Public Goods as Drivers of Private Investment

Public Goods as Drivers of Private Investment

Ram Singh

February 2^nd, 2018

The Model

The Model

We begin with a comparison of two localities (l,h) over an infinite time horizon.

Each locality is owned by a representative individual owning fixed amount of landL

Welfare of localityi at timet isaⁱ_t ∈[0,a]⊂R+, wheret ∈Z+

acaptures the various amenities provided by the locality to it’s residents, witha0representing the initial amenities ofbothlocalities at time 0 Land can only be used to provide housing facilities and housing of localityi at timet, given byH_tⁱ(Lⁱ,K_tⁱ) is

H_tⁱ(Lⁱ,K_tⁱ) = (Lⁱ)^α(K_tⁱ)^1−αwhereα∈(0,1) andi∈ {l,h} (0.1) Note that land has no time subscript—it is fixed on a per-locality basis.

The Model

Per period profit from housing in each locality is

πⁱ_t(aⁱ_t−1) =p_tⁱ(aⁱ_t−1)H_tⁱ−rK_tⁱ withaⁱ₀given fori∈ {l,h} (0.2) Here,

r is the rental rate of capital

pⁱ_t(aⁱ_t−1) is the per unit price of housing services. It is dependant on the amenities in the localityin the previous period.

Agents look back at the state of the locality yesterday in determining their willingness to pay for housing today.

Specifically, we consider

pⁱ_t(aⁱ_t−1) = r

1−α aⁱ_t−1^α

(0.3) Given our assumptions, this is both increasing and concave inaⁱ_t−1

The Model

Optimal Capital Investment

Substituting the value ofH_tⁱ from (0.1) in (0.2), the FOC wrt toK_tⁱ is (1−α)p_tⁱ(aⁱ_t−1)

Lⁱ K_tⁱ

^α

−r = 0

⇒K_tⁱ =

(1−α)pⁱ_t(aⁱ_t−1) r

α¹

Lⁱ

⇒ K_tⁱ

Lⁱ ≡k_tⁱ(aⁱ_t−1) =

(1−α)p_tⁱ(aⁱ_t−1) r

α¹

Here,k_tⁱ is the optimal capital-land ratio in localityi at timet. From (0.3), we can see that

k_tⁱ(aⁱ_t−1) =aⁱ_t−1 (0.4) Investment in land today is simply the level of amenities yesterday. Clearly, investment is increasing in amenities.

The Model

Amenities and Investment

We now model amenities and investment. We have the followinglocality-specific amenity function which depends on the current level of investment per unit of land.

aⁱ_t(k_tⁱ) =







log(1 +ck_tⁱ), i=l B^(1+k_1−m^ti)1−m−1, i=h

(0.5) Here,c,mandB are positive constants withB,c,m>1 and^B/c>1 +a.

The richer locality has a natural advantage due to higher pre-existing public investment (infrastructure, health, education)

Because of this, amenities are more responsive to private investment (K_tⁱ) Operation of this mechanism is depicted by the use of log for localityl and iso-elastic (CRRA) for localityh.

Amenities and Investment

Amenities and Investment

Both amenity functions areincreasing andconcave, but the log function dampens much quicker and is thus flatter than the iso-elastic functions.

k k

a^l =k a=k

log(1 +ck_tⁱ) B^(1+kit_1−m^)1−m−1

k_tⁱ ai t

A Steady State

A Steady State

Define ¯k ask :a^h(k) = ¯a, i.e. the level of investment per unit land that results in maximum amenities for the high locality.

Assume that capital employed beyond ¯k doesn’t translate into higher amenities (as amenities are bounded above by ¯a), thus capital will be bounded above by ¯k. Therefore we have

a^l_t: [0,k]→[0,k] defined by

a^l_t = log(1 +ck_t^l) (0.6)

Since

the domain iscompactandconvex

and the amenity function iscontinuousover the domain We can useBrouwer’s fixed point theoremto conclude that:

∃k ∈[0,k] :a^l_t(k) =k ∀ t (0.7)

A Steady State

A Steady State (cont)

Clearly 0 is one such trivial fixed point.

But we are interested in another. Either

Choosec=^ek_k⁻¹>1 for somek ∈(0,k), which guarantees the existence of such a fixed point

Alternatively, we can find the steady state by equatinga^l_t+1=a^l_t k_t+1^l = log(1 +ck_t^l)

At steady state,k_t^l =k_t+1^l =k

⇒k= log(1 +ck)

⇒e^k = 1 +ck

⇒e^k −1

k =c

Given our assumption onc, this implies thatk =k

Proofs

Proposition 1

Given(0.4),(0.6),(0.7), the sequence of amenities for locality l will converge to k which is a stable level of capital

Proof.

Case 1 : a₀^l <k

Define the mappingA^l_t≡a^l_t◦k_t^l ∀t. Thus given previous period’s amenity a^l_t−1, A^l_t gives the current period’s amenity (a^l_t).

For the given (restricted) domain,A^l_t has the following form:

A^l_t(a^l_t−1)≡a^l_t◦k_t^l(a^l_t−1) = log(1 +ca_t−1^l )∀t

LetA^l_t(a_t−1^l ) be denoted simply asA^l_t. We thus have a sequence{A^l_t}^∞_t=0 of amenities withA^l₀=a₀^l given.

Our task is to show that this sequence converges tok

Proofs

at−2 at−1 at at+1 ¯al

At−1 At At+1

Proofs

Proof. (Cont.)

By construction, ifa^l_t<k then log(1 +ck_t^l)>k_t^l.

We have thus shown that if the initial amenity level is belowk then the sequence of amenities is an increasing sequence as long as each member of the sequence is less thank

Sincek is a fixed point of the amenity function and the amenity function is increasing and concave, we have thatk is theLeast Upper Bound of the sequence {A^l_t}^∞_t=0 i.e. A^l_t ≤k ∀t.

Proofs

Proof. (Cont.)

Since every increasing and bounded above sequence converges to it’ssupremum, thusA^l_t →k

Case 2 : k <a^l₀

Now log(1 +ck_t^l)<k_t^l i.e. A^l_t <A^l_t−1∀t

Sincek is a stable steady state, the decreasing sequence{A^l_t}^∞_t=0 converges to it’sgreatest lower bound k

Now, letγ(a^l_t)≡^at+1l _a^−a_l ^tl

t

be the rate of growth of amenities for localityl.

We have already shown thata^l_t+1≶a^l_t fora_t^l ≷a^l and therefore, γ(a^l_t) = a^l_t+1

a^l_t −1≶0 ,a^l_t ≷a^l

Therefore, ¯al is the steady state level of amenity for localityl.

Proofs

We now restrict attention to the other locality and look at amenity function in equation (0.5).

a_t^h: [0,k]→[0,k] defined as a^h_t(k_t^h) =B(1 +k_t^h)^1−m−1

1−m (0.8)

Since it iscontinuouson a compactandconvexdomain, byBrouwer’s fixed point theoremwe have a fixed point.

ais the desirable level of amenity forany localityand localityhwill achieve it in the long run. To ensure this, we fixB at

B = a(1−m)

(1 +a)^1−m−1 (0.9)

so thata^h_t(a) =a.

Sincea^h_t(.) isincreasingandconcave, there exists a unique, non-trivial fixed point atk=a.

Proofs

Proposition 2

Given(0.4),(0.5),(0.8),(0.9)and that a^h₀∈(0,a), locality h will achieve the best possible amenity in the long run and this steady state is stable.

Proof.

DefineA^h_t ≡a^h_t ◦k_t^h∀ t.

A^h_t(a^h_t−1) =B(1 +a^h_t−1)^1−m−1 1−m

A^h_t is the amenity of the locality at timet and it is a function of the previous period’s amenity of the locality.

Assumption (0.9) and the fact thata is the unique fixed point of an increasing, concavefunction guarantee that {A^h_t}^∞_t=0 is an increasing

sequence which converges to it’ssupremum a.