A New Theory on Business Cycle and Economic Growth

Abstract

Starting from an investigation into the nature of business cycles, Chapter 4 illustrates a new theory to explain business cycle and economic growth. This theory is subsequently derived rigorously by using economic models. The implications of the new theory are also discussed. The chapter ends with a demonstration of empirical evidence relevant to the new theory.

Appendices

Appendix 1 (for Section 5.4): An Economic Modelling Approach to the New Theory

To sharpen the focus, the multi-commodity model used in this section is a static one, but a dynamic upgrade is provided for interested readers.

5.1.1 A Static General Equilibrium Model

The economy in the model consists of one representative household and n representative firms. For simplicity, the government is not included in the model but the function of government is implicitly included in the broad definition of households. Government spending and investment are similar to those for households. The function of government taxation and social welfare influences income distribution, which is reflected in household income distribution. Since income inequality is not the focus of the study, only one representative household is used in the model. This means that income inequality, as well as lending and borrowing, are not explicitly considered in the model.² However, they are indirectly included in the income distribution parameter. Lending and borrowing can lead to temporarily more equitable income distribution, so they can be expressed as a change in income distribution parameter. Also for simplicity, a closed economy is assumed, so international trade and finance are not included in the model.

The basic transactions in the model are as follows. The household provides labour and capital to all firms and obtains wages and capital rentals in return. The household also uses its income to purchase goods from firms for consumption purposes and supplies its savings to firms for investment purposes. Under the zero economic profit condition, each firm uses labour, capital and technology to produce a unique type of commodity for the economy, and decides on its requirements for labour, capital, and investment in production. We express the role of each agent in mathematical form.

5.1.1.1 Household Consumption and Savings

The ultimate goal of a society is to maximize household utility (other goals such as investment, accumulation and development are parts of household utility in the future). This means that household utility is a crucial part of an economy-wide model. The utility function described in Sect. 5.3.1 requires further modification before it is used in the model. First, since commodity demand includes both consumption demand and investment demand, we use ‘c_i’ to replace ‘x_i’ in the utility function in the previous section to explicitly indicate consumption demand. Second, we need to consider the fact that there are a large number of households in an economy and that the distributional effect is an important factor in household consumption and utility. It is desirable to develop a multi-household model to include the distributional effect. This would, however, complicate the model and thus interfere with the main purpose of this chapter. Instead, the author adds a distributional effect parameter in the utility function of the representative household. Finally, the varieties of commodities may increase due to product innovation, so n + ∆n is used to reflect this effect. For simplicity, this study does not model the determinants of product innovation, so n + ∆n is assumed as exogenous. The new utility function is as follows:

$$ \begin{aligned} U = & \,U(c_{1} ,c_{2} , \ldots ,c_{n + \Delta n} ,\,\,{\text{Savings}}) \\ = & \,\sum\limits_{i = 1}^{n + \Delta n} {\alpha_{i} (2\theta m_{i} c_{i} - c_{i}^{2} )} + \alpha_{S} *\,{\text{Savings}} \\ \end{aligned} $$

(5.7)

where α_i and α_S are weights for consumption and savings, respectively. θ is the distributional parameter, 0 < θ ≤ 1. θ = 1 indicates that every household in the household group has the same level of income. When income distribution is not equal, some households cannot reach their consumption saturation point due to a lack of income support. In other words, their consumption ceilings are practically lowered due to income constraint. This effect is captured by making θ < 1.

The household budget can be expressed as:

$$ Y = \sum\limits_{i = 1}^{n + \Delta n} {P_{i} *c_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {P_{i} *S_{i} } $$

where c_i is consumption of each commodity, S_i is the amount of commodity saved, and P_i is the commodity price.

To obtain aggregate real savings, we need a weighting parameter for aggregation. Letting it be w_i, we have aggregate real savings as:

$$ {\text{Savings}} = \sum w_{i} *S_{i} . $$

Let P_s be the price of aggregate savings, we have:

$$ \sum\limits_{i = 1}^{n + \Delta n} {P_{i} *S_{i} } = P_{\text{S}} *{\text{Saving}}. $$

Defining δ_i as the share of each commodity saved (S_i) in total savings, i.e. δ_i = S_i/Savings, we can obtain the price for aggregate savings as:

$$ P_{\text{S}} = {{\left( {\sum\limits_{i = 1}^{n + \Delta n} {P_{i} *S_{i} } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 1}^{n + \Delta n} {P_{i} *S_{i} } } \right)} {\text{Saving}}}} \right. \kern-0pt} {\text{Saving}}} = \sum\limits_{i = 1}^{n + \Delta n} {P_{i} *\delta_{i} } . $$

As such, the optimal consumption problem for households can be expressed as:

$$ \begin{aligned} {\text{Maximize}}\,\,U = & \,U(c_{1} ,c_{2} , \ldots ,c_{n + \Delta n} ,{\text{Savings}}) \\ = & \,\sum\limits_{i = 1}^{n + \Delta n} {\alpha_{i} (2\theta m_{i} c_{i} - c_{i}^{2} )} + \alpha_{\text{S}} *{\text{Savings}} \\ \end{aligned} $$

Subject to $Y = \sum\limits_{i = 1}^{n + \Delta n} {P_{i} *c_{i} } + P_{\text{S}} *{\text{Savings}}$

Setting up a Lagrangian expression:

$$ \ell = U + \lambda (Y - \sum\limits_{i = 1}^{n + \Delta n} {P_{i} *c_{i} } - P_{S} *{\text{Savings}}) $$

Using the first order condition we can derive the optimal consumption of good i as follows:

$$ c_{i} = \theta m_{i} - \frac{{\alpha_{S} }}{{2\alpha_{i} }}\frac{{P_{i} }}{{P_{\text{s}} }} $$

(5.8)

$$ {\text{Savings}} = {{\left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{\text{S}} }}{{2\alpha_{i} P_{\text{S}} }}} } \right)} \mathord{\left/ {\vphantom {{\left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{\text{S}} }}{{2\alpha_{i} P_{\text{S}} }}} } \right)} {P_{\text{S}} }}} \right. \kern-0pt} {P_{\text{S}} }} $$

(5.9)

5.1.1.2 Firm’s Investment and Unsold Stock

Since the firm’s investment decision has an impact on its production, we discuss the firm’s investment first. It is assumed that the firm can identify both the consumption ceilings and the impact of the distributional effect on consumption, so the firm can invest a proportion of the perceived consumption growth potential, i.e. a proportion of the gap between the constrained consumption ceiling and the current consumption. Letting the investment demand for commodity i be proportionally related to consumption growth potential and to the propensity to invest after being discounted by interest rates, we have the following investment demand function for each commodity:

$$ I_{i} = \frac{B}{1 + r}(\theta m_{i} - c_{i} ) $$

where I_i is investment demand, B indicates the propensity to invest, r is interest rate, m_i is the maximum amount of consumption on good i, c_i is the actual amount of consumption of good i.

It is assumed that 0 ≤ B ≤ 1. When B = 1, the firm invests the highest amount in production to produce a maximum amount of goods which will be purchased by the household.

To obtain the aggregate real investment, we need a weighting parameter for aggregation. We use the same weighting w_i as that for aggregating savings because, in the case of the existence of market clearance, the same weighting ensures that the amounts of investment and saving at both aggregate and disaggregate levels are equal, namely, I_i = S_i and I = Saving. As such, we have

$$ I = \sum w_{i} *I_{i} ,\,\,{\text{or}} $$

$$ I_{{}} = \frac{B}{1 + r}\left( {\sum\limits_{i = 1}^{n + \Delta n} {\theta m_{i} } w_{i} - \sum\limits_{i = 1}^{n + \Delta n} {c_{i} w_{i} } } \right) $$

(5.10)

Based on the investment functions at both disaggregated and aggregate levels, we can calculate the share of each commodity in total investment: $\beta_{i} = I_{i} /I.$

It is worth mentioning that, for simplicity, we use the same parameter B for the investment for all commodities so we have the same overall propensity to invest in the total investment demand function. This treatment does not lose generality. If one uses different parameters as the propensity of investment demand for different commodities, the parameter for propensity of overall investment demand will be the weighted average of the parameters for all commodities. The only difference is that calculation of weight average is required to obtain the parameter for the overall propensity to invest.

This investment demand is financed by household savings. The uninvested household saving (the gap between saving and investment demand) equals the unsold stock (S) or inventory at firm. Although the firm has not cleared its inventory, it has paid the household the value of the inventory as wages or capital rentals. The household spends money on consumption and saves the rest. Household saving can be divided into two parts: the part invested equals the value of investment goods and the part uninvested equals the value of firm’s unsold stock. As such, we have.

$$ S = {\text{Savings}}\,\,\,{-}\,I = \sum {w_{i} S_{i} } - \sum {w_{i} I_{i} } $$

In a static model for a closed economy, the unsold stock S in the above equation should be non-negative because investment demand must be financed by savings. However, the unsold stock S can be negative when the economy is open or when the model has multiple periods. The additional finance in this case can come from overseas. In considering the accumulated past savings (e.g. wealth), the savings in a dynamic model can be negative, i.e. dissaving.

5.1.1.3 Firm’s Input Demand

To depict the firm’s production, the following Cobb–Douglas function is used for the purpose of simplicity:

$$ x_{i} = (A_{i} + \Delta A_{i} )*L_{i}^{{\gamma_{i} }} *K_{i}^{{1 - \gamma_{i} }} $$

where L means labour, K capital, A the level of technology, ∆A technological changes, and γ the share of labour in total inputs.

The optimal production problem can be expressed as:

Minimize ${\text{Cost}} = P_{\text{L}} *L_{i} + P_{\text{K}} *K_{i}$

Subject to ${\text{Output}} = x_{i} = (A_{i} + \Delta A_{i} )*L_{i}^{{\gamma_{i} }} *K_{i}^{{1 - \gamma_{i} }}$

Setting up a Lagrangian expression:

$$ \ell = P_{\text{L}} *L_{i} + P_{\text{K}} *K_{i} + \lambda \left[ {x_{i} - (A_{i} + \Delta A_{i} )*L_{i}^{{\gamma_{i} }} *K_{i}^{{1 - \gamma_{i} }} } \right] $$

Using the first order condition we can show the optimal demand for labour and capital as follows:

$$ L_{i} = \left( {\frac{{x_{i} }}{{A_{i} + \Delta A_{i} }}} \right)\left( {\frac{{\gamma_{i} P_{\text{K}} }}{{(1 - \gamma_{i} )P_{\text{L}} }}} \right)^{{1 - \gamma_{i} }} $$

(5.11)

and

$$ K_{i} = \left( {\frac{{x_{i} }}{{A_{i} + \Delta A_{i} }}} \right)\left( {\frac{{(1 - \gamma_{i} )P_{L} }}{{\gamma_{i} P_{K} }}} \right)^{{\gamma_{i} }} $$

(5.12)

These results link the firm’s demand for labour and capital (L_i and K_i) to the firm’s output x_i. More generally, the results show that the factor market is closely related to the commodity market.

5.1.1.4 Resource Constraints and Market Clearance Condition

The resources constraint in a closed economy can be expressed as

Savings are not less than investment: S = Savings − I = ∑ w_i S_i − ∑ w_i I_i ≥ 0.

Labour supply is not less than labour demand: $L \ge \sum {L_{i} }$

Capital supply is not less than capital demand: $K \ge \sum {K_{i} } .$

Next, we consider the market clearance condition. The total supply of commodity x_i in the economy is the sum of both the consumed and the unconsumed commodity, namely, $x_{{{\text{S}}i}} = c_{i} + S_{i}$. On the other hand, the total demand for commodity x_i comprises the consumption demand and the investment demand, so that the total demand for x_i can be expressed as $x_{{{\text{D}}i}} = c_{i} + I_{i}$. Thus, the excess demand function for x_i is: ${\text{ED}}_{i} = x_{{{\text{D}}i}} - x_{{{\text{S}}i}} = I_{i} - S_{i}$.

The conditions for market clearance require that, for each commodity i, $ED_{i} = x_{{{\text{D}}i}} - x_{{{\text{S}}i}} = I_{i} - S_{i} = 0$ or I_i = S_i. Aggregating all commodities, we have market clearance condition: $\sum {w_{i} I_{i} } = \sum {w_{i} S_{i} } ,\,\,{\text{or}}\,\,I = {\text{Savings}}.$

In a traditional general equilibrium model, investment is always equal to savings because neoclassical economics simplistically and idealistically assumes that all savings are invested, so I = Savings is guaranteed by presumption. With this guarantee, a general equilibrium is achievable at any time: if I_i ≠ S_i for some or all commodity types, the price mechanism will kick in and adjust any difference between investment and savings in all commodity types.

However, in our static model, investment is determined by the consumption growth potential (the difference between current consumption and the maximum consumption) while saving is determined by the utility maximization procedure, so there is no guarantee that total investment equals total savings—only the resource constraint (Savings ≥ I) is applied to savings and investment. As a result, the general equilibrium is not guaranteed: if Savings > I, the price mechanism cannot work out the solution for S_i = I_i because in this case there is a positive net saving and thus an overall oversupply in the economy.

5.1.2 Static Result Interpretation: A Demand-Side Perspective

At this point, we assume the prices in the static model are fixed so that we can derive some intuitive but essential results from the model. For an economy to grow without a recession, the total supply of a commodity must be cleared by the market, i.e. excess demand for any commodities must be nonnegative: ${\text{ED}}_{i} = x_{{{\text{D}}i}} - x_{{{\text{S}}i}} = I_{i} - S_{i} \ge 0$, or ${\text{ED}} = \sum\nolimits_{i = 1}^{n + \Delta n} {I_{i} w_{i} } - \sum\nolimits_{i = 1}^{n + \Delta n} {S_{i} w_{i} } = I - {\text{Saving}} \ge 0$.

Recalling the investment equation (Eq. 5.10), we can express the condition to avoid a recession as:

$$ {\text{ED}} = \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\theta m_{i} w_{i} } - \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {c_{i} w_{i} } - {\text{Saving}} \ge 0 $$

Plugging the consumption equation (Eq. 5.8) and the saving equation (Eq. 5.9) into the above inequality, we have:

$$ \begin{aligned} & \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\theta m_{i} } w_{i} - \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\left( {\theta m_{i} w_{i} - \frac{{\alpha_{S} P_{i} w_{i} }}{{2\alpha_{i} P_{\text{S}} }}} \right)} \\ & \quad - {{\left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{\text{S}} }}{{2\alpha_{i} P_{\text{S}} }}} } \right)} \mathord{\left/ {\vphantom {{\left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{\text{S}} }}{{2\alpha_{i} P_{\text{S}} }}} } \right)} {P_{\text{S}} \ge 0}}} \right. \kern-0pt} {P_{\text{S}} \ge 0}} \\ \end{aligned} $$

$$ Y \le \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{BP_{i} \alpha_{S} w_{i} }}{{2(1 + r)\alpha_{i} }}} - \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} $$

(5.13)

This inequality shows that, to avoid a recession, the household income must be below a certain level! To allow income to increase without a ceiling, one may increase θ (i.e. improving the equality in income distribution) or increase B (the propensity to invest) or decrease r (the interest rate), but the effect of these efforts is limited because the maximum value of both θ and B is 1 and the minimum value of r is 0. The only way to allow the income level to increase unrestrictedly is to increase ∆n, i.e. inventing new products. Since θ * m_i is much larger compared with P_iα_s/2α_iP_s (for an economy, the consumption ceiling m_i is generally very high compared with other items here), when ∆n increases, the increase in the first term on the right-hand side will outweigh the increase in the third term and thus the cap on Y will be lifted.

Household supply of labour and capital is determined by household willingness to obtain income, which in turn is determined by consumption and savings. So, the household will supply the amount of labour and capital to produce the amount of output of good i that is equal to the sum of the consumed and the saved by the household. In this reasoning, we substitute x_i= c_i+ S_i into Eqs. (5.11) and (5.12) and obtain the amount of labour and capital supplied for the production of good x_i:

$$ L_{{{\text{S}}i}} = \left( {\frac{{c_{i} + S_{i} }}{{A_{i} + \Delta A_{i} }}} \right)\left( {\frac{{\gamma_{i} P_{\text{K}} }}{{(1 - \gamma_{i} )P_{\text{L}} }}} \right)^{{1 - \gamma_{i} }} $$

(5.14)

$$ K_{{{\text{S}}i}} = \left( {\frac{{c_{i} + S_{i} }}{{A_{i} + \Delta A_{i} }}} \right)\left( {\frac{{(1 - \gamma_{i} )P_{\text{L}} }}{{\gamma_{i} P_{\text{K}} }}} \right)^{{\gamma_{i} }} $$

(5.15)

On the other hand, the demand for labour and capital is determined by final demand c_i+ I_i. Therefore, substituting x_i = c_i+ I_i into Eqs. (5.11) and (5.12) we have the labour and capital demand functions:

$$ L_{{{\text{D}}i}} = \left( {\frac{{c_{i} + I_{i} }}{{A_{i} + \Delta A_{i} }}} \right)\left( {\frac{{\gamma_{i} P_{\text{K}} }}{{(1 - \gamma_{i} )P_{\text{L}} }}} \right)^{{1 - \gamma_{i} }} $$

(5.16)

$$ K_{{{\text{D}}i}} = \left( {\frac{{c_{i} + I_{i} }}{{A_{i} + \Delta A_{i} }}} \right)\left( {\frac{{(1 - \gamma_{i} )P_{\text{L}} }}{{\gamma_{i} P_{\text{K}} }}} \right)^{{\gamma_{i} }} $$

(5.17)

Excess demand in the factor market is the sum of the excess demand for labour and capital in producing each commodity, namely:

$$ {\text{ED}}_{\text{L}} = \sum\limits_{i = 1}^{n + \Delta n} {{\text{ED}}_{{{\text{L}}i}} } = \sum\limits_{i = 1}^{n + \Delta n} {L_{{{\text{D}}i}} } - \sum\limits_{i = 1}^{n + \Delta n} {L_{{{\text{S}}i}} } $$

(5.18)

$$ {\text{ED}}_{\text{K}} = \sum\limits_{i = 1}^{n + \Delta n} {{\text{ED}}_{{{\text{K}}i}} } = \sum\limits_{i = 1}^{n + \Delta n} {K_{{{\text{D}}i}} } - \sum\limits_{i = 1}^{n + \Delta n} {{\text{K}}_{{{\text{S}}i}} } $$

(5.19)

Substituting Eqs. (5.14) to (5.17) into the Eqs. (5.18) and (5.19) and utilizing the saving share δ_i and investment share β_i, we have:

$$ \begin{aligned} {\text{ED}}_{\text{L}} = & \sum\limits_{i = 1}^{n + \Delta n} {\left( {\frac{{I_{i} - S_{i} }}{{A_{i} }}} \right)\left( {\frac{{\gamma_{i} P_{\text{K}} }}{{(1 - \gamma_{i} )P_{\text{L}} }}} \right)^{{1 - \gamma_{i} }} } \\ = & \sum\limits_{i = 1}^{n + \Delta n} {\mu_{i} I_{i} } - \sum\limits_{i = 1}^{n + \Delta n} {\mu_{i} S_{i} } = \sum\limits_{i = 1}^{n + \Delta n} {\mu_{i} \beta_{i} I} - \sum\limits_{i = 1}^{n + \Delta n} {\mu_{i} \delta_{i} S} \\ \end{aligned} $$

(5.20)

$$ \begin{aligned} {\text{ED}}_{\text{K}} = & \sum\limits_{i = 1}^{n + \Delta n} {\left( {\frac{{I_{i} - S_{i} }}{{A_{i} }}} \right)\left( {\frac{{(1 - \gamma_{i} )P_{\text{L}} }}{{\gamma_{i} P_{\text{K}} }}} \right)^{{\gamma_{i} }} } \\ = & \sum\limits_{i = 1}^{n + \Delta n} {\nu_{i} I_{i} } - \sum\limits_{i = 1}^{n + \Delta n} {\nu_{i} S_{i} } = \sum\limits_{i = 1}^{n + \Delta n} {\nu_{i} \beta_{i} I} - \sum\limits_{i = 1}^{n + \Delta n} {\nu_{i} \delta_{i} S} \\ \end{aligned} $$

(5.21)

$$ \begin{aligned} & {\text{where}} \\ & \mu_{i} = {{\left( {\frac{{\gamma_{i} P_{\text{K}} }}{{(1 - \gamma_{i} )P_{\text{L}} }}} \right)^{{1 - \gamma_{i} }} } \mathord{\left/ {\vphantom {{\left( {\frac{{\gamma_{i} P_{\text{K}} }}{{(1 - \gamma_{i} )P_{\text{L}} }}} \right)^{{1 - \gamma_{i} }} } {(A_{i} + \Delta A_{i} )}}} \right. \kern-0pt} {(A_{i} + \Delta A_{i} )}},\quad \nu_{i} = {{\left( {\frac{{(1 - \gamma_{i} )P_{\text{L}} }}{{\gamma_{i} P_{\text{K}} }}} \right)^{{\gamma_{i} }} } \mathord{\left/ {\vphantom {{\left( {\frac{{(1 - \gamma_{i} )P_{\text{L}} }}{{\gamma_{i} P_{\text{K}} }}} \right)^{{\gamma_{i} }} } {(A_{i} + \Delta A_{i} )}}} \right. \kern-0pt} {(A_{i} + \Delta A_{i} )}} \\ \end{aligned} $$

The above two equations show that the excess demand for both labour and capital is the difference between weighted total investment and weighted total savings. This is very similar to the excess demand function in the commodity market. Thus, if the investment demand is greater than savings, there is an excess demand in commodity markets, there will be an excess demand for labour and capital. The size of excess demand in different markets will, however, differ due to the weights. Since the demand for primary factors closely links to the demand for commodities, the reasons for excess supply in the commodity market can also explain the excess supply in the factor market.

5.1.3 Static Result Interpretation: A Supply-Side Perspective

The above demand-side approach gives us an intuitive picture of commodity and factor markets. However, this picture is not a high-resolution one because the prices for the commodities and for factors are set as exogenous. In fact, the prices in the model are related to each other and they can change when the economy goes from disequilibrium to equilibrium, so we allow the prices be endogenous in this section. In determining the prices of commodities and factors, we can link the production side (or supply side) to the consumption side (or demand side).

First of all, by using Eqs. (5.11) and (5.12), we can obtain the price linkage between labour and capital.

$$ P_{\text{L}} = \frac{{\gamma_{i} }}{{(1 - \gamma_{i} )}}\frac{{K_{i} }}{{L_{i} }}P_{\text{K}} $$

Based on the zero economic profit assumption, the cost of producing X_i will determine the price of X_i, so,

$$ P_{i} X_{i} = P_{\text{L}} L_{i} + P_{\text{K}} K_{i} = \frac{{\gamma_{i} }}{{(1 - \gamma_{i} )}}\frac{{P_{\text{K}} K_{i} }}{{L_{i} }}L_{i} + P_{\text{K}} K_{i} = \frac{{P_{\text{K}} K_{i} }}{{(1 - \gamma_{i} )}} $$

or,

$$ P_{i} = \frac{{P_{\text{K}} K_{i} }}{{(1 - \gamma_{i} )X_{i} }} $$

Normalizing the price level of the economy to 1, we have

$$ \begin{aligned} & 1 = \frac{{\sum\nolimits_{i = 1}^{n + \Delta n} {P_{i} X_{i} } }}{{\sum\nolimits_{i = 1}^{n + \Delta n} {X_{i} } }} = \frac{{P_{\text{K}} \sum\nolimits_{i = 1}^{n + \Delta n} {\frac{{K_{i} }}{{1 - \gamma_{i} }}} }}{{\sum\nolimits_{i = 1}^{n + \Delta n} {X_{i} } }}\,\,\,{\text{or}} \end{aligned} $$

$$ \begin{aligned} P_{\text{K}} = \frac{{\sum\nolimits_{i = 1}^{n + \Delta n} {X_{i} } }}{{\sum\nolimits_{i = 1}^{n + \Delta n} {\frac{{K_{i} }}{{1 - \gamma_{i} }}} }} = \frac{{\sum\nolimits_{i = 1}^{n + \Delta n} {(A_{i} + \Delta A_{i} )L_{i}^{{\gamma_{i} }} K_{i}^{{1 - \gamma_{i} }} } }}{{\sum\nolimits_{i = 1}^{n + \Delta n} {\frac{{K_{i} }}{{1 - \gamma_{i} }}} }}, \end{aligned} $$

And thus,

$$ \begin{aligned} P_{i} = & \frac{{K_{i} }}{{(1 - \gamma_{i} )X_{i} }}\frac{{\sum\nolimits_{i = 1}^{n + \Delta n} {X_{i} } }}{{\sum\nolimits_{i = 1}^{n + \Delta n} {\frac{{K_{i} }}{{1 - \gamma_{i} }}} }} \\ = & (1 - \gamma_{i} )^{ - 1} (A_{i} + \Delta A_{i} )^{ - 1} L_{i}^{{ - \gamma_{i} }} K_{i}^{{\gamma_{i} }} \frac{{\sum\nolimits_{i = 1}^{n + \Delta n} {(A_{i} + \Delta A_{i} )L_{i}^{{\gamma_{i} }} K_{i}^{{1 - \gamma_{i} }} } }}{{\sum\nolimits_{i = 1}^{n + \Delta n} {\frac{{K_{i} }}{{1 - \gamma_{i} }}} }} \\ \end{aligned} $$

(5.22)

Based on this P_i, we can easily calculate Ps as $P_{s} = \sum\nolimits_{i = 1}^{n + \Delta n} {P_{i} \delta_{i} } .$ Plugging P_i and P_s into Eq. (5.8), we can assess (albeit a bit complicated) the impact of a change in inputs (K and L) on consumption.

The excess supply of commodities is the difference between commodities saved and commodities invested, i.e.

$$ \begin{aligned} {\text{ES}}_{i} & = S_{i} - I_{i} = S\delta_{i} - I\beta_{i} \\ & = {{\delta_{i} \left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} } \right)} \mathord{\left/ {\vphantom {{\delta_{i} \left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} } \right)} {P_{\text{S}} }}} \right. \kern-0pt} {P_{\text{S}} }} \\ & \quad - \frac{B}{1 + r}\beta_{i} \left( {\sum\limits_{i = 1}^{n + \Delta n} {\theta m_{i} } w_{i} - \sum\limits_{i = 1}^{n + \Delta n} {\left( {\theta m_{i} w_{i} - \frac{{P_{i} \alpha_{S} w_{i} }}{{2\alpha_{i} P_{S} }}} \right)} } \right) \\ & = {{\delta_{i} \left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} } \right)} \mathord{\left/ {\vphantom {{\delta_{i} \left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} } \right)} {P_{\text{S}} - \frac{B}{1 + r}\beta_{i} \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i} \alpha_{S} w_{i} }}{{2\alpha_{i} P_{S} }}} }}} \right. \kern-0pt} {P_{\text{S}} - \frac{B}{1 + r}\beta_{i} \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i} \alpha_{S} w_{i} }}{{2\alpha_{i} P_{S} }}} }} \\ \end{aligned} $$

The excess commodity supply will reduce the commodity supply in the next period, so the factors contributing to the excess commodity supply will not be employed in the next period. This causes unemployed labour and unutilized capital in the factor markets.

To avoid economic stagnation, it is necessary that there is no overall excess supply, i.e. ∑ES_i ≤ 0. Summarizing all ES_i and using the fact that ∑δ_i = 1 and ∑β_i = 1, we have:

$$ \sum\limits_{i = 1}^{n + \Delta n} {{\text{ES}}_{i} } = {{\left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} } \right)} \mathord{\left/ {\vphantom {{\left( {Y - \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{S} }}{{2\alpha_{i} P_{S} }}} } \right)} {P_{S} - \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i} \alpha_{S} w_{i} }}{{2\alpha_{i} P_{S} }}} \le 0}}} \right. \kern-0pt} {P_{S} - \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i} \alpha_{S} w_{i} }}{{2\alpha_{i} P_{S} }}} \le 0}} $$

$$ Y \le \sum\limits_{i = 1}^{n + \Delta n} {\theta P_{i} m_{i} } + \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i} \alpha_{\text{S}} w_{i} }}{{2\alpha_{i} }} - \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{2} \alpha_{\text{S}} }}{{2\alpha_{i} P_{\text{S}} }}} } $$

This is the same results as Eq. (5.13) derived for the income ceiling from the demand-side perspective. However, the prices for commodities and for savings in the above equation are determined by the amount of capital and labour inputs through Eq. (5.22). Considering this, the ceiling on income is not fixed.

To be more accurate, the income derived above is measured by money so it is a nominal income. In real terms, we must leave out price P_i in above equation, so the real income in the equilibrium should be the goods consumed and invested, i.e.,

$$ Y_{\text{real}} = \sum\limits_{i = 1}^{n + \Delta n} {\theta m_{i} } + \frac{B}{1 + r}\sum\limits_{i = 1}^{n + \Delta n} {\frac{{\alpha_{\text{S}} w_{i} }}{{2\alpha_{i} }}} - \sum\limits_{i = 1}^{n + \Delta n} {\frac{{P_{i}^{{}} \alpha_{\text{S}} }}{{2\alpha_{i} P_{\text{S}} }}} $$

(5.23)

In this equation, the prices of commodities (and thus the prices of savings, labour and capital) only affect the negative item, so the fixed ceiling still exists for real income.

Equation (5.23) can also be used to demonstrate a case of government tax policy. Although there is no government in the model, the effect of a lump sum tax can be demonstrated indirectly. If the government imposes this tax and distributes it to poor households, the distributional parameter θ in Eq. (5.23) increases with other things being equal, and this leads to an increase in real income. This is consistent with intuition: income redistribution to improve equality will encourage consumption and thus stimulate economic growth. If the government use the tax to boost budget balance (i.e. tax revenue is unspent), this indicates an increased preference to save, so α_S increase while α_i decreases in Eq. (5.23). The consequence of this is an increase in the absolute size of both the positive investment effect (the second term at the right of the equation) and the negative saving effect (the last term at the right of the equation). The enlarged negative saving effect results from the decreased current consumption level due to government tax; and the enlarged positive investment effect is due to the increased gap between the consumption ceiling and the decreased current consumption. The overall effect of this lump sum tax depends on the relative size of changes in both the saving effect and the investment effect. Generally speaking, the investment effect is smaller because the value of B/(1 + r) is less than one, so the overall effect would be a decrease in real income.

5.1.4 A Recursive Dynamic Model

The model presented so far is a static one, but it can be easily upgraded by adding a time frame and by considering dynamics in technology, capital, and wealth.

Since wealth is accumulated savings, so the wealth dynamic can be described by equalling W_t+1 (wealth in time t + 1) to W_t (wealth in time t) plus savings in time t.

$$ W_{t + 1} = W_{t} + {\text{Savings}}_{t} $$

The technological and capital dynamics for industry i can be expressed as

$A_{i,\,t + 1} = \, A_{i,\,t} + \Delta A_{i,\,t}$,

where ∆A_i,t is the change of technology in industry i at time t.

$K_{i,\,t + 1} = \, (1 - \delta )K_{i,\,t} + \, I_{i,\,t}$,

where I_i,t is the change of capital in industry i at time t, δ is the depreciation rate.

The household utility function at time t can be written as:

$$ \begin{aligned} U_{t} = & \,U(c_{1,t} ,c_{2,t} , \ldots ,c_{{n + \Delta n_{,t} }} ,{\text{Savings}}_{t} ) \\ = & \,\sum\limits_{i = 1}^{n + \Delta n} {\alpha_{i,t} (2\theta_{t} m_{i,t} c_{i,t} - c_{i,t}^{2} )} + \alpha_{S,t} *{\text{Savings}}_{t} \\ \end{aligned} $$

This utility function is subject to a budget:

$W_{t} + Y_{t} \ge$ $\sum {P_{i,t} c_{i,t} } + \, P_{S,t} \sum {S_{i,t} } .$

The optimal production problem can be expressed as:

Maximize

$x_{i,t} = A_{i,t} *L_{i,t}^{{\gamma_{i} }} *K_{i,t}^{{1 - \gamma_{i} }},$

subject to production cost $= P_{L,t} *L_{i,t} +$ $P_{K,t} *K_{i,t} .$

The investment demand function is as follows:

$$ I_{t} = \frac{{B_{t} }}{{1 + r_{t} }}\left( {\sum\limits_{i = 1}^{n + \Delta n} {\theta_{t} m_{i,t} w_{i,t} } - \sum\limits_{i = 1}^{n + \Delta n} {c_{i,t} w_{i,t} } } \right) $$

(5.24)

In a similar fashion to the static model, the household uninvested savings equal the unsold stock (i.e. inventory) at firm, which are calculated as the difference between savings and investment demand,

$$ S_{t} = {\text{Savings}}_{t} \,{-}\,I_{t} $$

Investment demand is financed by savings; any uninvested savings (or stocks) will be accumulated as wealth; and any excess investment demand over savings will be drawn from wealth.

The above equations transform the static model to a recursive dynamic model. This recursive model can generate the equilibrium for each period. The results from the previous period will have an influence on the results in the next period. For example, I_t (the investment in time t) will affect K_t+1 (the capital in time t + 1), which in turn will affect a number of variables in time t + 1 such as output level (x_t+1), savings (S_t+1), consumption (c_t+1), and investment (I_t+1). However, the mechanism determining the equilibrium or disequilibrium in each period is the same as in the static model.

5.1.5 An Intertemporal General Equilibrium Model

The above recursive model is unable to determine either the intertemporal equilibrium or the optimal time path for the economy, so an intertemporal model is needed. Since an intertemporal equilibrium model with multi-commodity is very complex, we have to be content with a Ramsey/Solow-style one-commodity model, but the essence of including multi-commodity in the static model—consumption ceiling—will be reflected in the intertemporal model. Moreover, to reduce the number of variables, we eliminate the variable labour by measuring capital, output, utility, and consumption in per capita term. In a traditional Ramsey/Solow model,³ all savings are assumed to be invested. This assumption is implausible thus it has to be relaxed. The three axioms featured in the static model will also be used in the intertemporal equilibrium model.

Since we are considering an intertemporal equilibrium model, we have to use continuous time, which is different from the discrete time in the recursive model. Also, because all variables in the intertemporal model are measured in per capita terms, we use the lower case for most variables so as to differentiate them from the aggregate variables, e.g. using k for capital, c for consumption, and s for savings.

In per capita term, the function can be written as:

y = a * f(k),

where a is technology, k is capital per worker, and f(k)′ > 0.

The household utility function in per capita terms can be written as:

$u\left( c \right) = \, \alpha_{C} *\left( {2mc - c^{2} } \right) + \alpha_{S} *\left( {af\left( k \right) - c} \right)$,

where, α_C is the weighting for utility from consumption, α_S is the weighting for utility from saving, m is consumption ceiling, c is actual consumption, c ≤ m.

The investment per capita is proportional to the gap between the consumption ceiling and the actual consumption level, so it can be written as

i = b(m − c),

where i is investment and b is an interest-rate-discounted investment propensity parameter, b = B/(1 + r), 0 < B<1; as before, B is the propensity to invest, r is the interest rate.

Based on this investment demand function, the per capita capital dynamics can be written as:

$k^{\prime } = \Delta k = b\left( {m - c} \right) - \delta k - nk$,

where δ is the capital depreciation rate and n is the growth rate of population (or labour force).

The dynamics of per capita assets are determined by uninvested household saving or unsold stock (inventory) at the firm:

$s^{\prime } = \Delta s = {\text{Savings}} = y - c - i = af\left( k \right) - c - b\left( {m - c} \right),$

where s stands for stock.

Considering a time preference rate (i.e. future discount rate) of θ, the optimal control problem can be expressed as:

Maximize

$V = \int_{0}^{\infty } {u(c)e^{ - \theta t} dt}$

Subject to $s^{\prime } = af\left( k \right) - c - b\left( {m - c} \right),k^{\prime } = b\left( {m - c} \right) - \,\delta k - nk,$

$s\left( 0 \right) = 0,$ $s\left( \infty \right) \ge \, 0,k\left( 0 \right) = 0,k\left( \infty \right) \ge 0,\,{\text{and}}\,0 \le c\left( t \right) \le m.$

The standard Hamiltonian function is

$$ H = u\left( c \right)e^{ - \theta t} + \lambda_{1} \left[ {af\left( k \right) - c - b\left( {m - c} \right)} \right] \\ + \lambda_{2} \left[ {b\left( {m - c} \right) - \delta k - nk} \right]. $$

From this Hamiltonian function, it is easy to see that if we impose a condition that λ₁ = λ₂, then the model collapses to the traditional Ramsey/Solow model.

The current-value form of Hamiltonian function is

$H_{c} = u\left( c \right) + \eta_{1} \left[ {af\left( k \right) - c - b\left( {m - c} \right)} \right] + \eta_{2} \left[ {b\left( {m - c} \right) - \delta k - nk} \right],$ ${\text{where}}\,\eta_{1} = \lambda_{1} e^{\theta t} ,{\text{and}}\,\eta_{2} = \lambda_{2} e^{\theta t} .$

The necessary condition for an optimal solution is that, at each t,

• (i) $\partial H_{\text{c}} /\partial c = 0$

• (ii) $\eta_{1}^{\prime } = - \partial H_{\text{c}} /\partial s \, + \, \theta \eta_{1} .$

• (iii) $\eta_{2}^{\prime } = - \partial H_{\text{c}} /\partial k \, + \, \theta \eta_{2} .$

• (iv) $s^{\prime } = \partial H_{\text{c}} /\partial \eta_{1}$

• (v) $k^{\prime } = \partial H_{\text{c}} /\partial \eta_{2}$

• (vi) $s\left( 0 \right) = 0,s\left( \infty \right) \ge 0,k\left( 0 \right) = 0,k\left( \infty \right) \ge 0,0 \le c\left( t \right) \le m.$

Condition (ii) gives: $\eta_{1}^{\prime } = - \partial H_{c} /\partial s + \theta \eta_{1} = \theta \eta_{1}$. The obvious solution for η₁ is $\eta_{1} = e^{\theta t}$.

Condition (i) gives $u^{\prime } + \eta_{1} \left( { - 1 + b} \right) + \eta_{2} \left( { - b} \right) = 0, {\text{or}}\,\,\eta_{2} = \eta_{1}$ $\left( {b - 1} \right)/b + u^{\prime } /b.$

Considering $\eta_{1} = e^{\theta t} \,{\text{and}}\,u = \alpha_{C} *\left( {2mc - c^{2} } \right) + \, \alpha_{S} *\left( {af\left( k \right) - c} \right),$ we have $\eta_{2} = \, e^{\theta t} \left( {b - 1} \right)/b + \left( {2\alpha_{\text{C}} m - 2\alpha_{\text{C}} c - \alpha_{\text{S}} } \right)/b,$ and thus $\eta_{2}^{\prime } = \theta e^{\theta t} \left( {b - 1} \right)/b + 2 \, \alpha_{\text{C}} c^{\prime } /b.$

Condition (iii) gives:

$$ \begin{aligned} \eta_{2}^{\prime } = & \, - \partial H_{\text{c}} /\partial k + \theta \eta_{2} = -\left[ {\eta_{1} af\left( k \right)^{\prime } -\, \eta_{2} \left( {\delta + n} \right)} \right] + \theta \eta_{2} \\ = & \, - {\eta_{1} af\left( k \right)^{\prime } + \eta_{2} \left( {\delta + n + \theta } \right)} \\ = & \, - e^{\theta t} af\left( k \right)^{\prime } + \left[ {e^{\theta t} \left( {b - 1} \right)/b \, } \right. \\ & \quad \left. { + \left( {2 \, \alpha_{\text{C}} *m - 2 \, \alpha_{\text{C}} *c - \, \alpha_{\text{S}} } \right)/b} \right]\left( {\delta + n + \theta } \right). \\ \end{aligned} $$

Based on the above two equations, we have:

$$ \begin{aligned} \eta_{2}^{\prime } = & \,\theta e^{\theta t} \left( {b - 1} \right)/b \, + 2\alpha_{\text{C}} *c^{\prime } /b \\ = & \, - e^{\theta t} af\left( k \right)^{\prime } \\ & \quad + \, \left[ {e^{\theta t} \left( {b - 1} \right)/b + \left( {2 \, \alpha_{\text{C}} *m - 2 \, \alpha_{\text{C}} *c - \alpha_{\text{S}} } \right)/b} \right]\left( {\delta + n + \theta } \right), \\ \end{aligned} $$

or,

$$ \begin{aligned} c^{\prime } = & \, - 0.5\theta e^{\theta t} \left( {b - 1} \right) - 0.5e^{\theta t} abf\left( k \right)^{\prime } \\ & \quad + \, 0.5\left[ {e^{\theta t} \left( {b - 1} \right) + \left( {2 \, \alpha_{C} *m - 2 \, \alpha_{C} *c - \alpha_{S} } \right)} \right]\left( {\delta + n + \theta } \right), \\ \end{aligned} $$

or,

$$ \begin{aligned} c^{\prime } = & - \, 0.5e^{\theta t} \left[ {abf\left( k \right)^{\prime } + \left( {1 - b} \right)\left( { \, \delta + n} \right)} \right] \\ & \quad \quad + \left( {\alpha_{\text{C}} *m - \alpha_{\text{C}} *c - 0.5 \, \alpha_{\text{S}} } \right)\left( {\delta + n + \theta } \right) \\ \end{aligned} $$

(5.25)

Conditions (iv) and (v) gives the growth rate of stock and capital respectively:

$$ s^{\prime } = af\left( k \right) - c - \, b\left( {m - c} \right) $$

(5.26)

$$ k^{\prime } = b\left( {m - c} \right) - \delta k - nk $$

(5.27)

These conditions will be used in the next section.

5.1.6 Result Interpretation: A Dynamic Perspective

The mechanism in the recursive model is essentially the same as that in the static model, so the equilibrium and disequilibrium in each period in the recursive model will be very similar to those in the static model. However, the investment function (5.24) plays a key role in patterns of economic growth. As the investment at time t is proportional to the potential of consumption growth (i.e. the gap between current consumption and consumption ceiling), the change of potential of consumption growth leads to cyclic the investment behaviour. When the gap between the current consumption and the consumption ceiling is large, investors perceive the high potential of increase in sales in the future and thus invest more in production. This leads to a large increase in aggregate demand and pushes the economy into the boom phase. As the gap between the current consumption and the consumption ceiling gets smaller (assuming no new product is invented), stagnancy or very slow growth of consumption is coupled with an investment decrease. This leads to a decrease in aggregate demand and thus an economic recession.

During a recession, the firm cannot find a chance to increase sales because the consumption level is close to the consumption ceiling. Under this circumstance, firms have no choice but to invest in research and innovation, hoping to invent a new product. Once the innovation succeeds, the new product will lift the consumption ceiling and the gap between the current consumption and the consumption ceiling increases. As a result, both consumption and investment increase and the economy enters a recovery phase which is followed by an expansion phase. This cyclical growth will continue as long as there is no mechanism to stimulate product innovation.

From a different perspective, the intertemporal equilibrium model in the previous section can demonstrate the same point: a steady or balanced growth is not achievable in the long run if product innovation cannot keep pace with production growth.

We start with a task to find out the steady-state conditions as well as the optimal economic growth path. At the steady state, the growth of stock and capital becomes zero. Using Eqs. (5.26) and (5.27), we have:

$$ s^{\prime } = \, af\left( k \right) - c - \, b\left( {m - c} \right) = 0 $$

$$ k^{\prime } = b\left( {m - c} \right) - \delta k - nk = 0 $$

Combining the above two equations, we have:

$$ k^{\prime } = af\left( k \right) - c - \delta k - nk $$

(5.28)

Setting k′ = 0, we have

$$ c = af\left( k \right) - \left( {\delta + n} \right)k $$

(5.29)

Equation (5.29) defines a k′ = 0 curve.

The condition for optimal consumption is ∂c/∂k = 0. This gives the same golden rule as in the Ramsey/Solow model:

$$ af\left( k \right)^{\prime } = \left( {\delta + n} \right) $$

(5.30)

At a steady state, the growth of consumption must also be zero. Setting Eq. (5.25) to zero, we have:

$$ \begin{aligned} c^{\prime } = & - 0.5e^{\theta t} \left[ {abf\left( k \right)^{\prime } + \left( {1 - b} \right)\left( {\delta + n} \right)} \right] \\ & \quad + \left( { \, \alpha_{\text{C}} *m - \alpha_{\text{C}} *c - 0.5\alpha_{\text{S}} } \right)\left( {\delta + n + \theta } \right) = 0 \\ \end{aligned} $$

(5.31)

Equation (5.31) defines a c′ = 0 curve.

Combining Eqs. (5.29) and (5.31), we can solve for the steady state E(c*, k*). However, this steady state will not be steady if the consumption ceiling (m) is fixed: due to the term $e^{\theta t}$, c′ will reduce over time. In other words, to maintain a steady state, m has to increase over time.

If the steady state is optimal, it must coincide with the optimal consumption point.⁴ We apply the golden rule by substitute Eqs. (5.29) and (5.30) into Eq. (5.31),

$$ c^{\prime } = - 0.5e^{\theta t} \left( {\delta + n} \right) + \left( {\alpha_{\text{C}} *m - \alpha_{\text{C}} *c - 0.5 \, \alpha_{\text{S}} } \right)\left( {\delta + n + \theta } \right) = 0 $$

(5.32)

$$ {\text{This}}\,{\text{gives:}}\,c = m - 0.5e^{\theta t} \left( {\delta + n} \right)/\left[ {\alpha_{\text{C}} \left( {\delta + n + \theta } \right)} \right] - 0.5\alpha_{\text{S}} /\alpha_{\text{C}} $$

(5.33)

The condition to achieve a steady consumption is:

$$ dc/dt = dm/dt - 0.5\theta e^{\theta t} \left( {\delta + n} \right)/\left( {\delta + n + \theta } \right) = 0,\,\,{\text{or}}, $$

$$ m = 0.5e^{\theta t} \left( {\delta + n} \right)/\left[ {\alpha_{C} \left( {\delta + n + \theta } \right)} \right] + {\text{constant}} $$

(5.34)

These results can be shown in Fig. 5.15. Panel (a) shows the space view of the phase diagram. The k′ = 0 curve is given by setting k′ = af(k) − c − (δ + n) k = 0, i.e. equation (5.29). Based on Eq. (5.28), ∂k′/∂c = −1 < 0, k′ and c move in opposite directions, i.e. as c increase, k′ decrease. This necessitates k′ > 0 within the k′ = 0 curve and k′ < 0 outside the k′ = 0 curve. Putting it differently, k will increase within the k′ = 0 curve and k will decrease outside the k′ = 0 curve. This movement is indicated by the solid arrows accompanied by letter ‘k’ (Fig. 5.20).

Fig. 5.20

Economic growth in an intertemporal optimization model

The steady state is at E (c*, k*), which is the intersection of the k′ = 0 curve and the c′ = 0 curve. For simplicity, we assume that the steady state is at optimal. Since c′ = 0, we have c = c* (this assumption can be relaxed and the analysis is similar, but the graphs will be more complicated). According to Eq. (5.32), c′ and c move in opposite directions. As a result, c′ > 0 when c < c*, and c′ < 0 when c > c*. In other words, c will increase when c < c*, and c will decrease when c > c*. The movement of c is indicated by the dotted arrows accompanied by letter ‘c’.

The phase diagram for c and k indicates that the economy can converge to a steady state E either (a) when c < c*, k < k*, and within the k′ = 0 curve, or (b) when k > k* and outside of the k′ = 0 curve.

Panel (b) of Fig. 5.15 shows the evolution of the economy over time. k′ = 0 is now a curve space and c′ = 0 is a vertical plane at c = c*. The tangent line of these two spaces SS shows the steady state of the economy. An economy at point A can reach the steady state SS through a path PP.

However, the e^θt term in Eq. (5.34) shows that the existence of the steady state is conditional on the lifting of the consumption ceiling over time. If the consumption ceiling (m) is fixed, the term e^θt in Eq. (5.32) necessitates that c will decrease over time. As a result, the c′ = 0 plane will be inclined to the k axis and intersect with k′ = 0 space on two curves SS₁ and SS₂. The economic equilibrium will evolve along either curve. The time path of economic growth is shown as the dotted arrow PP₁ or PP₂. Since the c and k on either SS₁ or SS₂ will change over time, there is no steady state—the consumption will decrease continuously.

In short, the intertemporal equilibrium model of single-commodity with consumption ceiling shows that, thanks to the fixed consumption ceiling, the economy will not reach a steady state—the consumption will keep falling in the long run. To reach a steady state, the consumption ceiling must keep increasing. Since the consumption of any commodity has a fixed ceiling according to Axiom 1, the only way to increase the consumption ceiling for the economy is to increase the variety of commodities. That is, product innovation is the key to reaching a steady state, or balanced growth.

Appendix 2 (for Section 5.6.2): On Investment-Consumption Dependency

Our new theory is rested on an assumption that the amount of investment depends on the expected future consumption. Here we use empirical data to examine the investment-consumption dependency axiom.

The standard econometric approach is to identify all factors related to private investment and use them to do a multi-regression. However, there are two shortcomings with this approach. One is that it is impossible to include all relevant variables and exclude irrelevant variables. The other is that nobody knows the correct function form for regression. Most econometricians simply assume a linear or log function for convenience. Thus, this approach is subject to data distortion if irrelevant variables are included or if the function form is incorrect. Since we are interested only in the correlation between investment and consumption, we use two variable regressions to avoid data distortion. This approach implies that the impact of other variables is negligible, so the results from this approach, just like results from other econometric approaches, are only indicative.

First, we examine the linkage between investment and consumption by using the 1929–2017 yearly US private consumption and private investment data, which are freely available from the website of the Bureau of Economic Analysis (BEA). The regression shows that private consumption is highly correlated with investment in all periods, i.e., current, lagged and leading investments. For example, the result of regressing of the current private investment ‘investpriv’ on current private consumption ‘consump’ using Stata software is shown in Fig. 5.21.

Fig. 5.21

Results of regressing US investment on consumption

The low standard error and high t-value show the extremely high significance of the consumption variable. The R-squared value 0.9795 is close to the maximum value of 1, which indicates the extremely high power of the consumption variable in explaining the behaviour of private investment. The results of regressing lagged investment on current consumption and the results of regression of leading investment on current consumption are not displayed here but they are very similar to those in Fig. 5.21. These results let us wonder about the types of investors’ behaviours: rational expectation, adaptive expectation, or adjusting investment immediately according to current consumption?

Actually, this regression suffers from a serious defect. Both private investment and consumption demonstrate a growing trend (see Fig. 5.22). This growing trend may be caused by other factors, e.g. the increased size of the US economy over time. Any two time series with the same trend will have a strong positive correlation but will indicate no causality between them.

Fig. 5.22

The US consumption and investment from 1927 to 2017

To avoid the problem of the common trend, we use the first difference (i.e. the yearly change) of private investment and consumption to conduct regression. The regression results are showing in Fig. 5.23.

Fig. 5.23

Results of regressing changes in yearly investment and consumption

In Fig. 5.23, we used a prefix of d for all variables (e.g. dinvestpriv, dconsump) to indicate that they are differenced values. The results show that, although the correlation between differenced investment and differenced consumption are very significant in all three regressions (showing by the high t-value, low standard error and very low p-value), the R-squared reduced significantly, compared with those in Fig. 5.21. For example, the R-squared for differenced investment and consumption (0.3752) decreased by more than half, compared with the value of R-squared of 0.9795 in Fig. 5.21. This is because we have excluded the misleading correlation due to a growth trend. Nevertheless, the reduced values of R-squared in Fig. 5.23 are still reasonably high considering that we omitted other variables which may affect investment. The R-squared value for the regression involved in the differenced lagged investment (dinvestlag1) and differenced lagged consumption (dconsulag1) are 0.2245 and 0.0774, which are much smaller than 0.3752. This indicates that the correlation between the differenced current investment and the differenced current consumption is much higher, so we may conclude that investors tend to adjust their investment immediately based on the current consumption situation.

However, this explanation is still not rigorous. One reason is that both the change in current consumption and in current investment may result from other common factors (e.g. a change in GDP level will affect consumption and investment in a similar way), so the correlation between differenced current consumption and differenced current investment may not indicate any causality between them. The other reason is related to the yearly data. One year is a long time frame for investors. If investors adjust their investment decision in the later part of the year according to the consumption data in the earlier part of the year, the yearly data would mask this investment behaviour.

To avoid the shortcoming in the above regression, we employ the quarterly US consumption and investment data from 1947Q1 to 2017Q4, provided by the Federal Reserve Bank of St Louis. Again, we use the first-differenced data to avoid high correlation caused by the trend of time series. The results of regressing current private investment, investment lagged by 1 period and investment with 1-period lead on private consumption are shown in Fig. 5.24.

Fig. 5.24

Results of regressing changes in quarterly investment and consumption

The regression results show the significance of consumption in all cases. This is not surprising because we only have one explanatory variable here. The R-squared values suggest that the correlation between the leading investment and current consumption are significantly higher than those other two cases (regression between current investment and current consumption, and regression between lagged investment and current consumption). By trying a different number of lags and leads, we find the highest R-squared value between current consumption and investment with a lead of one quarter shown in Fig. 5.24 is highest. These empirical results tend to suggest that investors engage in adaptive expectation so the consumption lagged by one quarter has a strong influence on investment decisions.

Appendix 3 (for Section 5.6.3): On R&D Investment, Innovations, and Economic Growth

R&D investment is supposed to have a positive impact on innovations and thus on economic growth, so we examine the relationship between R&D investment, innovation, and economic growth. Considering the complications caused by many variables explained in Appendix 2, we use two variables models and focus on the correlation between the pairs.

We use the real GDP as an indicator of the performance of the economy. The 1959–2015 US real GDP data is obtained from the BEA. The US R&D data from 1959 to 2007 are also available from the innovation account on the BEA website. The number of innovations is indicated by the number of patent applications and patent approvals. The data on US patent application and patent approvals from 1963 to 2015 are obtained from the US Patent and Trademark Office. Since all data display positive trends, we use the first-differenced data (changes over previous year) to avoid misleading high correlations.

Figure 5.25 shows the results of regressing different periods of real GDP on the number of patent applications with US origins. Since we are dealing with first-differenced data for all variables, we omit the prefix ‘d’, which were used in Figs. 5.23 and 5.24. The R-squared value of 0.3444 for the regression between current GDP (gdp) and current US patent applications (patappus) indicates the high relevance between these two variables. However, this correlation cannot be interpreted as the positive contribution of innovation (indicated by patent application) to economic growth (indicated by GDP) because there is a significant time gap between patent application and implementation of the innovation. On the contrary, it is more likely that the condition of the economy has an influence on patent application: a better economic performance indicated by a higher GDP means that the patent applicants have more resources to file patent applications.

Fig. 5.25

Results of regressing changes in US GDP and patent applications

Due to the time lag between the patent application and the implementation of patent technology, we can examine the impact of patent technology on economic performance by regressing current GDP on lagged patent application numbers or, alternatively, by regressing leading GDP on current patent application numbers. As leads of GDP increase, we found the R-squared value decreases; e.g., the R-squared value for GDP of 2 leads (gdplead2) is 0.1576, which is less than half of that for regression of current GDP on current patent application numbers. However, as the number of leads of GDP increases to 5 (i.e. gdplead5), the R-squared value increases to 0.3851, which is even higher than that from regressing of current GDP on current patent application numbers. When we increase the number of leads for GDP, the R-squared value starts to decrease again. The high R-squared value for GDP of 5 leads cannot be explained as the impact of GDP on patent applications, so it tends to indicate a causality from patent application to GDP. Namely, the patent applications have a significant impact on real GDP, but with about a 5-year lag.

Next, we examine the impact of R&D investment on innovation by regressing the various periods of US-origin patent application numbers on R&D investment in the USA. The regression results are shown in Fig. 5.26.

Fig. 5.26

Results of regressing changes in patent applications and investment

The R-squared value of 0.3247 between current total R&D (rndtotal) and patent application numbers (patappus) indicates that the two variables are highly correlated. However, this correlation may not indicate any causality between these two variables because the causality running from R&D investment to patent application requires a significant time gap; meanwhile, it is implausible that the number of patent applications will instantly affect R&D investment. The correlation in the current period is most likely caused by a common factor: good economic conditions mean the firms have more money to invest in research and also have more money to file patent applications.

As the leads for patent applications increase, the R-squared value decreases sharply. For example, the R-squared value for patent application of 4 leads (patappusf4) becomes as little as 0.0593. However, as the leads increase further, the R-squared value peaks at 0.3016 when the lead number is 6 (patappusf6), and then it starts to decline again. This result may be interpreted as the impact of current R&D on the number of innovations with a 6-year lag.

However, Fig. 5.27 shows the results of regressing different periods of real GDP on total R&D investment in USA. The very high R-squared value of 0.7156 indicates a very strong correlation between current GDP (gdp) and current R&D investment (rndtotal). Again, this correlation is more likely due to the impact of GDP on R&D investment. R&D investment cannot affect GDP instantly, so the only plausible explanation is that firms may have more money to invest during good economic times. As the number of leads for GDP increases, the R-squared value decreases.

Fig. 5.27

Results of regressing changes in US GDP and R&D investment

Figure 5.27 also shows that the R-squared value for GDP of 3 leads (gdplead3) decreased to 0.3308, less than half of the R-squared value for current GDP on current R&D. Increasing the GDP leads further, we find that the R-squared values started to increase and peaked at 0.5564 when the lead number is 9 (gdplead9). This high R-squared tends to indicate that the impact of R&D on the GDP has a lag of about 9 years.

Although the above interpretation is plausible, there may be alternative explanations about the results. It may be argued that the correlation within the time series (i.e. autocorrelation) may be responsible for the high correlation between patent application numbers and GDP of 5 leads, between R&D investment and patent applications of 6 leads, and between R&D investment of GDP of 9 leads.

The correlogram tests indeed indicate that there are autocorrelations for each time series. Judged by the 95% confidence level, the autocorrelations are significant between the current GDP and the GDP of 1–3 lags, between the current R&D and the R&D of 1 or 2 lags, and between the current patent applications and the patent applications of 5 or 6 lags. The autocorrelations between the current patent applications and those of around 5 lags may have contributed to the high R-squared value for regression between the current patent applications and the GDP of 5 leads, and it may also have contributed to the correlation between the current R&D and the patent applications of 6 leads. However, the autocorrelation of 1–3 lags in time series GDP and the autocorrelation of 1–2 lags in time series R&D can lead to a correlation between current R&D and the GDP of maximum 5 leads, so they cannot explain the high R-squared value between current R&D and GDP of 9 leads. A more plausible explanation is as follows. R&D investment has a positive impact on innovations with about 5-year lags. Meanwhile, innovation indicated by patent applications have a positive impact on real GDP after about 5 years. Consequently, R&D influences GDP through innovation after about 10 years.