Dynamic inefficiency and fiscal interventions in an economy with land and transaction costs

Proof of Proposition 3.1.

I first show that, under the specified assumptions, for any E, a, π, L 0 , the equation

has a unique solution p ∗ ( E , a , π , L 0 ). This equation is equivalent to the equation

If p is close to zero, the left-hand side of (A.2) is close to u ′ ( E ), and the right-hand side of (A.2) is very large. If p is close to (but less than) E / L 0 , the left-hand side of (A.2) is very large, and the right-hand side of (A.2) is close to v ′ ( a L 0 + ( 1 − π ) E ) · ( a + ( 1 − π ) E / L 0 ). Because both sides of (A.1) are continuous in p, the intermediate-value theorem implies that (A.2) has a solution. Under the given assumptions, the left-hand side of (A.2) is increasing in p, and the right-hand side is decreasing in p. Therefore the solution to (A.2) is unique.

Next I note that, if p t + 1 = p t = p, then a necessary and sufficient condition for a solution to the maximization problem of generation t is given by the equation

in combination with the constraints in (2.2). Hence if p is equal to the solution p ∗ ( E , a , π , L 0 ) to equation (A.1), a necessary and sufficient condition for a solution to the maximization problem of generation t is to have

as well as

and

At this point, it is easy to see that the triples ( c 2 t − 1 , c 1 t , L t ) that are given by (A.4)–(A.6) satisfy conditions (2.3) and (2.4) for all t. Hence, by setting p t = p ∗ ( E , a , π , L 0 ) for all t and using (A.4)–(A.6) to specify an allocation, one obtains a stationary equilibrium. Uniqueness follows upon observing that the equilibrium condition (2.4) implies (A.4) for all t, so (A.3) is equivalent to (A.1), which has p ∗ ( E , a , π , L 0 ) as its only solution.

To prove the claimed comparative-statics properties, I note that under the stated assumptions, an increase in E makes the left-hand side of (A.2) go down without affecting the right-hand side. Since, by the above argument, the difference between the left-hand side and the right-hand side is increasing in p. Following an increase in E, therefore, an increase in p is required to restore equality. An increase in L 0 makes the left-hand side of (A.2) go up and the right-hand side go down, so a decrease in p is needed to restore equality.

To see that lim E → ∞ p ∗ ( E , a , π , L 0 ) = ∞, it suffices to observe that, if p ∗ ( E , a , π , L 0 ) were bounded as E goes out of bounds, then, for p = p ∗ ( E , a , π , L 0 ), the left-hand side of (A.1) would converge to zero and the right-hand side would converge to a positive limit. Similarly, if p ∗ ( E , a , π , L 0 ) were bounded as L 0 becomes small, the left-hand side of (A.1) would converge to u ′ ( E ) and the right-hand side of (A.1) would go out of bounds. In either case, the validity of (A.1) at p = p ∗ ( E , a , π , L 0 ) would be violated. Hence p ∗ ( E , a , π , L 0 ) must satisfy (3.1).  □

Proof of Proposition 3.3 (a).

The line of argument is the same as in the proof of the First Welfare Theorem for competitive equilibria in a complete market system. Given that, for any t ≥ 1, the triple ( c 1 t , c 2 t , L t ) maximizes (2.1) subject to (2.2), it must also be the case that, for any t ≥ 1, the pair ( c 1 t , c 2 t ) maximizes (2.1) subject to

Given the stationarity of the equilibrium, with p t = p ∗ ( E , a , π , L 0 ) for all t, and given the strict monotonicity of (2.1), this constraint may be rewritten as

where

If statement (a) of the proposition is false, there exists an alternative allocation { c ˆ 2 t − 1 , c ˆ 1 t } t = 1 ∞ of nonnegative consumption levels satisfying (3.8) for all t such that

and, for t = 1 , 2 , … ,

and at least one of the inequalities in (A.10) and (A.11) is strict. Because, for t ≥ 1, the pair ( c 1 t , c 2 t ) maximizes (2.1) subject to (A.8), it follows that, for t ≥ 1,

and at least one of the inequalities in (A.10) and (A.12) is strict. The inequalities in (A.12) are equivalent to the inequalities

Upon adding these inequalities over t = 1 , 2 , … and adding (A.10), using the fact that at least one of the inequalities is strict, one obtains

where the infinite sums are well defined because r > 0 and the consumption variables are uniformly bounded by E + a L 0 . Upon reordering sums, one finds that (A.14) is equivalent to the inequality

which is incompatible with (3.8) holding for all t ≥ 1. The assumption that statement (a) of the proposition is false has thus led to a contradiction.  □

Proof of Proposition 3.3 (b).

Given the stationarity of the equilibrium, let c 1 ∗ , c 2 ∗ be the common values of c 1 t , c 2 t − 1 , t = 1 , 2 , … By the first-order condition (3.5) and the assumption that a p ∗ ( E , a , π , L 0 ) − π < 0, there exists a pair ( c ˆ 1 ∗ , c ˆ 2 ∗ ) such that

Upon setting ( c ˆ 2 t − 1 , c ˆ 1 t ) = ( c ˆ 2 ∗ , c ˆ 1 ∗ ) for all t, one obtains an allocation { c ˆ 2 t − 1 , c ˆ 1 t } t = 1 ∞ of consumption levels that satisfies (3.8) for all t and that Pareto dominates the equilibrium allocation { c 2 t − 1 , c 1 t } t = 1 ∞ . Therefore the equilibrium allocation { c 2 t − 1 , c 1 t } t = 1 ∞ is not weakly efficient.  □

Proof of Proposition 4.1.

For uniqueness and for the comparative statics with respect to E, as well as the limit (4.8), the argument is the same as in the proof of Proposition 3.1, with equation (A.2) replaced by the equation

The details are left to the reader. For the comparative statics with respect to T and S, it suffices to observe that increases in T and S make the difference between the left-hand side and the right-hand side of (A.19) go up, so a decrease in p is needed to restore equality in (A.19).  □

Proof of Proposition 4.2.

Dropping the dependence on other parameters, for any T, let p ( T ), c ˆ 1 ( T ) and c ˆ 2 ( T ) be the common values of p t , c ˆ 1 t and c ˆ 2 t − 1 , t = 1 , 2 , … , in the stationary equilibria with parameters T , S = ( 1 − σ ) T , E , a , π , L 0 . By the implicit function theorem to (A.19), p ( T ) is continuously differentiable; its derivative is:

Thus, under the given assumptions on u and v,

for all T.

Since

and

it follows that c ˆ 1 ( T ) is decreasing and c ˆ 2 ( T ) is increasing in T.

To assess the welfare implications of the fiscal intervention, I note that people born in date 0, i. e., the old generation at date 1, benefit from the change because their consumption is equal to c ˆ 2 ( T ), which goes up with T. For people born in t ≥ 1, the effects of T on the lifetime

are computed as

Upon using (A.19) to substitute for u ′ ( c ˆ 1 ( T ) ), one further obtains

The second term on the right-hand side is positive because p ′ ( T ) < 0. The first term on the right-hand side is also positive if

or, equivalently,

The proposition follows immediately.  □

Proof of Proposition 4.3.

The argument is the same as in the proof of Proposition 3.1, with equation (A.2) replaced by the equation

where R ( τ , s , p ) = ( 1 − τ ) r ( p ) + s. The left-hand side of (A.27) is increasing in p. The term R ( τ , s , p ) on the right-hand side is decreasing in p; the term R ( τ , s , p ) p L 0 inside the marginal utility on the right-hand side is increasing in p, so the right-hand side altogether is decreasing in p. Therefore equation (A.27) has at most one solution. Existence of a solution follows from boundary behaviour and the intermediate-value theorem, as in the proof of Proposition 3.1. For monotonicity with respect to E and the limit in (4.15), the argument is also the same as in the proof of Proposition 3.1. The details are left to the reader.  □

Lemma A.1.

Let ( c 1 ( R ) , c 2 ( R ) ) maximize u ( c 1 ) + v ( c 2 ) under the constraint c 1 + 1 R c 2 = E. Then c 2 ( R ) is increasing in R; c 1 ( R ) is increasing in R if v ′ ( c ) + c v ″ ( c ) ≤ 0 for all c and decreasing in R if v ′ ( c ) + c v ″ ( c ) > 0 for all c.

Proof.

The first-order condition for ( c 1 ( R ) , c 2 ( R ) ) is

Total differentiation yields

From the constraint, we also have

Upon combining (A.29) and (A.30), one obtains

and

The lemma follows immediately.  □

Proof of Propositions 4.4 and 4.5.

Dropping the dependence on other parameters, for any τ, let p ( τ ), c ˆ 1 ( τ ) and c ˆ 2 ( τ ) be the common values of p, c ˆ 1 t , c ˆ 2 t − 1 , t = 1 , 2 , … , in the stationary equilibria with parameters τ , s = ( 1 − σ ) τ , E , a , π , L 0 . I first show that, if r ( p ( τ ) ) < 1 − σ, then R ∗ ( τ , p ( τ ) ) is increasing in τ. For suppose that, for some τ and Δ > 0, we have

Because, with r ( p ( τ ) ) < 1 − σ, R ∗ ( τ , p ) is increasing in τ and decreasing in p, it follows that p ( τ + Δ ) > p ( τ ). Then also R ∗ ( τ + Δ , p ( τ + Δ ) ) · p ( τ + Δ ), and therefore

Since c ˆ 2 ( τ + Δ ) = c 2 ( R ∗ ( τ + Δ , p ( τ + Δ ) ) ) and c ˆ 2 ( τ ) = c 2 ( R ∗ ( τ , p ( τ ) ) ), by Lemma A.1, (A.32) implies that R ∗ ( τ , p ( τ ) ) < R ∗ ( τ + Δ , p ( τ + Δ ) ), contrary to (A.31). The assumption that R ∗ ( τ , p ( τ ) ) fails to be increasing in τ has thus led to a contradiction and must be false. By Lemma A.1, it follows that c ˆ 2 ( τ ) must also be increasing in τ. The first statement of Proposition 4.5 is thereby proved.

Proposition 4.4 follows because, with R ∗ ( τ , p ( τ ) ) is increasing in τ, all generations t ≥ 1 are made better off by the change, and with c ˆ 2 ( τ ) increasing in τ, generation 0 is also made better off.

The second statement of Proposition 4.5 follows from the observation that, as R ∗ ( τ , p ( τ ) ) is increasing in τ, by Lemma A.1 and the curvature condition (4.23), c ˆ 1 ( τ ) is nondecreasing, and E − c ˆ 1 ( τ ) is nonincreasing in τ. Spending on land, p ( τ ) L 0 = ( 1 − τ ) ( E − c ˆ 1 ( τ ) ), is therefore decreasing in τ. Thus p ( τ ) goes down and r ( p ( τ ) ) goes up as τ is increased.  □

Proof of Proposition 5.1.

Since L 0 and M 0 are strictly positive, any stationary monetary equilibrium satisfies L t > 0 and M t > 0 for all t. In every period, therefore, the rates of return on land and money must be the same, i. e., the prices p t , q t , t = 1 , 2 , … must satisfy the equation

for all t. With stationarity, this equation takes the form

or p = a π , as claimed in the proposition. The equilibrium net real rate of return on land, a p − π must therefore be equal to zero. With this rate of return, utility maximization requires that, for any t ≥ 1, ( c 1 t , c 2 t ) must maximize u ( c 1 ) + v ( c 2 ) under the constraint c 1 + c 2 = E. Moreover, the first-period savings E − c 1 must be equal to the sum p L t + q M t = p L 0 + q M 0 of the values of land and money held. Since p = a π , it follows that q = [ E − c 1 − a π L 0 ] / M 0 , as claimed in the proposition.

For a monetary equilibrium, q must be positive, i. e. we must have E − c 1 > a π L 0 . I claim that this is not possible if a p ∗ − π ≥ 0, where, for simplicity, I have dropped the arguments from p ∗ = p ∗ ( E , a , π , L 0 ). To prove this claim, suppose that we have E − c 1 > a π L 0 and a p ∗ − π ≥ 0. Then, trivially, a π ≥ p ∗ and therefore

By (A.34), r a π = 1. By (A.35) and the constraint c 1 + c 2 = E, it follows that

By the budget constraints in (2.2), r ( p ∗ ) · p ∗ · L 0 = c 2 ( r ( p ∗ ) ), where, for any r, c 2 ( r ) maximizes u ( E − 1 r c 2 ) + v ( c 2 ). Thus, (A.36) implies c 2 > c 2 ( r ( p ∗ ) ). However, one easily verifies that the function c 2 ( · ) must be nondecreasing. Since c 2 = c 2 ( 1 ) and, by (A.34) and (A.35), 1 = r ( a π ) ≤ r ( p ∗ ), it follows c 2 ≤ c 2 ( r ( p ∗ ) ). The assumption that we can have E − c 1 > a π L 0 and a p ∗ − π ≥ 0 has thus led to a contradiction and must be false.

Conversely, suppose that a p ∗ − π < 0, where again p ∗ is shorthand for p ∗ ( E , a , π , L 0 ). Then a π < p ∗ and therefore

Using (A.34), (A.37) and the constraint c 1 + c 2 = E, one now obtains

so q = [ E − c 1 − a π L 0 ] / M 0 is strictly positive. One easily verifies that the specified price sequence and allocation is indeed a stationary monetary equilibrium.

To see that this equilibrium Pareto-dominates the stationary equilibrium under laissez-faire without money, it suffices to observe that second-period consumption is higher – so generation 0 is better off – and the equilibrium rate of return on assets is also higher – so later generations are also better off.

Pareto efficiency follows by the arguments of Okuno and Zilcha (1980) for the case when the interest rate is equal to the growth rate of the economy.  □

Lemma B.1.

Under the given assumptions about utility functions and about the random returns on real investments, for any r > 0, the functions φ and ψ in (B.6) satisfy

with ψ ( r ) = 1 for r ≥ E θ ˜ t + 1 . At any r ∈ ( 0 , E θ ˜ t + 1 ), φ ( r ) is decreasing and ψ ( r ) is increasing in r.

Proof.

By (B.6) and (B.7), the first term in (B.5) is equal to ln ( E 2 ) and the second term equal to

The pair ( φ ( r ( p ) ) , ψ ( r ( p ) ) ) must maximize (B.8) under the constraint (B.7). By the strict concavity of the logarithm, it follows that r ( p ) ≥ E θ ˜ t + 1 implies φ ( r ( p ) ) = 0 and ψ ( r ( p ) ) = 1.

If r ( p ) < E θ ˜ t + 1 , then, at the point φ = 0, ψ = 1, the derivatives of (B.8) with respect to φ and ψ are

so an increase in φ combined with an equal-sized decrease in ψ raises (B.8), proving that the pair ( φ , ψ ) = ( 0 , 1 ) does not maximize (B.8) subject to (B.7) and that, therefore, φ ( r ( p ) ) > 0 and ψ ( r ( p ) ) < 1 if r ( p ) ∈ ( 0 , E θ ˜ t + 1 ).

Maximization of (B.8) also requires ψ ( r ( p ) ) > 0 whenever r ( p ) > 0; if we had r ( p ) > 0 and ψ ( r ( p ) ) = 0, then, because Pr { θ ˜ t + 1 = 0 } > 0, the derivative of (B.8) with respect to ψ would be unboundedly positive.

At any r ( p ) ∈ ( 0 , E θ ˜ t + 1 ), we thus have φ ( r ( p ) ) > 0 and ψ ( r ( p ) ) > 0, so the pair ( φ ( r ( p ) ) , ψ ( r ( p ) ) ) must satisfy the first-order condition for an interior maximum,

Because the logarithmic function exhibits decreasing absolute risk aversion, by standard arguments, (B.9) implies that φ ( r ) is decreasing and ψ ( r ) is increasing in r at r = r ( p ) ∈ ( 0 , E θ ˜ t + 1 ).  □

Proposition B.2.

For given E, a, π, L 0 , the model with real investments and with logarithmic utility has a unique stationary equilibrium. The equilibrium land price is increasing in the endowment E, with

If E is sufficiently large, the stationary-equilibrium net real rate of return on land,

is negative, and the stationary equilibrium is not weakly efficient.

Proof.

Market clearing requires that the (stationary) demand for land L be equal to the available stock L 0 . By (B.6), this condition holds if and only if the equilibrium land price satisfies the equation

If p is close to zero, r ( p ) is very large so the right-hand side of (B.12) is E 2 L 0 > p. If p is very large, the right-hand side of (B.12) is no larger than E 2 L 0 , which is less than p. Existence of a market-clearing land price follows by the intermediate-value theorem. Uniqueness follows by the monotonicity of the difference p − 1 L 0 · ψ ( r ( p ) ) · E 2 in p. Monotonicity of the equilibrium land price in E also follows by the monotonicity of the difference p − 1 L 0 · ψ ( r ( p ) ) · E 2 . Finally, (B.10) follows from the observation that, by definition, r ( p ) > 1 − π and therefore ψ ( r ( p ) ) > ψ ( 1 − π ) for all p. Thus

for all π, and (B.10) follows because, by Lemma B.1, ψ ( 1 − π ) > 0.

The last statement of the proposition follows by the same argument as in Propositions 3.2 and 3.3.  □

Remark B.3.

In any stationary equilibrium, whether efficient or not, the expected rate of return on real investment exceeds the real growth rate, i. e., E θ ˜ t + 1 > 1 for all t.

Proposition B.4.

Suppose that E is large enough so that the stationary-equilibrium net real rate of return on land (B.11) under laissez-faire is negative, i. e., p ∗ ( E , a , π , L 0 ) > a π . Then, for

there exists a Pareto-dominating stationary equilibrium of the model with real investments and with logarithmic utility, with a lump-sum tax T in the first period of people’s lives and a lump-sum transfer S in the second period, such that the stationary equilibrium land price is p ˆ = a π , with r ( p ˆ ) = 1, and the stationary-equilibrium real investment is I ˆ = φ ( 1 ) E 2 .

Proof.

Given T, S, p ˆ, and r ( p ˆ ) = 1, the problem of any generation t ≥ 1 is to maximize (2.2) under the constraints

and

This is equivalent to the problem of maximizing (2.2) under the consolidated constraint

The consolidated constraint is exactly the same as in a situation without the lump-sum tax and transfer, but with the price p ˆ and rate of return on land r ( p ˆ ) = 1. The solution to the problem of maximizing (2.2) subject to (B.16) is therefore also the same, namely c 1 t = E 2 , I t = φ ( 1 ) · E 2 , and c ˜ 2 t = ( θ ˜ t + 1 − 1 ) φ ( 1 ) · E 2 + E 2 . By (B.14) and (B.13), the demand for land is

which is equal to the stock of land L 0 if and only if p = a π . Pareto dominance of the equilibrium with lump-sum taxes and transfers over the laissez-faire equilibrium follows because the rate of return on land is higher and second-period expected utility is also higher.  □