX(t)

LX (t)

]=

σ (µ− 1)

kR (1− sR)+

[σ(2− η) + (η − 1)]

kR

> 0

since µ > 1 and 1 ≤ η < 2.

– (D7) is an unstable differential equation for which the only economically sensible

solution is LX (t) = 0. It follows that:

L− LX

kR

=σ(µ− 1)(L/kR)− σρ (1− sR)

σ (µ− 1) + [σ (2− η) + η − 1] (1− sR)

– using this result in (D6) we find the growth rate for N , C , and Y :

γN =σ(µ− 1)(L/kR)− σρ (1− sR)

σ (µ− 1) + [σ (2− η) + η − 1] (1− sR)> 0

γC = γY = (η − 1)γN


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