“The” Optimal Growth Path for the Economy and Optimal Discount Rates for Investment Decisions

Abstract

Chapters 1, 2, 3, and 4 refers to a two-sector growth model for a closed and an open economy, which is the basic frame of reference for our theoretical criticism of the European Treaties.

We present the structure of the consumption/investment goods market and the assets market in the case of a closed economy, a two-country open economy and a small-country open economy.

The government is assumed to operate with standard fiscal and monetary tools. In addition, the government is also assumed to allocate its expenditure between the current account (consumption) and investment. In this respect, government expenditure is a part of this and affects the demand side of the economy.

However, government investments are assumed to enter the macro production function, improving total factor productivity and affecting the supply side of the economy. So, government expenditure does not only affect the economic system on the demand side, but, due to a component of the investments into total government expenditure, it also affects the supply side of the economy by increasing, together with private investments, the capital accumulation process of the whole economy. Direct reference here can easily be made not only to material infrastructure investments, such as energy and transport networks (gas and electricity, railways, highways, ports, airports, logistics etc.) but also to intangible infrastructure investments, such as information and communication technology (ICT) (internet, broadband connections etc.).

Therefore, since different compositions of government expenditure and taxes lead to a different growth path for the economy, the impact of the government budget on the economy cannot exclusively relate to deficit and debt.

The parameters established in the Maastricht Treaty and the ones given in the ECB Statute are “static”, while within a dynamic framework, deficit and debt are directly affected by the level and the composition of government expenditure and revenue through their effects on the growth path. Therefore, they need to be adjusted and updated according to what a consolidated theory would suggest.

Appendix: Optimal Growth Path in the Case that Both Consumption and Government Investments Enter the Welfare Function

Government intervention in the economy in the pursuit of “social” targets has long been, and still is, a hotly debated issue both in theory and practice.

Many contributions have refused to attach any particular benefit to public policy, seeing it as causing a distortive reallocation of resources within a market system. Under a static framework, competitive equilibrium has been proved to represent a Paretian optimum solution, defined according to the original proposal of Pareto and Barone.⁹ Government policy may then be called to guarantee the competitive framework, or to deal with the presence of “externalities”, or to meet an income distribution target. In the first two cases the necessary conditions on the convexity of the functions both within the consumption and production sectors are not verified, and public policy can be assigned the goal of filling the gap. In the last case, the target is completely external to Paretian lines since for “any” given income distribution a Pareto-optimum solution can be proved to exist.

As it is well-known, the validity of this approach depends on the existence of a stable competitive equilibrium. Such conditions cannot always be met. Some authors would rather support the idea that instability is the most general rule.

Further, if the Keynesian case of under-employment equilibrium is referred to, government policy is urgently needed for the system to make fuller use of its resources. Within a dynamic framework, further arguments can be made for the evaluation of public intervention. First, the Ramseyan criterion, used in the previous section, beyond the interpersonal measuring of utility, may also refer to subjective or social parameters. Second, an exact equivalent between dynamic competitive equilibrium and social optimum cannot be proven to exist, nor can a competitive economy necessarily enter optimal paths spontaneously.

Placed on such a basis, fiscal and monetary policies have, by and large, represented “the” tool for achieving long-run growth targets, and for the fine tuning of short-run stabilization. Government investments have very seldom been used.

We have, however, shown that such a policy tool presents additional possibilities. Previous chapters examined this case within a very general economic framework. Welfare conditions under pure consumption maximization were also explored.

However, government intervention in the economy may not be limited to the simple long-run target of consumption maximization. Several other parameters may, in fact, be considered, and government agencies may appropriately be called upon to pursue them.

This appendix, therefore, analyses the alternative paths that an economy might follow to maximize welfare conditions, given by a multi-parameter target function.

Several combinations of private and social targets could be of interest. We will limit ourselves to considering only those targets that are relevant to the role that government investment might play.

First, consider government expenditure for consumption goods entering the welfare function in a way different from per-capita private consumption. In this case, the traditional trade-off between private and public consumption is met, within the particular framework we introduced. Indeed, we saw how government investment expenditure can influence per-capita steady-state private consumption. Therefore, in our analysis, the direct impact depends on the allocation of government expenditure to consumption and investment. As long as a social utility is granted to public consumption, the first flow directly affects welfare conditions. On the other hand, the second flow, investments, has its impact through making available a greater amount of private consumption goods. A further hypothesis refers to the case in which utility is also granted to government investments, per se. Indeed, they might be assigned particular targets resulting in the attainment of welfare gains.

The function to be maximized would then include three different parameters: private consumption; public consumption; and government investments. Both government budget and national income identity would then be operative.

For the sake of simplicity, we assume that government and private consumption enter the welfare function in the same way, i.e. only total per-capita consumption will be referred to.

The welfare function is then given by:

$$ \underset{0}{\overset{\infty }{\int }}{e}^{-\varrho t}U\left({q}_c,{k}^G\right)dt $$

(A.4.1)

which has to be maximized with respect to the following constraints:

$$ \dot{k}={q}_I\left({k}^T,{p}_k\right)-\left( he/{p}_k\right)-nk $$

(A.4.2)

$$ he=\beta {p}_k{q}_i\left({k}^T,{p}_k\right) $$

(A.4.3)

$$ {k}^G={k}^T\beta $$

(A.4.4)

$$ {k}^T=k+{k}^G $$

(A.4.5)

$$ {q}_c\left({k}^T,{p}_k\right)-\left(1-h\right)e=\left(1-s\right)\left[q\left({k}^T,{p}_k\right)+\left(d,{\pi}_mg\right){p}_m-e\right] $$

(A.4.6)

$$ \dot{g}=d- ng $$

(A.4.7)

Now, we can solve Eq. (A.4.6) for “d” and substitute it into Eq. (A.4.1) to obtain:

$$ \dot{g}=\frac{s{q}_c+\beta {p}_k{q}_I- se-\left(1-s\right){q}_I}{\left(1-s\right){p}_m}-\left(n+{\pi}_m\right)g $$

(A.4.7′)

where the constraint Eq. (A.4.3) is also considered. Further, the relations (A.4.3), (A.4.4) and (A.4.5) can be used to transform Eq. (A.4.2) into:

$$ \dot{k}=\left(1-\beta \right){q}_I-n\left(1-\beta \right){k}^T $$

(A.4.2′)

The problem is now the maximization of Eq. (A.4.1) subject to Eqs. (A.4.2′) and (A.4.7′).

The Lagrangian can then be expressed as:

$$ L=U\left({q}_c,\beta {k}^T\right)+{\lambda}_1\left(1-\beta \right){q}_I-n\left(1-\beta \right){k}^T+{\lambda}_2\left[\frac{s{q}_c+\beta {p}_k{q}_I-se-\left(1-s\right){q}_I}{\left(1-s\right){p}_m}-\left(n+{\pi}_m\right)g\right] $$

(A.4.8)

Now, we can assign to fiscal policy the target of stabilizing the rate of inflation, π _m , at some value, ${\pi}_m^{\ast }$. As mentioned before, monetary policy has to meet certain target levels of the price of capital, ${p}_k^{\ast }$.

Therefore, given the private propensity to save and the steady-state variables, k ^T and g, the optimal growth path will be determined by using the instrument β, i.e. the intensity of government capital obtained through the government propensity to invest, h, as in Eq. (A.4.3).

The first first-order condition is:

$$ \frac{\partial L}{\partial \beta }=0={U}_{\beta }{k}^T-{\lambda}_1\left[{q}_I-n{k}^T\right]+{\lambda}_2\frac{p_k{q}_I}{\left(1-s\right){p}_m} $$

which gives:

$$ {\lambda}_1=\frac{U_{\beta }+{\lambda}_2\left[{p}_k{q}_I/\left(1-s\right){p}_m\right]}{\left({q}_I-n{k}^T\right)} $$

(A.4.9)

$$ \frac{\partial L}{\partial g}=0=-{\lambda}_2\left(n+{\pi}_m\right)\kern1.25em {\lambda}_2=0 $$

(A.4.10)

which shows a zero shadow price of government debt, since no constraints are so far considered in the growth of g.

The second first-order condition is:

$$ \frac{\partial L}{\partial {k}^T}={U}_{k^T}\beta +{U}_{q_c}\frac{\partial {q}_c}{\partial {k}^T}+{\lambda}_1\left[\left(1-\beta \right)\left(\frac{\partial {q}_I}{\partial {k}^T}-n\right)\right] $$

by which we obtain, through Eq. (A.4.10),

$$ {\lambda}_1=-\frac{U_{k^T}\beta +{U}_{q_c}\frac{\partial {q}_c}{\partial {k}^T}}{\left(1-\beta \right)\left(\frac{\partial {q}_I}{\partial {k}^T}-n\right)} $$

(A.4.11)

Now, by equalizing (A.4.9) and (A.4.11), we obtain:

$$ \beta =\frac{\left[\left(r/{p}_k-n\right){U}_{\beta }{k}^T-\left({q}_I-n{k}^T\right)U{q}_cr\right]}{\left[\left(r/{p}_k-n\right){U}_{\beta }{k}^T-\left({q}_I-n{k}^T\right){U}_{k^T}\right]} $$

(A.4.12)

At any point in time along any optimal path, and for any level of consumption goods and share of government capital, β, the marginal utility of consumption has to be equal to the marginal utility of government capital. Therefore, at each instant we can verify that:

$$ {U}_{q_c}\frac{\partial {q}_c}{\partial {k}^T}={U}_{k^T}={U}_{\beta }{k}^T $$

Hence, the optimal solution for β in a steady state will be at a unity level, i.e. the whole stock of physical assets has to be owned by the government.

This result also means that an economy where any price is controlled by the government is equivalent to a fully centralized economy.

Such a conclusion may well be surprising, but it can easily be explained. We argued that fiscal and monetary policies can control the price of money, p _m , and the price of capital, p _k , both expressed in terms of the price of consumption goods, p _m , taken as numeraire. Further, we did not constrain the expansion of government debt, g. Therefore, its shadow price turned out to be zero. In fact, two upper limits can be considered as constraints on g. The first is met within the consumption goods market. Indeed, provided the rate of inflation is not zero, the effects due to the so-called inflation tax on disposable income have to be considered. We may correctly refer to a minimum level of private income related to some level of minimum consumption. We would then meet a constraint, such as:

$$ {\pi}_mg\le {\pi}_m^{\ast }{g}^{\ast } $$

Clearly, such a constraint is not met if the government maintains the price of money constant, i.e. the rate of inflation at zero.

The second constraint is met within the asset market. Remember that in our model the debt/money ratio “x”, is supposed to be moved to maintain a stock equilibrium in that market. However, a “liquidity trap” can limit the issue of money, or an aversion to government bonds can limit the issue of debt. These two cases can be expressed by a traditional “LM” curve, either perfectly elastic or totally inelastic to the rate of interest. If either of these situations is met, an additional constraint is added to the previous Eqs. (A.4.2′) and (A.4.7′). Such a constraint is given by:

$$ g\le {g}^{\ast } $$

(A.4.13)

Our problem can then be expressed as:

$$ \mathit{\operatorname{Max}}\underset{0}{\overset{\infty }{\int }}{e}^{-\varrho t}U\left({q}_c,\beta {k}^T\right)dt $$

subject to: Eqs. (A.4.2′), (A.4.8′) and (A.4.13).

The new expression for the Lagrangian is then:

$$ L=U\left({q}_c,\beta {k}^T\right)+{\lambda}_1\left[\left(1-\beta \right)-n\left(1-\beta \right){k}^T\right]+{\lambda}_2\left[\frac{s{q}_c+{\beta p}_k{q}_I-se\left(1-s\right){q}_I}{\left(1-s\right){p}_m}\right]-\left(n+{\pi}_m\right)g+{\lambda}_3\left(g-{g}^{\ast}\right) $$

(A.4.8′)

where, again we have two state variables k ^T and g, and one instrument.

Now, the three first-order conditions for maximization are given by:

$$ \frac{\partial L}{\partial \beta }=0={U}_{\beta }{k}^T-{\lambda}_1\left({q}_I-n{k}^T\right)+{\lambda}_2\left({p}_k{q}_I\right)/\left(1-s\right){p}_m $$

from which:

$$ {\lambda}_1=\frac{U_{\beta }{k}^T}{\left({q}_I-n{k}^T\right)}+{\lambda}_2\frac{p_k{q}_I}{\left({q}_I-n{k}^T\right)\left(1-s\right){p}_m} $$

(A.4.14)

and:

$$ \frac{\partial L}{\partial g}=0\kern1em {\lambda}_2\left(n+{\pi}_m\right)={\lambda}_3 $$

(A.4.15)

and:

$$ \begin{array}{l}\frac{\partial L}{\partial {k}^T}=0={U}_{q_c}r+{U}_{k^T}\beta +{\lambda}_1\left(1-\beta \right)\left(r/{p}_k-n\right)\hfill \\ {}{+\lambda}_2\left[\left( sr+\beta r-\left(1-s\right)\frac{r}{p_k}\right)\right]/\left(1-s\right){p}_m\hfill \end{array} $$

(A.4.16)

Now, we can substitute Eqs. (A.4.15) into (A.4.14) and (A.4.16):

$$ {\lambda}_1=\frac{U_{\beta }{k}^T}{\left({q}_I-n{k}^T\right)}+{\lambda}_3\frac{p_k{q}_I}{n\left({q}_I-n{k}^T\right)\left(1-s\right){p}_m} $$

(A.4.14′)

$$ \begin{array}{ll}{\lambda}_1=& \frac{U_{q_c}r}{\left(1-\beta \right)\left[n-\left(r/{p}_k\right)\right]}+\frac{{\beta U}_{k^T}}{\left(1-\beta \right)\left[n-\left(r/{p}_k\right)\right]}\hfill \\ {}& +\frac{\lambda_3}{\left(1-\beta \right)\left[n-\left(r/{p}_k\right)\right]n\left(1-s\right){p}_m}\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]\hfill \end{array} $$

(A.4.16′)

Further, λ ₁ can be eliminated by equalizing (A.4.14′) to (A.4.16′):

$$ \frac{U_{\beta }{k}^T}{\left({q}_I-n{k}^T\right)}+{\lambda}_3\frac{p_k{q}_I}{\left({q}_I-n{k}^T\right)n\left(1-s\right){p}_m}=\frac{U_{q_c}r}{\left(1-\beta \right)\left[n-\left(r/{p}_k\right)\right]}+\frac{{\beta U}_{k^T}}{\left(1-\beta \right)\left[n-\left(r/{p}_k\right)\right]}{+\lambda}_3\left[\frac{sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)}{\left(1-\beta \right)\left[n-\left(r/{p}_k\right)\right]n\left(1-s\right){p}_m}\right] $$

(A.4.17)

Therefore, given p _k and p _m , achieved by monetary policies, s and n as exogenously determined, the government intensity of capital, β, is left as a function of the total intensity of capital, k ^T, and of the shadow price of the government debt, g.

Now, a system given by the three first-order conditions can be proven to be recursive. Indeed, given λ ₁ and λ ₂, λ ₃ is determined. The relation (A.4.16) can also be expressed as:

$$ \begin{array}{ll}{\lambda}_1=& \frac{1}{\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]}\times \left[{\lambda}_2\frac{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]}{\left(1-s\right){p}_m}-{U}_{q_c}r-{U}_{k^T}\beta \right]\hfill \end{array} $$

which can be substituted into Eq. (A.4.14) to obtain:

$$ {\lambda}_2={f}_1\left({k}^T,\beta \right)\kern1em \mathrm{or}\kern1em \beta ={f}_2\left({k}^T,{\lambda}_2,{\lambda}_2\right) $$

Hence:

$$ {\lambda}_2\frac{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]}{\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]\left(1-s\right){p}_m}={U}_{q_c}r+{U}_{k^T}\beta +\frac{U_{\beta }{k}^T}{\left({q}_I-n{k}^T\right)}+{\lambda}_2\frac{p_k{q}_I}{\left({q}_I-n{k}^T\right)\left(1-s\right){p}_m} $$

which can be solved for λ ₂ as:

$$ \begin{array}{l}{\lambda}_2=\frac{\left[{U}_{q_c}r+{U}_{k^T}\beta +{U}_{\beta }{k}^T/\left({q}_I-n{k}^T\right)\right]\dots \left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]\left(1-s\right){p}_m}{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]\left[{q}_I-n{k}^T\right]-{p}_k{q}_I\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]}\hfill \end{array} $$

Now, we can substitute it into Eq. (A.4.14):

$$ \begin{array}{ll}{\lambda}_1=& \frac{U_{\beta }{k}^T}{\left({q}_I-n{k}^T\right)}\hfill \\ {}& +\frac{\left[{U}_{q_c}r+{U}_{k^T}\beta +{U}_{\beta }{k}^T/\left({q}_I-n{k}^T\right)\right]\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]{p}_k{q}_I}{\left\{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]\left[{q}_I-n{k}^T\right]-{p}_k{q}_I\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]\right\}}\hfill \end{array} $$

After substituting the values of λ ₁, λ₂, λ ₃ given by the first-order conditions into the Lagrangian, we have:

$$ \begin{array}{ll}L=& U\left({q}_c,\beta {k}^T\right)+\left\{\frac{U_{\beta }{k}^T}{\left({q}_I-n{k}^T\right)}\right.\hfill \\ {}& +\left.\frac{\left[{U}_{q_c}r+{U}_{k^T}\beta +{U}_{\beta }{k}^T/\left({q}_I-n{k}^T\right)\right]\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]{p}_k{q}_I}{\left\{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]\left[{q}_I-n{k}^T\right]-{p}_k{q}_I\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]\right\}}\right\}\hfill \\ {}& \times \left(1-\beta \right)\left[{q}_I-n{k}^T\right]\hfill \\ {}& +\left\{\frac{\left[{U}_{q_c}r+{U}_{k^T}\beta +{U}_{\beta }{k}^T/\left({q}_I-n{k}^T\right)\right]\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]\left(1-s\right){p}_m}{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]\left[{q}_I-n{k}^T\right]-{p}_k{q}_I\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]}\right\}\hfill \\ {}& \times \left\{\left[\frac{s{q}_c+{\beta p}_k{q}_I- se-\left(1-s\right){q}_I}{\left(1-s\right){p}_m}-\left(n+{\pi}_m\right)g\right]+\left(n+{\pi}_m\right)\left(g-{g}^{\ast}\right)\right\}\hfill \end{array} $$

(A.4.18″)

which can be simplified in:

$$ L=U\left({q}_c,\beta {k}^T\right)+{U}_{\beta }{k}^T\left(1-\beta \right)+X\left[{p}_k{q}_I+s{q}_c- se-\left(1-s\right){q}_I-n{g}^{\ast}\left(1-s\right){p}_m\right] $$

where:

$$ \begin{array}{l}X=\frac{\left[{U}_{q_c}r+{U}_{k^T}\beta +{U}_{\beta }{k}^T/\left({q}_I-n{k}^T\right)\right]\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]\left({q}_I-n{k}^T\right)}{\left[ sr+\beta r-\left(1-s\right)\left(r/{p}_k\right)\right]\left({q}_I-n{k}^T\right)-{p}_k{q}_I\left(1-\beta \right)\left[\left(r/{p}_k\right)-n\right]}\hfill \end{array} $$

Now, by making use of Pontryagin’s Maximum Principle we have:

$$ {\dot{\lambda}}_1={\alpha \lambda}_1+\left(-\frac{\partial L}{\partial {k}^T}\right) $$

$$ {\dot{\lambda}}_2={\alpha \lambda}_2+\left(-\frac{\partial L}{\partial g}\right) $$

or:

$$ {\dot{\lambda}}_1={\alpha \lambda}_1+{U}_{k^T}+{U}_{\beta }{k}^T\left(1-\beta \right)+\left(\frac{\partial X}{\partial {k}^T}\right)\times \left[{p}_k{q}_I+s{q}_c- se-\left(1-s\right){q}_c+\left(n{g}^{\ast }+{\pi}_mg\right)\left(1-s\right){p}_m\right]+X\left[\left( rs\left(1+{p}_k\right)\right)/{p}_k\right] $$

(A.4.18)

$$ {\dot{\lambda}}_2={\alpha \lambda}_2 $$

(A.4.19)

which together with Eqs. (A.4.2′) and (A.4.8′) form a complete dynamic system where the instrument β follows from Eq. (A.4.17).

Glossary of Symbols for Chaps. 1–4

a ^T

total wealth

a

private wealth

b

bonds

C

consumption goods

d

per capita government deficit

e

per capita government expenditure

G

government debt

g

(G/N), per capita government debt

h

government propensity to save

I

investment goods

k _c

(K _c /N _c ), capital intensity of the consumption goods sector

k _I

(K _I /N _I ), capital intensity of the investment goods sector

k ^T

(K ^T/N), capital intensity of the economy

k _c

input of capital in consumption goods’ production

k _I

input of capital in investment goods’ production

k ^T

stock of capital

k ^G

government capital stock

k ^G

(K ^G/N), intensity of government capital

K

private capital stock

k

(K/N), private capital intensity

m

money

N _c

input of labour in consumption goods’ production

i

interest in government bonds

N _I

input of labour in investment goods’ production

N

labour force population

n

rate of growth of population

p _c

consumption goods’ price

p _k

price of capital

p _m

price of money

q

gross national product = qc + q _I p _k

q _I

production of investment goods

q _c

production of consumption goods

r

rental price of capital

w

wage rate

x

(g/m), debt/money ratio

z

net government transfer

ϒ

(gp _m ), per capita government debt

Π _m

expected rate of deflation

π _k

expected rate of change in the price of capital

Θ

ġ/g

Stars are for “exogenously given levels of variables”. Dots are for time derivatives.