Optimal Planning of Asian Expressway Network with Dynamic Interregional Input–Output Programming Model

Abstract

The optimal assignments of the exogenously given amount of capital funds for transportation infrastructures and social overhead capitals are treated in Chap. 6. The themes are serious agendas in that day and analyzed with an empirical application of the interregional input–output programming model. As an extension of the shipment activities initiated by Moses in order to include the transportation sector explicitly, shipment activities among the regions and sectors are specified based on the interregional input–output table of competitive-import type. Namely, interregional trades are endogenously determined responding to the relative advantage of regional economies that are mainly dependent on the accessibility to the markets. Usually, the trade patterns which are endogenously and therefore optimally determined should be critically different from the presupposed trade pattern inherent to, for example, Isard-type model. The optimality of the assignment of capital funds for potential investment targets is obtained thanks to the opportunity cost criteria that is intrinsically built into the algorithm for solving the linear programming model. However, the optimality obtained is a static one (Hirschman 1968; Kohno 1991a, b, c, 1992, 1994; Kohno et al. 1987).

Appendices

Appendix 1: Mathematical Expression of the Model

7.1.1 Index, Set of Indices, and Index Function

• I≡a set of sector codes (indices).

• Q≡ a set of indices for varieties of housing quality.

• J≡ a set of zone/region codes in China.

• $\hat{J}\equiv$ a set of zone codes outside China.

• JJ≡ a set of all zone codes $\equiv J\cup \hat{J}.$

• I_S≡ a set of service sector codes.

• I_o≡ a set of other service sector codes.

• I_I≡ a set of non-service sector codes.

• I_ν≡ a set of transport sector codes.

• ν₁≡ a set of truck transport sector codes.

• ν₂≡ a set of railway transport sector codes.

• ν₃≡ a set of coastal/water shipment sector codes.

• ν₄≡ a set of harbor distribution sector codes.

• I ≡ I_I ∪ I_o ∪ I_ν ≡ I_I ∪ I_S.

• I_S ≡ I_o ∪ I_ν.

• MT(i, j, r)≡ a set of (network) route codes, each of which connects regions i and j using transportation infrastructure links of rank 1 to rank r (i, j ϵ JJ, 1 ≤ r ≤ 4).

• ML≡ a set of transportation infrastructure codes.

• MM≡ a set of indices of transportation infrastructure facility ≡{1, 2, 3, 4, 5}; 1: expressway; 2: railway; 3: infrastructure for coastal and water shipment; 4: ordinary road infrastructure; 5: harbor infrastructure.

• JCT(j)≡ a set of transportation infrastructure (link) codes which (partially or totally) belong to zone j.

• MS(j)≡ an index function that converts transportation infrastructure (link) code into 1, 2, 3, 4, or 5 as follows:

MS(j)	=	1	if	101 ≤ j ≤ 199;
		2	if	201 ≤ j ≤ 299;
		3	if	301 ≤ j ≤ 399;
		4	if	401 ≤ j ≤ 499; and
		5	if	501 ≤ j ≤ 599.

• NR(t) ≡ a rank index that specifies transportation infrastructures (links) available at period t.

• JH(k, r) ≡ {(i, j, l)∣ expressway link in zone k constitutes routes l which connect zones i and j using transportation infrastructures (links) of rank 1 to rank r}.

• JRL(k, r) ≡ {(i, j, l)∣ railway link in zone k constitutes routes l which connect zones i and j using transportation infrastructures (links) of rank 1 to rank r}.

• JS(k, r) ≡ {(i, j, l)∣ coastal/water shipment infrastructure (link) in zone k constitutes routes l which connect zones i and j using transportation infrastructures (links) of rank 1 to rank r}.

• JR(k, r) ≡ {(i, j, l)∣ ordinary road infrastructure (link) in zone k constitutes routes l which connect zones i and j using transportation infrastructures (links) of rank 1 to rank r}.

• JP(j)≡ a set of harbor codes which is located in zone j.

• MOOD(k, r)≡ {(i, j, l)∣ transportation infrastructure (link) k constitutes routes l which connect zones i and j using transportation infrastructures (links) of rank 1 to rank r}.

• JTH≡ {(i, j)∣ routes that connect between zones (regions/countries) i and j use expressway link ($i, j\epsilon \hat{J}$)}.

• MLC≡ a set of codes of transportation infrastructures (link) which (totally or partially) belong to China.

7.1.2 Variables

• Y_ij(t): total production of sector i in zone j at period t (i ϵI, jϵJ, 0 ≤ t ≤ N_h).

• ΔK_ij(t): industrial capital formation of sector i in zone j at period t (i ϵI, jϵJ, 0 ≤ t ≤ N_h).

• ΔH_ij(t): housing capital formation of housing quality type i in zone j at period t

  (i ϵQ, jϵJ, 0 ≤ t ≤ N_h).

• ΔSK_j(t): social (overhead) capital formation in zone j at period t (jϵJ, 0 ≤ t ≤ N_h).

• DW_ij(t): workers employed by (employable for) sector i in zone j at period t

  (i ϵI, jϵJ, 0 ≤ t ≤ N_h).

• ${CF}_{ij}^{kl}(t):$ shipment of commodity k from zones i to j using route l at period t

  (i, jϵJJ, kϵI_I, lϵMT(i, j, NR(t)), 0 ≤ t ≤ N_h).

• C_kj(t): consumption of goods k in zone j which exceeds the minimum permissible level at period t

  (jϵJ, kϵI_I ∪ I_o, 0 ≤ t ≤ N_h ).

• ΔR_p(t): capital formation of transportation infrastructure (link) p at period t

(pϵML, 0 ≤ t ≤ N_h).

• ${EX}_i^k(t):$ export of goods k from China to zone i outside China at period t ($i\epsilon \hat{J}, k\epsilon {I}_I$, 0 ≤ t ≤ N_h).

• ${IM}_i^k(t):$ import of goods k from zone i outside China into China at period t

($i\epsilon \hat{J}, k\epsilon {I}_I$, 0 ≤ t ≤ N_h).

• BEX(t): international balance of payments of China at period t (0 ≤ t ≤ N_h).

• K_ij(t): industrial capital stock for sector i in zone j at period t (i ϵI, jϵJ, 0 ≤ t ≤ N_h).

• H_ij(t): stock of housing capital of quality i in zone j at period t (iϵQ, jϵJ, 0 ≤ t ≤ N_h).

• R_P(t): stock of transportation infrastructure (link) p at period t (pϵML, 0 ≤ t ≤ N_h).

• SK_j(t): stock of social (overhead) capital in zone j at period t (jϵJ, 0 ≤ t ≤ N_h).

• GH_ij(t): natural growth of workers employable by sector i in zone j at period (t + 1)

  (i ϵI, jϵJ, 0 ≤ t ≤ N_h − 1).

• ISH_ij(t): social increase in workers employable by sector i in zone j at period (t + 1)

  (i ϵI, jϵJ, 0 ≤ t ≤ N_h − 1).

• OSH_ij(t): social decrease in workers employable in sector i in zone j at period (t + 1)

  (i ϵI, jϵJ, 0 ≤ t ≤ N_h − 1).

• SSK_j(t): stock of social (overhead) capital available in zone j at period t exceeding the minimum permissible level (jϵJ, 0 ≤ t ≤ N_h).

• GNP(t):gross national product (income) of China at period t (0 ≤ t ≤ N_h).

• NNP(t): net national product of China at period t (0 ≤ t ≤ N_h).

• INV(t): gross investment of China at period t (0 ≤ t ≤ N_h).

• NI(t): net investment of China at period t (0 ≤ t ≤ N_h).

• CC(t): total consumption of China at period t (0 ≤ t ≤ N_h).

7.1.3 Parameters

• $\left({a}_{ki}^j\right)$: input–output coefficient matrix in zone j (jϵJ, kεI_I ∪ I_o, iϵI).

• (μ_ki): industrial capital formation coefficient matrix (kϵI_I ∪ I_o, iϵI).

• (λ_ki): housing capital formation coefficient matrix (kϵI_I ∪ I_o, iϵQ).

• (ψ_k): social overhead capital formation coefficient matrix (kϵI_I ∪ I_o).

• σ_k(t): minimum permissible consumption level of goods k per worker at period t (kϵI_I ∪ I_o, 0 ≤ t ≤ N_h).

• Y^d: coefficient for conversion from variable per day to variable per period.

• $\left({\tau}_i^k\right)$: capital formation matrix of transportation infrastructure mode i (iϵMM, kϵI_I ∪ I_o).

• ω_p(j): ratio of the part of transportation infrastructure (link) p which belongs to zone j to the total length of link p (pϵML, jϵJJ).

• DDR(p): geographical length of transportation infrastructure (link) p (pϵML).

• ${a}_i^k:$ coefficient of production of transport/distribution sector i per one unit distance transportation of goods k (kϵI_I, iϵI_ν).

• DH(p, q, k, l): economic distance of expressway links located in zone k if any that constitute route l from zone p to zone q (kϵJ, p, qϵJJ, lϵMT(p, q, 4)).

• DRL(p, q, k, l): economic distance of railway links located in zone k if any that constitute route l from zone p to zone q (kϵJ, p, qϵJJ, lϵMT(p, q, 4)).

• DS(p, q, k, l): economic distance of coastal/water shipment links located in zone k if any that constitute route l from zone p to zone q (kϵJ, p, qϵJJ, lϵMT(p, q, 4)).

• DR(p, q, k, l): economic distance of ordinary road links located in zone k if any that constitute route l from zone p to zone q (kϵJ, p, qϵJJ, lϵMT(p, q, 4)).

• ρ: social discount rate.

• $\hat{\rho}$: borrowing interest rate.

• $\overset{\sim }{\rho }$: growth rate of the permissible level of consumption.

• ρ^S: growth rate of the permissible level of social overhead capital stock.

• m^k: markup ratio in order to add expressway toll onto charge for utilization of truck (freight) transport service which is used for the transportation of goods k in China in case shipment of goods k from foreign zone to foreign zone uses Asian Expressway in China.

• $\overline{UEX_k}(t)$: upper limit to export of China to zone (country) k at period t ($k\epsilon \hat{J},0\le t\le {N}_h\Big)$.

• $\overline{LEX_k}(t)$: lower limit to export of China to zone (country) k at period t ($k\epsilon \hat{J},0\le t\le {N}_h\Big)$.

• $\overline{UIM_k}(t)$: upper limit to import of China from zone (country) k at period t ($k\epsilon \hat{J},0\le t\le {N}_h\Big)$.

• $\overline{LIM_k}(t)$: lower limit to import of China from zone (country) k at period t ($k\epsilon \hat{J},0\le t\le {N}_h\Big)$.

• $\overline{LBEX}$(t): upper limit to the accumulated international deficit from period 0 to t (0 ≤ t ≤ N_h).

• $\overline{THEX_{pq}}$(t): upper limit to the total export from zones p to q using expressway in China at period t (pqϵJTH, 0 ≤ t ≤ N_h).

• α_ij: output-capital ratio of sector i in zone j (iϵI, jϵJ).

• β_ij: output-labor ratio of sector i in zone j (iϵI, jϵJ).

• d^H: reciprocal of the number of workers per household.

• ${g}_i^k$: coefficient which converts transportation of goods k to load to transport infrastructure of mode i (kϵI_I, iϵMM).

• d^s(t): minimum permissible level of social overhead capital per worker at period t ( 0 ≤ t ≤ N_h).

• ${\hat{\delta}}_i^K$: depreciation rate for industrial capital stock of sector i (iϵI).

• ${\hat{\delta}}_i^H$: depreciation rate for housing capital of quality i (iϵQ).

• ${\hat{\delta}}^S$: depreciation rate for social overhead capital.

• ${\hat{\delta}}_i^R$: depreciation rate for transport infrastructure of mode i (iϵMM).

• ${\hat{n}}^h$: natural growth rate of worker.

• r^IS: ratio of the upper limit on the total movement of workers between sectors to the total worker.

• N_h: planning horizon

• WC(j): ratio of the part of transportation infrastructure (link) j which belongs to China to the total length of link j (jϵMLC).

• ${\nu}_i^R$: coefficient which converts transportation infrastructure (link) stock in terms of [capacity × (link length in km)] into monetary value (iϵMM).

• δ^NH(t):it is 1 if 0 ≤ t ≤ N_h − 1, otherwise it is 0.

• ${\alpha}_{kj}^C$: valuation coefficient of consumption of goods k, of which amount exceeds the minimum permissible level.

• ${\alpha}_j^S$: valuation coefficient of consumption of social overhead capital, of which amount exceeds the minimum permissible level.

7.1.4 Structural Equation and Objective Function

7.1.4.1 Market Flow Condition 1: Non-service

$$ {\sum}_{i\epsilon I}{\alpha}_{ki}^j\bullet {Y}_{i j}(t)+{\sum}_{i\epsilon I}{\mu}_{ki}\bullet \Delta {K}_{i j}(t)+{\sum}_{i\epsilon Q}{\lambda}_{ki}\bullet {\Delta H}_{i j}(t)+{\psi}_k\bullet \Delta {SK}_j(t)+\sum \limits_{i\epsilon I}{\sigma}_k(t)\bullet {DW}_{i j}(t) $$

$$ +{C}_{kj}(t)+\sum \limits_{s\epsilon JCT(j)}{\tau}_{MS(s)}^k\bullet {\omega}_s(j)\bullet DDR(s)\bullet {\nu}_{MS(s)}^R\bullet \Delta {R}_s(t) $$

$$ -\sum \limits_{i\epsilon J J}{\sum}_{l\epsilon MT\left(i,j, NR(t)\right)}{CF}_{i j}^{kl}(t)\le 0\ \left( k\epsilon {I}_I, j\epsilon J,0\le t\le {N}_h\right) $$

(7.49)

7.1.4.2 Market Flow Condition 2: Other Service

• $$ {\sum}_{i\epsilon I}{\alpha}_{ki}^j\bullet {Y}_{i j}(t)+{\sum}_{i\epsilon I}{\mu}_{ki}\bullet \Delta {K}_{i j}(t)+{\sum}_{i\epsilon Q}{\lambda}_{ki}\bullet {\Delta H}_{i j}(t)+{\psi}_k\bullet \Delta {SK}_j(t)+\sum \limits_{i\epsilon I}{\sigma}_k(t)\bullet {DW}_{i j}(t) $$
• $$ +{C}_{kj}(t)+\sum \limits_{s\epsilon J CT(j)}{\tau}_{MS(s)}^k\bullet {\omega}_s(j)\bullet DDR(s)\bullet {\nu}_{MS(s)}^R\bullet \Delta {R}_s(t)-{Y}_{kj}(t)\le 0\ \left( k\epsilon {I}_o,\kern0.5em j\epsilon J,\kern0.5em 0\le t\le {N}_h\right) $$

  (7.50)

7.1.4.3 Market Flow Condition 3: Transport/Distribution Service

7.1.4.3.1 Truck (Freight) Transportation Service

• $$ \sum \limits_{s\epsilon {I}_I}{a}_k^s\Big\{\sum \limits_{\left(p,q,l\right)\epsilon JH\left(j, NR(t)\right)} DH\left(p,q,j,l\right)\bullet {CF}_{pq}^{s l}(t) $$

$$ +\sum \limits_{\left(p,q,l\right)\epsilon JR\left(j, NR(t)\right)} DR\left(p,q,j,l\right)\bullet {CF}_{pq}^{sl}(t)\Big\}-{Y}_{kj}(t)\le 0\ \left( k\epsilon {\nu}_1, j\epsilon J,0\le t\le {N}_h\right) $$

(7.51)

7.1.4.3.2 Railway Transportation Service

$$ \sum \limits_{s\epsilon {I}_I}\ \sum \limits_{\left(p,q,l\right)\epsilon JRL\left(j, NR(t)\right)}{a}_k^s DRL\left(p,q,j,l\right)\ {CF}_{pq}^{s l}(t)-{Y}_{kj}(t)\le 0\ \left( k\epsilon {\nu}_2, j\epsilon J,0\le t\le {N}_h\right) $$

(7.52)

7.1.4.3.3 Coastal/Water Shipment Service

$$ \sum \limits_{s\epsilon {I}_I}\ \sum \limits_{\left(p,q,l\right)\epsilon JS\left(j, NR(t)\right)}{a}_k^s DS\left(p,q,j,l\right)\ {CF}_{pq}^{s l}(t)-{Y}_{kj}(t)\le 0\ \left( k\epsilon {\nu}_3, j\epsilon J,0\le t\le {N}_h\right) $$

(7.53)

7.1.4.3.4 Harbor Distribution Service

$$ \sum \limits_{s\epsilon {I}_I}{\sum}_{i\epsilon J P(j)}{\sum}_{\left(p,q,l\right)\epsilon MOOD\left(i, NR(t)\right)}{a}_k^s\ {CF}_{pq}^{s l}(t)-{Y}_{kj}(t)\le 0\ \left( k\epsilon {\nu}_4, j\epsilon J,0\le t\le {N}_h\right) $$

(7.54)

7.1.4.4 Shipment Balance Equation 1: Shipment from Zones in China

$$ \sum \limits_{j\epsilon J J}\ \sum \limits_{l\epsilon MT\left(i,j, NR(t)\right)}{CF}_{ij}^{kl}(t)-{Y}_{ki}(t)\le 0\ \left( k\epsilon {I}_I, i\epsilon J,0\le t\le {N}_h\right) $$

(7.55)

7.1.4.5 Shipment Balance Equation 2

7.1.4.5.1 Export of China

$$ \sum \limits_{k\epsilon {I}_I}\ \sum \limits_{j\epsilon J}\ \sum \limits_{l\epsilon MT\left(j,p, NR(t)\right)}{CF}_{j p}^{k l}(t)-{EX}_p(t)=0, $$

$$ \overline{LEX_p}(t)\le \overline{EX_p}(t)\le \overline{UEX_p}\ \left(\mathrm{t}\right)\ \left(\ p\epsilon \hat{J},0\le t\le {N}_h\right) $$

(7.56)

7.1.4.5.2 Import of China

$$ \sum \limits_{k\epsilon {I}_I}\ \sum \limits_{j\epsilon J}\ \sum \limits_{l\epsilon MT\left(p,j, NR(t)\right)}{CF}_{pj}^{k l}(t)-{IM}_p(t)=0, $$

$$ \overline{LIM_p}(t)\le \overline{IM_p}(t)\le \overline{UIM_p}\ \left(\mathrm{t}\right)\ \left(\ p\epsilon \hat{J},0\le t\le {N}_h\right) $$

(7.57)

7.1.4.5.3 Balance of International Payments: Cumulative Deficit Constraint

• $$ \sum \limits_{\tau =1}^{t+1}{\left(1+\hat{\rho}\right)}^{\tau -1}\ \Big\{\ \sum \limits_{k\epsilon {I}_I}{\sum}_{i\epsilon \hat{J}}{\sum}_{j\epsilon J}{\sum}_{l\epsilon MT\left(i,j, NR\left(\tau -1\right)\right)}{CF}_{i j}^{k l}\left(\tau -1\right) $$
• $$ -\sum \limits_{p\epsilon J}\sum \limits_{k\epsilon {I}_I}\sum \limits_{ij\epsilon J TH}\sum \limits_{l\epsilon MT\left(i,j, NR\left(\tau -1\right)\right)}\left(1+{m}^k\right)\bullet {a}_1^k\bullet DH\left(i,j,p,l\right)\bullet {CF}_{ij}^{k l}\left(\tau -1\right) $$
• $$ -\sum \limits_{k\epsilon {I}_I}\sum \limits_{i\epsilon J}\sum \limits_{j\epsilon \hat{J}}\sum \limits_{l\epsilon MT\left(i,j, NR\left(\tau -1\right)\right)}{CF}_{i j}^{k l}\left(\tau -1\right)\Big\}- BEX(t)=0 $$

$$ \mathrm{BEX}\left(\mathrm{t}\right)\le \overline{LBEX}(t)\ \left(\ 0\le t\le {N}_h\right) $$

(7.58)

7.1.4.6 Shipment Balance Equation 3: Between Foreign Countries (Regions)

$$ \sum \limits_{k\epsilon {I}_I}\sum \limits_{l\epsilon MT\left(i,j, NR(t)\right)}\ {CF}_{ij}^{k l}(t)\le \overline{THEX_{ij}}(t)\ \left(\ ij\epsilon JTH,0\le t\le {N}_h\right) $$

(7.59)

7.1.4.7 Balance Between Demand and Supply of Industrial Capital

$$ {Y}_{ij}(t)-{\alpha}_{ij}{K}_{ij}(t)\le 0\ \left(\ j\epsilon J, i\epsilon I,0\le t\le {N}_h\right) $$

(7.60)

7.1.4.8 Balance Between Demand and Supply of Labor

$$ {Y}_{ij}(t)-{\beta}_{ij}{DW}_{ij}(t)\le 0\ \left(\ j\epsilon J, i\epsilon I,0\le t\le {N}_h\right) $$

(7.61)

7.1.4.9 Balance Between Demand and Supply of Housing Stock

$$ {d}^H\sum \limits_{i\epsilon I}{DW}_{i j}(t)-\sum \limits_{i\epsilon Q}{H}_{i j}(t)\le 0\ \left(\ j\epsilon J,0\le t\le {N}_h\right) $$

(7.62)

7.1.4.10 Balance Between Demand Against and Supply of Transportation Infrastructure

$$ \sum \limits_{k\epsilon {I}_I}\sum \limits_{\left(i,j,l\right)\epsilon MOOD\left(p, NR(t)\right)}\ {g}_{MS(p)}^k\ {CF}_{ij}^{k l}(t)-{Y}^d\ {R}_p(t)\le 0\ \left(\ p\epsilon ML,0\le t\le {N}_h\right) $$

(7.63)

7.1.4.11 Balance Between Demand and Supply of Social (Overhead) Capital (Other Social Capitals)

$$ {d}^S(t)\sum \limits_{i\epsilon I}{DW}_{i j}(t)-{SK}_j(t)+{SSK}_j(t)=0\ \left(\ j\epsilon J,0\le t\le {N}_h\right) $$

(7.64)

7.1.4.12 Formation of Capital Stock

7.1.4.12.1 Formation of Industrial Capital

$$ {K}_{ij}\left(t+1\right)-\left\{1-{\hat{\delta}}_i^k\right\}{K}_{ij}(t)-\Delta {K}_{ij}(t)=0\ \left( i\epsilon I, j\epsilon J,0\le t\le {N}_h-1\right) $$

(7.65)

7.1.4.12.2 Formation of Housing Capital

$$ {H}_{ij}\left(t+1\right)-\left\{1-{\hat{\delta}}_i^H\right\}{H}_{ij}(t)-\Delta {H}_{ij}(t)=0\ \left( i\epsilon Q, j\epsilon J,0\le t\le {N}_h-1\right) $$

(7.66)

7.1.4.12.3 Formation of Social Capital (Other Capitals)

$$ {SK}_j\left(t+1\right)-\left\{1-{\hat{\delta}}^S\right\}{SK}_j(t)-\Delta {SK}_j(t)=0\ \left(\ j\epsilon J,0\le t\le {N}_h-1\right) $$

(7.67)

7.1.4.12.4 Formation of Transportation Infrastructure

$$ {R}_j\left(t+1\right)-\left\{1-{\hat{\delta}}_{MS(j)}^R\right\}{R}_j(t)-\Delta {R}_j(t)=0\ \left(\ j\epsilon MLC,0\le t\le {N}_h-1\right) $$

(7.68)

7.1.4.13 Population Growth and Migration

7.1.4.13.1 Natural Growth

$$ \sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{GH}_{i j}(t)-{\hat{n}}^h\bullet \sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{DW}_{i j}(t)=0\ \left(0\le t\le {N}_h-1\right) $$

(7.69)

7.1.4.13.2 Migration: Social Population Growth

$$ \sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{ISH}_{i j}(t)-\sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{OSH}_{i j}(t)=0\ \left(0\le t\le {N}_h-1\right) $$

(7.70)

$$ \sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{OSH}_{i j}(t)-{r}^{IS}\bullet \sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{DW}_{i j}(t)=0\ \left(0\le t\le {N}_h-1\right) $$

(7.71)

7.1.4.13.3 Population Growth: Natural Plus Social Growth

$$ {DW}_{ij}\left(t+1\right)-{DW}_{ij}(t)-{GH}_{ij}(t)-{ISH}_{ij}(t)+{OSH}_{ij}(t)=0\kern0.50em \left( i\epsilon I, j\epsilon J,0\le t\le {N}_h-1\right) $$

(7.72)

7.1.4.14 National Income Accounting of China

7.1.4.14.1 Gross National Income (GNP/GNI)

• $$ {\sum}_{j\epsilon J}{\sum}_{i\epsilon I}{Y}_{i j}(t)-{\sum}_{j\epsilon J}{\sum}_{i\epsilon I}{\sum}_{k\epsilon {I}_I\cup {I}_o}{a}_{k i}^j\bullet {Y}_{i j}(t) $$
• $$ +{\sum}_{p\epsilon J}{\sum}_{k\epsilon {I}_I}{\sum}_{ij\epsilon J TH}{\sum}_{l\epsilon MT\left(i,j, NR(t)\right)}{m}^k\bullet {a}_1^k\bullet DH\left(i,j,p,l\right)\bullet {CF}_{ij}^{k l}(t) $$
• $$ -{\sum}_{j\epsilon J}{\sum}_{i\epsilon I}\left({\sum}_{k\epsilon {I}_I\cup {I}_o}{\mu}_{k i}-1\right)\bullet \Delta {K}_{i j}(t)-{\sum}_{j\epsilon J}{\sum}_{i\epsilon Q}\left({\sum}_{k\epsilon {I}_I\cup {I}_o}{\lambda}_{k i}-1\right)\bullet \Delta {H}_{i j}(t) $$
• $$ -{\sum}_{j\epsilon J}\left({\sum}_{k\epsilon {I}_I\cup {I}_o}{\psi}_k-1\right)\bullet \Delta {SK}_j(t) $$
  $$ -{\sum}_{s\epsilon MLC}\left({\sum}_{k\epsilon {I}_I\cup {I}_o}{\tau}_{MS(s)}^k-1\right)\bullet WC(s)\bullet DDR(s)\bullet {\nu}_{MS(s)}^R\Delta {R}_s(t)- GNP(t)=0\ \left(0\le t\le {N}_h\right) $$

  (7.73)

7.1.4.14.2 Net National Product (NNP)

$$ GNP(t)-{\sum}_{j\epsilon J}{\sum}_{i\epsilon I}{\hat{\delta}}_i^K\bullet {K}_{i j}(t)-{\sum}_{j\epsilon J}{\sum}_{i\epsilon Q}{\hat{\delta}}^H\bullet {H}_{i j}(t)-{\sum}_{j\epsilon J}{\hat{\delta}}^S\bullet {SK}_j(t) $$

$$ -{\sum}_{j\epsilon MLC} WC(j)\bullet {\hat{\delta}}_{MS(j)}^R\bullet DDR(j)\bullet {\nu}_{MS(j)}^R\bullet {R}_j(t)- NNP(t)=0\ \left(0\le t\le {N}_h\right) $$

(7.74)

7.1.4.14.3 Gross Investment (INV)

• $$ \sum \limits_{j\epsilon J}\sum \limits_{i\epsilon I}\left(\sum \limits_{k\epsilon {I}_I\cup {I}_o}{\mu}_{k i}\right)\bullet \Delta {K}_{i j}(t)+\sum \limits_{j\epsilon J}\sum \limits_{i\epsilon Q}\left(\sum \limits_{k\epsilon {I}_I\cup {I}_o}{\lambda}_{k i}\right)\bullet \Delta {H}_{i j}(t) $$
• $$ +\sum \limits_{j\epsilon J}\left(\sum \limits_{k\epsilon {I}_I\cup {I}_o}{\psi}_k\right)\bullet \Delta {SK}_j(t)+\sum \limits_{s\epsilon MLC}\left(\sum \limits_{k\epsilon {I}_I\cup {I}_o}{\tau}_{MS(s)}^k\right)\bullet WC(s)\bullet DDR(s)\bullet {\nu}_{MS(s)}^R\bullet \Delta {R}_s(t) $$

$$ - INV(t)=0\ \left(0\le t\le {N}_h-1\right) $$

(7.75)

7.1.4.14.4 Net Investment (NI)

$$ \sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}\left\{{K}_{i j}\left(t+1\right)-{K}_{i j}(t)\right\}+\sum \limits_{i\epsilon Q}\sum \limits_{j\epsilon J}\left\{{H}_{i j}\left(t+1\right)-{H}_{i j}(t)\right\} $$

$$ +\sum \limits_{j\epsilon J}\left\{{SK}_j\left(t+1\right)-{SK}_j(t)\right\}+\sum \limits_{s\epsilon MLC} WC(s)\bullet DDR(s)\bullet {\nu}_{MS(s)}^R\bullet \left\{{R}_s\left(T+1\right)-{R}_s(t)\right\} $$

$$ - NI(t)=0\ \left(0\le t\le {N}_h-1\right) $$

(7.76)

7.1.4.14.5 Consumption (CC)

• $$ \sum \limits_{j\epsilon J}\sum \limits_{i\epsilon I}\sum \limits_{k\epsilon {I}_I\cup {I}_o}{\sigma}_k(t)\bullet {DW}_{i j}(t)+\sum \limits_{j\epsilon J}\sum \limits_{k\epsilon {I}_I\cup {I}_o}{C}_{k j}(t)- CC(t)=0 $$
• $$ \left(0\le t\le {N}_h\right) $$

  (7.77)

7.1.4.15 Objective Function

7.1.4.15.1 GNP(GNI) to Be Maximized

$$ \max . FV=\sum \limits_{t=0}^{N_h}{\left(\frac{1}{1+\rho}\right)}^t\mathrm{GNP}(t) $$

(7.78)

7.1.4.15.2 NNP to Be Maximized

$$ \max . FV=\sum \limits_{t=0}^{N_h}{\left(\frac{1}{1+\rho}\right)}^t\mathrm{NNP}(t) $$

(7.79)

7.1.4.15.3 Welfare Maximization with Lower Constraint on the Accumulation of Industrial Capital Stock at the End of Period

• $$ \mathit{\max}. FV=\sum \limits_{j\epsilon J}\sum \limits_{k\epsilon {I}_I\cup {I}_O}{\alpha}_{k j}^c(t)\bullet {C}_{k j}(t)+\sum \limits_{j\epsilon J}{\alpha}_j^S(t)\bullet {SSK}_j(t)+\sum \limits_{j\epsilon Q}\sum \limits_{j\epsilon J}{\alpha}_{ij}^H(t){H}_{ij}(t), $$

  (7.80)
• $$ \mathbf{st}.\sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{K}_{i j}\left({N}_h\right)\geqq \overline{K} $$

  (7.81)

7.1.4.15.4 Consumption Maximization with Lower Constraint on the Accumulation of Industrial Capital Stock at the End of Period

$$ \max . FV=\sum \limits_{t=0}^{N_h}{\left(\frac{1}{1+\rho}\right)}^t CC(t) $$

(7.82)

$$ \mathrm{st}.\sum \limits_{i\epsilon I}\sum \limits_{j\epsilon J}{K}_{i j}\left({N}_h\right)\geqq \overline{K} $$

(7.83)

7.1.4.16 Initial Conditions

K_ij(0), H_ij(0), R_p(0), SK_j(0), and DW_ij(0) are exogenously given with non-negative values.

Appendix 2: Dynamic Programming Model and Roundabout Production Through Space and Time

In order to explain the two concepts—roundabout production through space and time, we construct the following simple closed economy model: (1) only one kind of goods (e.g., composite goods) is produced⁴; (2) the economy is consisting of two regions, which are called—region 1 and region 2; (3) goods are produced in each region; (4) the production of goods is dependent on the capital stock and intermediate input of goods; (5) goods can be used for consumption, intermediate input, and capital stock formation in the region in which the goods are produced as well as in another region; (6) two regions are connected with each other through highway (transportation infrastructure) that is used for the shipment of goods to another region; (7) highway has the capacity that limits the shipment of goods between two regions.

The following equation system shows the essence of the roundabout production. Of course, all variables are nonnegative.

7.1.1 Production Function

• $$ {Y}_t^i={f}^i\left({K}_t^i,{X}_t^{ii}+{X}_t^{ji}\right)\ \left(i,j=1,2;i\ne j,t=0,1,2,\dots, N\right), $$

  (7.84)

in which: ${Y}_t^i$: production of goods in region i at period t; ${K}_t^i$: amount of capital stock in region i at period t; ${X}_t^{ij}$: intermediate input (shipment) of goods from region i to region j at period t (i, j = 1, 2; t = 0, 1, 2, …, N); fⁱ(∙): production function of goods in region i at period t, and N: the time horizon for the planning

7.1.2 Flow Condition of the Markets

• $$ {Y}_t^i={X}_t^{i1}+{X}_t^{i2}+{I}_t^{i1}+{I}_t^{i2}+{C}_t^{i1}+{C}_t^{i2}+{IR}_t^{i1}+{IR}_t^{i2}\ \left(i=1,2;t=0,1,2,\dots, N\right), $$

  (7.85)

in which: ${I}_t^{ij}$: investment (shipment) of goods from region i for the capital stock formation in region j at period t (i, j = 1, 2; t = 0, 1, 2, …, N); ${C}_t^{ij}$: shipment of goods from region i for the consumption in region j at period t (i, j = 1, 2; t = 0, 1, 2, …, N); and ${IR}_t^{ij}$: investment (shipment) of goods from region i for the improvements in the highway capacity in region j at period t (i, j = 1, 2; t = 0, 1, 2, …, N).

7.1.3 Stock Formation

$$ \Delta {K}_t^1={h}^1\left({I}_t^{11}+{I}_t^{21}\right)\ \left(t=0,1,2,\dots, N\right), $$

(7.86a)

$$ \Delta {K}_t^2={h}^2\left({I}_t^{12}+{I}_t^{22}\right)\ \left(t=0,1,2,\dots, N\right), $$

(7.86b)

$$ \Delta {R}_t^{12}={g}^{12}\left({IR}_t^{11}+{IR}_t^{21}\right)\ \left(t=0,1,2,\dots, N\right), $$

(7.86c)

$$ \Delta {R}_t^{21}={g}^{21}\left({IR}_t^{12}+{IR}_t^{22}\right)\ \left(t=0,1,2,\dots, N\right), $$

(7.86d)

in which: hⁱ(∙): function that converts investment of goods into increase in the capital stock in region i (i = 1, 2); g^ij(∙): function that converts investment of goods into increase in the stock of highway that can be only used for the shipment of goods from region i to region j (i, j = 1, 2; i ≠ j); and it is assumed that: (a) the capacity of lanes is differentiated between the lanes from region 1 to region 2 and from region 2 to region 1; (b) namely, the shipment of goods from region i to region j becomes loads on the highway capacity of the lane from region i to region j, and (c) increase in the capacity of highway lane from region i to region j can be made only by the investment of goods for the highway stock formation in region i ( i, j = 1, 2; i ≠ j).

7.1.4 Dynamic Equation

$$ {K}_t^i={K}_{t-1}^i+\Delta {K}_{t-1}^i\ \left(i=1,2;t=1,2,3,\dots, N\right) $$

(7.87)

$$ {R}_t^{12}={R}_{t-1}^{12}+\Delta {R}_{t-1}^{12}\ \left(t=1,2,3,\dots, N\right), $$

(7.88a)

$$ {R}_t^{21}={R}_{t-1}^{21}+\Delta {R}_{t-1}^{21}\ \left(t=1,2,3,\dots, N\right), $$

(7.88b)

$$ {K}_0^i\equiv {\overline{K}}^i\ \left(i=1,2\right), $$

(7.89)

$$ {R}_0^{ij}\equiv {\overline{R}}^{ij}\ \left(i,j=1,2;i\ne j\right), $$

(7.90)

in which: ${\overline{K}}^i$: initial capital stocks in region i (i = 1, 2), which are nonnegative constants; and ${\overline{R}}^i$: initial highway stocks in region i (i = 1, 2),which are nonnegative constants.

7.1.5 Highway Capacity Constraint

$$ {X}_t^{ij}+{I}_t^{ij}+{C}_t^{ij}+{IR}_t^{ij}\le {q}^i\left({R}_t^{ij}\right)\ \left(i,j=1,2;i\ne j\right), $$

(7.91)

in which: qⁱ(∙): function that converts the amount of highway stock into the highway capacity in terms of the shipment of goods; and it is assumed that the intra-regional shipment can be made with no highway capacity limit for simplicity of the explanation.

7.1.6 Definition of Vector Variable

We define the following vector variables for the simplicity of notation:

$$ {Y}^i\equiv \left({Y}_0^i,{Y}_1^i,{Y}_2^i,\bullet \bullet \bullet, {Y}_N^i\right)\ \left(i=1,2\right), $$

$$ {X}^{ij}\equiv \left({X}_0^{ij},{X}_1^{ij},{X}_2^{ij},\bullet \bullet \bullet, {X}_N^{ij}\right)\ \left(i,j=1,2;i\ne j\right), $$

$$ {I}^{ij}\equiv \left({I}_0^{ij},{I}_1^{ij},{I}_2^{ij},\bullet \bullet \bullet, {I}_N^{ij}\right)\ \left(i,j=1,2;i\ne j\right), $$

$$ {IR}^{ij}\equiv \left({IR}_0^{ij},{IR}_1^{ij},{IR}_2^{ij},\bullet \bullet \bullet, {IR}_N^{ij}\right)\ \left(i,j=1,2;i\ne j\right), $$

$$ {C}^{ij}\equiv \left({C}_0^{ij},{C}_1^{ij},{C}_2^{ij},\bullet \bullet \bullet, {C}_N^{ij}\right)\ \left(i,j=1,2;i\ne j\right), $$

$$ {C}^i\equiv {C}^{ii}+{C}^{ji}\ \left(i,j=1,2;i\ne j\right) $$

$$ {X}^i={X}^{ii}+{X}^{ji}\ \left(i,j=1,2;i\ne j\right) $$

$$ {K}^i\equiv \left({K}_0^i,{K}_1^i,{K}_2^i,\bullet \bullet \bullet, {K}_N^i\right)\ \left(i=1,2\right), $$

$$ {R}^{ij}\equiv \left({R}_0^{ij},{R}_1^{ij},{R}_2^{ij},\bullet \bullet \bullet, {R}_N^{ij}\right)\ \left(i,j=1,2;i\ne j\right), $$

$$ \Delta {K}^i\equiv \left(\Delta {K}_0^i,\Delta {K}_1^i,\Delta {K}_2^i,\bullet \bullet \bullet, \Delta {K}_N^i\right)\ \left(i=1,2\right),\mathrm{and} $$

$$ \Delta {R}^i\equiv \left(\Delta {R}_0^i,\Delta {R}_1^i,\Delta {R}_2^i,\bullet \bullet \bullet, \Delta {R}_N^i\right)\ \left(i=1,2\right). $$

A bundle, Yⁱ,is called—trajectory of the production in region i (i = 1, 2), a bundle, X¹², is called—trajectory of the shipment of goods from region 1 to region 2, and so on.

7.1.7 Feasible Trajectory of the Economy

We define bundles of vector variables, Tⁱ (i = 1, 2), as follow:

$$ {T}^1\equiv \left({Y}^1,{X}^{11},{X}^{21},{I}^{11},{I}^{21},{IR}^{11},{IR}^{21},{C}^{11},{C}^{21},{K}^1,{R}^{12},\Delta {K}^1,\Delta {R}^1\right),\mathrm{and} $$

$$ {T}^2\equiv \left({Y}^2,{X}^{12},{X}^{22},{I}^{12},{I}^{22},{IR}^{12},{IR}^{22},{C}^{12},{C}^{22},{K}^2,{R}^{21},\Delta {K}^2,\Delta {R}^2\right). $$

A bundle, Tⁱ, is called—trajectory of the economy of region i (i = 1, 2) and a bundle, T ≡ (T¹, T²), is called—a trajectory of the economy.

Suppose that the trajectory of consumption in each region is exogenously given as follows:

• $$ {C}^i={\overline{C}}^i\ \left(i=1,2\right), $$

  (7.92)

in which: ${C}^i\equiv \left({C}_0^i,{C}_1^i,{C}_2^i,\bullet \bullet \bullet, {C}_N^i\right)$ (i = 1,2) (namely, ${C}_t^i={C}_t^{ii}+{C}_t^{ji}\ \left(i,j=1,2;i\ne j;t=0,1,2,\bullet \bullet \bullet, N\right)$ by definition of Cⁱ above); ${\overline{C}}^i\equiv \left({\overline{C}}_0^i,{\overline{C}}_1^i,{\overline{C}}_2^i,\bullet \bullet \bullet, {\overline{C}}_N^i\right)$ (i = 1,2); and ${\overline{C}}_t^i$: given level of the consumption in region i at period t, which are nonnegative constants.

We symbolize a given trajectory of the consumption as follows:

• $$ \overline{C}=\left({\overline{C}}^1,{\overline{C}}^2\right). $$

A trajectory of the economy, T, which meets all the Eqs. (7.84)–(7.92), is called—a feasible trajectory of the economy with $\overline{C}$.

7.1.8 The Objective of the Planning

A set of feasible trajectories of the economy, $\overset{\sim }{T},$which satisfy the system of Eqs, (7.84)—(7.92), is the correspondence (a set of feasible trajectories of the economy) to $\overline{C}$ and it can be symbolized as $\overset{\sim }{T}\left(\overline{C}\right)$. A possible and most plausible objective of the planning is to find out an optimal consumption trajectory, $\hat{C} \equiv \left({\hat{C}}^1,{\hat{C}}^2\right)$, by solving the system of Eqs. (7.84)—(7.92), by changing components of $\overline{C}$ in terms of a given valuation function. A typical valuation function is given as follows:

• $$ W\left(\hat{C}\right)=W\left({\hat{C}}^1,{\hat{C}}^2\right)=\underset{\left\{\overset{\sim }{T}\left(\overline{C}\right),\overline{C}\right\}}{\max}\sum \limits_{t=0}^N{U}_t\left({\overline{C}}_t^1,{\overline{C}}_t^2\right) $$

  (7.93)

in which: W(∙): valuation function of the consumption trajectories of region 1 and region 2; U_t(∙): evaluation function of the consumption of goods in region 1 and region 2 at period t; and ${\hat{\boldsymbol{C}}}^{\boldsymbol{i}}$: (optimal) consumption trajectory of region i (i = 1, 2), which is a component of the optimal consumption trajectory, $\hat{C}.$

Specifically, U_t(∙) can be defined as follows:

$$ {U}_t\left(\bullet \right)\equiv \frac{1}{{\left(1+\rho \right)}^t}\ U\left(\bullet \right)\ \left(t=0,1,2,\bullet \bullet \bullet, N\right) $$

(7.94)

in which: ρ: given discount rate (ρ > 0); and U(∙): evaluation function of the consumption of goods in region 1 and region 2, of which functional form is constant over the time horizon.

The function, $\sum \limits_{t=0}^N{U}_t\left({\overline{C}}_t^1,{\overline{C}}_t^2\right)$, is called—objective function. Though it is empirically or practically tough work to give a concrete functional form of U_t(∙) or U(∙), we do not need to explicitly specify it as far as the substance of the roundabout production is concerned.

A set of trajectories of the economy which maximize the objective function is called—optimal set of trajectories of the economy and it can be represented as $\hat{T}\ \left(\hat{C}\right)$, which is a subset of $\left\{\overset{\sim }{T}\left(\overline{C}\right)\right|$ trajectory of the economy which meets the system of Eqs. (7.84)–(7.91) for any $\overline{C}\ge 0\Big\}$, by definition.

7.1.9 Necessary Conditions for the Optimality

7.1.9.1 Roundabout Production Through Time

Hereafter we consider only feasible trajectories of the economy. Namely, without specific notation of overline, “$\overline{\square}$,” tilde, $\overset{\sim }{\square }$, and hat, “$\hat{\square}$,” we use notation of T = (T¹, T²), representing a feasible trajectory of the economy that corresponds to the consumption trajectory, C = (C¹, C²).

Consider a perturbation, ε (>0), with, for example, ${X}_{\tau}^{11}$ and ${I}_{\tau}^{11}$ in the market flow condition of region 1 with a given feasible trajectory T at a certain period τ (0 ≤ τ ≤ N − 1) such that⁵:

• $$ {Y}_{\tau}^1-\omega =\left({X}_{\tau}^{11}-\varepsilon \right)+{X}_{\tau}^{12}+\left({I}_{\tau}^{11}+\varepsilon \right)+{I}_{\tau}^{12}+\left({C}_{\tau}^{11}-\omega \right)+{C}_{\tau}^{12}+{IR}_{\tau}^{11}+{IR}_{\tau}^{12} $$

  (7.95)

The perturbation causes a decrease in the production of goods in region 1 at period τ as the intermediate input of goods in region 1 at period τ is decreased by ε:

• $$ {Y}_{\tau}^1-\omega ={f}^1\left({K}_{\tau}^1,{X}_{\tau}^1-\varepsilon \right) $$

  (7.96)

in which: ω: decrease in the production of goods in region 1 at period τ, which is caused by a decrease, ε, in the intermediate input of goods into the production, compared to the given trajectory.

A decrease, ω, in the production can be borne by any term other than ${X}_{\tau}^{11}$ and ${I}_{\tau}^{11}$ in Eq. (7.85), it is supposed that a decrease in the production of goods, ω, is all borne by a decrease in ${C}_{\tau}^1$ by ω, which results in a decrease in the valuation of the consumption by γ:

• $$ U\left({C}_{\tau}^1,{C}_{\tau}^2\right)-\gamma =U\left({C}_{\tau}^1-\omega, {C}_{\tau}^2\right). $$

  (7.97)

On the other hand, the perturbation causes an increase in the investment by ε for the capital formation in region 1, which will result in an increase in the capital stock by δ, and will increase the production of goods by α_τ + 1 in region 1 at period τ + 1 compared to the given feasible trajectory of the economy, T:

$$ \Delta {K}_{\tau}^1+\delta ={h}^1\left({I}_{\tau}^{11}+\varepsilon +{I}_{\tau}^{21}\right), $$

(7.98)

$$ {K}_{\tau +1}^1+\delta ={K}_{\tau}^1+\Delta {K}_{\tau}^1+\delta, $$

(7.99)

$$ {Y}_{\tau +1}^1+{\alpha}_{\tau +1}={f}^1\left({K}_{\tau}^1+\delta, {X}_{\tau +1}^1\right). $$

(7.100)

An increase in the production in region 1 at period τ + 1, α_τ + 1, can be again used for the perturbation with any term in the market flow condition at period τ + 1. This means that there could be many (infinite number of) trajectories of the economy which could be different from the given feasible trajectory, T, with its components at period τ and later till period N. However, for a while, suppose that an increase in the production, α_τ + 1, is all used to increase the consumption in region 1, which results in an increase in the evaluation of the consumption by β_τ + 1 in region 1 at period τ + 1:

• $$ U\left({C}_{\tau +1}^1,{C}_{\tau +1}^2\right)+{\beta}_{\tau +1}=U\left({C}_{\tau +1}^1+{\alpha}_{\tau +1},{C}_{\tau +1}^2\right). $$

  (7.101)

We suppose that a perturbation of Eq. (7.95) is made only once at period τ and adjustments to the perturbation are made following Eq. (7.97) and Eq. (7.98). The supposition means that the triggered trajectory, which is generated by the perturbation in Eq. (7.95) and adjustments of Eq. (7.96)–(7.101), is different from the given feasible trajectory, T, with respect to its (T ‘s) components, ${X}_{\tau}^{11}$, ${I}_{\tau}^{11}$, ${C}_{\tau}^{11}$ (${C}_{\tau}^1\Big)$, ${Y}_{\tau}^1$, ${K}_t^1\ \left(t=\tau +1,\tau +2,\bullet \bullet \bullet, N\right)$, and ${Y}_t^1\ \left(t=\tau +1,\tau +2,\bullet \bullet \bullet, N\right)$ by −ε, ε, ε (ε), −ω, δ (t = τ + 1, τ + 2, ∙  ∙  ∙ , N), and α_t (t = τ + 1, τ + 2, ∙  ∙  ∙ , N), respectively. We may symbolize it as T^b and call it—perturbed trajectory of the economy. We now examine a necessary condition for the optimality of trajectory of the economy and we may put aside another perturbed trajectory of the economy with T^b for a while because we may apply a set of obtained necessary conditions to T^b, recursively. Of course, as we see previously, the triggered trajectory is a feasible trajectory with the following trajectory of the consumption:

• $$ {C}^{\prime }=\left({C}^{1\prime },{C}^2\right), $$

  (7.102)

in which:

• $$ {C}^{1\prime }=\left({C}_0^1,{C}_1^1,{C}_2^1,\bullet \bullet \bullet, {C}_{\tau -1}^1,{C}_{\tau}^1-\omega, {C}_{\tau +1}^1+{\alpha}_{\tau +1},{C}_{\tau +2}^1+{\alpha}_{\tau +2},\bullet \bullet \bullet, {C}_{N-1}^1+\kern0.5em {\alpha}_{N-1},{C}_N^1+{\alpha}_N\right) $$

  (7.103)

The trajectory of the difference in the evaluation of the consumption trajectory between C^′ and C can be calculated as follows:

• $$ \Delta W\left({C}^{\prime}\right)=W\left({C}^{1^{\prime }},{C}^2\right)-W\left({C}^1,{C}^2\right) $$

$$ =\frac{1}{{\left(1+\rho \right)}^{\tau }}\left\{-\gamma +\sum \limits_{t=1}^{N-\tau}\frac{1}{{\left(1+\rho \right)}^t}\ {\beta}_{\tau +t}\right\}. $$

(7.104)

Since fⁱ(∙) (i = 1, 2) and U(∙) are usually concave functions and α_t and β_t (t = τ + 1, τ + 2, ∙ ∙ ∙, N) decrease (not increase) as t increases. Therefore, ΔW(C^′) has an finite value in case N is infinite because $\frac{\mathbf{1}}{{\left(\mathbf{1}+\boldsymbol{\rho} \right)}^{\boldsymbol{t}}}$ will vanish as t increases to the infinity.

We can say that C^′ is superior to C if ΔW(C^′) >0. The perturbation of ε in Eq. (7.95) is taken as an example of the roundabout production through time in that a saving of consumption at a certain period results in the possibility of more consumption later on (namely, as time elapses). The roundabout production through time is possible by further accumulating the capital stock compared to the case in which the said saving in the consumption was not made. If ΔW(C^′) > 0, we can say that the roundabout production which generates the consumption trajectory, C^′, is successful and T(C) is not an optimal trajectory of the economy or we can say that T^′(C^′) is superior to T(C).

As already mentioned previously, there are many trajectories which can be generated and triggered by the perturbation of ε in Eq. (7.85): (i) to change timing from τ + 1 to τ + n (1 < n ≪ N), namely to postpone the timing of joy of the merit of the roundabout production of ε that is made at period τ while keeping the increasing merits to be used for the capital stock accumulation in region 1 till the merits are enjoyed as a further increase in the consumption in region 1, (ii) by changing the period τ, at which the perturbation itself is made, and so on. Also, (iii) a similar and different perturbation from that of Eq. (7.95) can be made by alternating region 1 and region 2 with region indices, and so on. Further, so many different trajectories of the economy can be generated and triggered with the combination and nesting of (i) and (ii) above with each other as well as the combination and nesting with the alternation (iii) above, and so on. We call these generated and triggered trajectories of the economy—perturbed trajectories of the economy through time.

We can obtain the following:

[Necessary condition-1 for the optimality through time⁶].

A necessary condition for a feasible trajectory of the economy, T(C), to be optimal is that there exists no perturbed trajectory of the economy through time, T^T(C^T), which is superior to T(C), namely there exists no T^T(C^T) that gives a larger value to the function, W(∙),than T(C).

7.1.9.2 Roundabout Production Through ‘Space’

Analogically with Eq. (7.85), we suppose the following perturbation of ε_t (ε_t > 0, t = τ, τ + 1, ∙ ∙ ∙, N) at period τ and all the periods that follow with the market flow condition:

• $$ {Y}_t^1-{\omega}_t^1=\left({X}_t^{11}-{\varepsilon}_t\right)+\left({X}_t^{12}+{\varepsilon}_t\right)+{I}_t^{11}+{I}_t^{12} $$
• $$ +\left({C}_t^{11}-{\omega}_t^1\right)+{C}_t^{12}+{IR}_t^{11}+{IR}_t^{12}\ \left(t=\tau, \tau +1,\bullet \bullet \bullet, N\right), $$

  (7.105)

in which:

${\omega}_t^1$ is defined as a solution to the following equation:

$$ {Y}_t^1-{\omega}_t^1={f}^1\left({K}_t^1,{X}_t^1-{\varepsilon}_t\right)\ \left(t=\tau, \tau +1,\bullet \bullet \bullet, N\right). $$

(7.106)

The perturbation will trigger the following adjustments:

• (1) decrease in the valuation of the consumption in region 1 at period t (t = τ, τ + 1, ∙ ∙ ∙, N):

$$ U\left({C}_t^1,{C}_t^2\right)-{\gamma}_t^1=U\left({C}_t^1-{\omega}_t^1,{C}_t^2\right)\ \left(t=\tau, \tau +1,\bullet \bullet \bullet, N\right); $$

(7.107)

• (2) as the shipment of the goods from region 1 to region 2 increases by ε_t and the capacity of highway from region 1 to region 2 needs to be increased in advanced, the following adjustments are further triggered:

  $$ {Y}_{\tau -1}^1-{\omega}_{\tau -1}^1=\left({X}_{\tau -1}^{11}-{\varepsilon}_{\tau -1}\right)+{X}_{\tau -1}^{12}+{I}_{\tau -1}^{11}+{I}_{\tau -1}^{12} $$

$$ +\left({C}_{\tau -1}^{11}-{\omega}_{\tau -1}^1\right)+{C}_{\tau -1}^{12}+\left({IR}_{\tau -1}^{11}+{\varepsilon}_{\tau -1}\right)+{IR}_{\tau -1}^{12}, $$

(7.108)

• $$ {Y}_{\tau -1}^1-{\omega}_{\tau -1}^1={f}^1\left({K}_{\tau -1}^1,{X}_{\tau -1}^1-{\varepsilon}_{\tau -1}\right), $$

  (7.109)
• $$ U\left({C}_{\tau -1}^1,{C}_{\tau -1}^2\right)-{\gamma}_{\tau -1}^1=U\left({C}_{\tau -1}^1-{\omega}_{\tau -1}^1,{C}_{\tau -1}^2\right), $$

  (7.110)
• $$ \Delta {R}_{\tau -1}^{12}+\delta ={g}^{12}\left({IR}_{\tau -1}^{11}+{\varepsilon}_{\tau -1}+{IR}_{\tau}^{21}\right), $$

  (7.111)
  $$ {R}_{\tau}^{12}+\delta ={R}_{\tau -1}^{12}+\Delta {R}_{\tau -1}^{12}+\delta, $$

  (7.112a)
  $$ {R}_{t+1}^{12}={R}_t^{12}+\Delta {R}_t^{12}+\delta\ \left(t=\tau, \tau +1,\dots, N\right), $$

  (7.112b)
• $$ {X}_t^{12}+{\varepsilon}_t+{I}_t^{12}+{C}_t^{12}+{IR}_t^{12}={q}^1\left({R}_t^{12}+\delta \right)\ \left(t=\tau, \tau +1,\bullet \bullet \bullet, N\right), $$

  (7.113)

in which: we may assume that the highway capacity constraint, Eq. (7.91), holds with equality by period τ with a given trajectory of the economy, T(C), as the (composite) goods must be a superior goods (as far as τ is enough large compared to given initial stock conditions);

• (3) in region 2, the following adjustments are made supposing an increase in the product is all used for an increase in the consumption at period τ and all the subsequent periods:

• $$ {Y}_t^2+{\alpha}_t^2={f}^2\left({K}_t^2,{X}_t^2+{\varepsilon}_t\right)\ \left(t=\tau, \tau +1,\tau +2,\dots, N\right), $$

  (7.114)
• $$ U\left({C}_t^1,{C}_t^2\right)+{\beta}_t^2=U\left({C}_t^1,{C}_t^2+{\alpha}_t^2\right)\ \left(t=\tau, \tau +1,\tau +2,\dots, N\right). $$

  (7.115)

The triggered trajectory, which is generated by the perturbation of Eq. (7.105) and adjustments of Eqs. (7.106)–(7.115), is different from the given feasible trajectory, T, with respect to its (T’s) components, ${X}_t^{11}$ (t = τ − 1, τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ${IR}_{\tau -1}^{11}$, ${X}_t^{12}$ (t = τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ${Y}_t^1$ (t = τ − 1, τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ${C}_t^{11}$(${C}_t^1\Big)$ (t = τ − 1, τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ${Y}_t^2\ \left(t=\tau, \tau +1,\tau +2,\bullet \bullet \bullet, N\right),$ ${C}_t^{12}$(${C}_t^2\Big)$ (t = τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), and ${R}_t^{12}\ \left(t=\tau, \tau +1,\tau +2,\bullet \bullet \bullet, N\right)$ by −ε_t (t = τ − 1, τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ε_τ − 1, ε_t (t = τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), $-{\omega}_t^1\ \left(-{\omega}_t^1\right)$ (t = τ − 1, τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), $-{\omega}_t^1\ \left(-{\omega}_t^1\right)$ (t = τ − 1, τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ${\alpha}_t^2$ (t = τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), ${\alpha}_t^2$ (α_t) (t = τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), and δ (t = τ, τ + 1, τ + 2, ∙  ∙  ∙ , N), respectively. We may assume that a shortage in the capacity, ${R}_t^{12}$ (t = τ + 1, τ + 2, ∙ ∙ ∙, N), due to growth of the economy and concavity of the function, q¹(∙), can be neglected as δ is small and $U\left({C}_t^1-{\omega}_t^1,{C}_t^2+{\alpha}_t^2\right)={\beta}_t^2-{\gamma}_t^1$ (t = τ, τ + 1, ∙ ∙ ∙, N). We may put aside another perturbed trajectory with thus obtained perturbed trajectory by the same reason above, too. Of course, as we see previously, the triggered trajectory is a feasible trajectory with the following trajectory of the consumption:

• $$ {C}^{\prime \prime }=\left({C}^{1\prime \prime },{C}^{2\prime \prime}\right), $$

  (7.116)
• $$ {C}^{1\prime \prime}\equiv \left({C}_0^1,{C}_1^1,\bullet \bullet \bullet, {C}_{\tau -1}^1-{\omega}_{\tau -1},{C}_{\tau}^1-{\omega}_{\tau },{C}_{\tau +1}^1-{\omega}_{\tau +1},\bullet \bullet \bullet, {C}_{N-1}^1-{\omega}_{N-1},{C}_N^1-{\omega}_N\right), $$

  (7.117)
• $$ {C}^{2\prime \prime}\equiv \left({C}_0^2,{C}_1^2,\bullet \bullet \bullet, {C}_{\tau -1}^2,{C}_{\tau}^2+{\alpha}_{\tau },{C}_{\tau +1}^1+{\alpha}_{\tau +1},\bullet \bullet \bullet, {C}_{N-1}^1+{\alpha}_{N-1},{C}_N^1+{\alpha}_N\right). $$

  (7.118)

$$ \Delta W\equiv W\left({C}^{\prime \prime}\right)-W(C)=\frac{1}{{\left(1+\rho \right)}^{\tau -1}}\ \left\{-{\gamma}_{\tau -1}^1+\sum \limits_{t=1}^{N-\tau}\frac{1}{{\left(1+\rho \right)}^t}\ \left({\beta}_t^2-{\gamma}_t^1\right)\right\}. $$

(7.119)

If ΔW > 0, the perturbed and triggered trajectory, T(C^′′), is superior to a given trajectory of the economy, T(C).

The perturbation of Eq. (7.105) and triggered adjustments can be taken as an example of the roundabout production through space, namely it aims to increase the total welfare of the economy through reallocation of resources from region 1 to region 2. However, due to the highway capacity constraint of Eq. (7.91), which usually holds with equality, an investment to the highway stock formation must be made in advance in order to enjoy merits of the roundabout production through space. This means that a roundabout production through time is necessary for the roundabout production through space and vice versa. With the supposed model, the main reason why a roundabout production through space can be a superior trajectory of the economy is that there possibly exists difference in the productivity between region 1 and region 2. If we assume traffic congestion on the highway, and specify a kind of traffic congestion function into the left-hand side of Eq. (7.91), the superiority of roundabout production must be examined in a complicated way. Increase or decrease in the congestion costs for all the traffic on the highway must be taken into the calculation of opportunity costs of diverting goods from region 1 to region 2 and investment into highway capital formation.

The triggered trajectories by the perturbation through space are many, too: (1) the timing of the perturbation can be changed over the time horizon for the planning (most probably in the early phase); (2) the timing of the investment into highway construction prior to the timing for the perturbation can be changed; (3) at a certain period τ^′ (τ < τ^′ < N)), the perturbation (reallocation of resource) over space can be stopped, namely ${\varepsilon}_{\tau^{\prime }}=0$; and (4) a similar and different perturbation from that of Eqs. (7.105) and (7.108) can be made by alternating region 1 and region 2 with region indices, and so on. Further, so many different trajectories of the economy can be generated and triggered with the combination and nesting of (1), (2), (3), and so on above with each other as well as the combination and nesting with the alternation (4) above, and so on. We call these generated and triggered trajectories of the economy—perturbed trajectories of the economy through space.

[Necessary condition-2 for the optimality through “space”⁷].

A necessary condition for a feasible trajectory of the economy, T(C), to be optimal is that there exists no perturbed trajectory of the economy through “space,” T^S(C^S), which is superior to T(C), namely there exists no T^S(C^S) that gives a larger value to the function, W(∙),than T(C).

7.1.9.3 Roundabout Production Through Space and Time

In the last two subsubsections, various ways of the perturbation and adjustment with the roundabout production through time and the roundabout production through “space” are separately illustrated in order to highlight the two concepts of the roundabout production. As readers may see, the merits of the roundabout production can be strengthened if the perturbation and triggered adjustments are made through space and time in the nested and combined ways. For example, a saving of the consumption at period τ in region 1 through saving of the intermediate input in region 1 can be shipped into region 2 for further capital stock formation in region 2. At a certain period τ^′ (τ < τ^′ < N), some of the products in region 2 can be shipped back into region 1 for the capital formation. In case of the multiple kinds of goods and therefore in case of multiple sectors, which is a more realistic case, if region 1 is less advantageous with the production of a certain goods that is the key for the capital formation and region 2 is relatively advantageous with the production of the key goods, such a return of the products from region 2 to region 1 for the capital formation in region 1 can be advantageous as far as region 1 is relatively advantageous in the production of other goods using the capital stock. This is the essential background of the so-called Turnpike Theorem by taking regions as sectors. We call all these perturbed and triggered trajectories—perturbed trajectories through space and time.

[Necessary condition-3 for the optimality through space and time⁸].

A necessary condition for a feasible trajectory of the economy, T(C), to be optimal is that there exists no perturbed trajectory of the economy through space and time, T^ST(C^ST), which is superior to T(C), namely there exists no T^ST(C^ST) that gives a larger value to the function, W(∙),than T(C).

7.1.10 Dynamic Optimality and Dynamic Model

Dynamic optimality should be dependent on the definition of “dynamic model.” The dynamic model in this book implies that the optimality of the state of the economy over the time horizon can be pursued using the said model by meeting the necessary conditions-3 for the optimality through space and time as far as the dynamic model has the spatial dimension in any sense. The optimality, thus obtained, is called—dynamic optimality. In order to pursue the dynamic optimality, the mechanism must be specified into the said model with which the (re-)allocation of resources over time as well as space can be made, that is, the equation for the capital accumulation, the network for the shipment of the goods which connects spatially dispersed economies, and so on. However, usually the specification of such mechanisms is not enough for the model alone to pursue the dynamic optimality. An algorithm such as the optimal control theory, the simplex algorithm, and so on is also required to pursue the dynamic optimality. In this sense, the dynamic model involves such an algorithm.

7.1.11 Dynamic Programming (Optimization) Model

The optimal trajectory of consumption, $\overline{\boldsymbol{C}}$, which maximizes the valuation (objective) function, Eq. (7.93), with the specific valuation function, Eq. (7.94), is substantially the solution to the following programming (optimization) problem:

• $$ \underset{\left\{\boldsymbol{T}\left(\boldsymbol{C}\right),\boldsymbol{C}\right\}}{\mathbf{\max}}\sum \limits_{\boldsymbol{t}=\mathbf{0}}^{\boldsymbol{N}}\frac{\mathbf{1}}{{\left(\mathbf{1}+\boldsymbol{\rho} \right)}^{\boldsymbol{t}}}\ \boldsymbol{U}\left({\boldsymbol{C}}_{\boldsymbol{t}}^{\mathbf{1}},{\boldsymbol{C}}_{\boldsymbol{t}}^{\mathbf{2}}\right), $$

  (7.120)

in which: T(C) is a feasible trajectory of the economy with C, namely T(C) is T = (T¹, T²) which satisfies Eqs. (7.84)–(7.91) with a given constant non-negative vector, C. We shall call the programming model of Eq. (7.120)—dynamic programming (optimization) model and we shall call the optimal trajectory of the economy⁹ which maximizes the objective function of Eq. (7.120)—optimal solution.

7.1.12 Bang-Bang Solution

The dynamic programming model has a defect in the sense that it sometimes gives an optimal solution in which the variables drastically change through time, which describes the state of the economy. For example, a feasible trajectory of the economy can be an optimal solution to the dynamic programming model in which the consumption is almost zero while the most of the products are devoted for the stock formation and intermediate inputs and the most or all the products are devoted to the consumption at several last periods of the time horizon. It could happen if the productivity of the capital stock in terms of the production function is higher than the discount rate and the roundabout production through time is much advantageous than the devotion of the produced goods to the instance consumption at each period till the period approaches to the end of time horizon. This has a kind of collateral effect such that the people belong to the current aged generation may relatively have disadvantage than the younger people who also may belong to the future generations to be able to enjoy a lavish consumption thanks to the roundabout production through time. The time horizon is the longer, the bigger is the inequity among the current and future generations.

The roundabout production through space (and time) has the same adverse effect such that, for example, the consumption of the people in region 1 is less or almost zero compared to the amount of the goods produced in region 1 as some or most of the products in region 1 are shipped into region 2 for the capital stock formation except for several last periods of the time horizon. It will cause an inequality among people who belongs to the same generations, and lives in different regions (inequality between region 1 and region 2) as well as an inequality between generations in the sense above. This type of inequality could happen irrespective of whether the valuation function takes into account the equality principle in any sense since it could happen due to differences in the technical condition such as the productivity with capital stock, efficiency in the capital stock formation, discount rate, and so on.

One possible way to eliminate and relieve the inequality between generations and regions is to set the lower constraints to the trajectory of consumption of the economy, which is typically given as follows:

• $$ {C}_t^i\ge {\overset{=}{C}}_t^i\ \left(i=1,2;t=0,1,2,3,\dots, N\right), $$

  (7.121)

in which: ${\overset{=}{C}}_t^i$: exogenously given minimum level of the consumption in region i at period t (i = 1, 2; t = 0, 1, 2, 3…, N).

The possible inequality with the optimal trajectory of consumption to the dynamic model which has no constraint of Eq. (7.121) (which shall be called—original dynamic model) can be eliminated in the optimal trajectory of consumption to the dynamic model which has the constraint of Eq. (7.121) (which shall be called—dynamic model of constrained consumption). However, as far as the original dynamic model gives the optimal trajectory of consumption that is inequal among generations and/or regions, the dynamic model of constrained consumption will still give an inequal trajectory of consumption to be a solution analogically as the technical condition that causes the inequality is not at all eliminated by the constrained consumption. With the example mentioned previously, the consumption will be kept minimum by making it as much as close to the lower constraint while the saved products will be invested for the capital formation for the most periods. Actually, it can be taken that the consumption variables of the original dynamic model have the lower constraints which are zero due to the non-negativity constraints. The degree of inequality in any sense can be relieved by increasing the values of ${\overset{=}{C}}_t^i$ from zero (0) to positive values. Dissatisfaction with inequality in any sense will exist till a perfect equality is realized forcibly.

It is tough to agree on the exogenously given values of ${\overset{=}{C}}_t^i$ (i = 1, 2; t = 0, 1, 2, 3…, N) and it cannot be totally neglected that: (a) the region having the more population will be assigned the higher values of ${\overset{=}{C}}_t^i$; and (b) the region having more products will be assigned the higher values of ${\overset{=}{C}}_t^i$. A compromise must be made between the former, which is based on a kind of the equality principle, and the latter, which is based on a kind of market principle.

7.1.13 Expanding the Production Possibility Frontier

It is tough at all to specify the functional form, U(∙), which is necessary for the practical planning because it should be directly linked with the compromise between the value judgment of (a) and (b) discussed in the previous subsection. To change the view, it can be another approach to pursue the maximized economic growth in terms of the gross domestic products (GDP) while the consumption is kept at an exogenously given level. The consumption level is still must be dependent on a value judgment. However, it can be calibrated to show a difference in the optimal trajectories of the economy to the stakeholders in order to smoothly obtain an agreement among them. The information given by the calibration would be useful for policy process. The dynamic model based on such approach is specified as follows:

$$ \underset{\left\{T(C)\right\}}{\max}\sum \limits_{t=0}^N\frac{1}{{\left(1+\rho \right)}^t}\ {\sum}_{i=1}^2{V}_t^i, $$

(7.122)

$$ \mathrm{subject}\ \mathrm{to}:{C}_t^i={\overset{=}{C}}_t^i\ \left(i=1,2;t=0,1,2,3,\dots, N\right), $$

(7.123)

in which: ${V}_t^i$: gross value-added in region i at period t and ${V}_t^i\equiv {Y}_t^i-\left({X}_t^{ii}+{X}_t^{ji}\right)\ \left(i,j=1,2;i\ne j\right)$.

7.1.14 Balanced Development

The dynamic model of Eqs. (7.122) and (7.123) will pursue the roundabout production through space and time, which will cause the same issue with the inequality between generations and regions. The issue of inequality in terms of the consumption is relieved due to Eq. (7.123) and the inequality between regions in terms of the capital stock accumulation may be raised. This is another issue of the balanced development of national land. Analogically, the following constraint can be added in order to relieve the issue:

• $$ {F}^k\left({K}_t^1,{K}_t^2\right)\ge 0\ \left(t=0,1,2,3,\dots, N;k=1,2,\dots, {n}^F\right), $$

  (7.124)

in which: n^F is the number of additional constraints.

An example of the specification of F^k(∙) is the following:

• $$ {\theta}_t\le \frac{K_t^2}{K_t^1}\le {\varphi}_t\ \left({K}_0^i\ne 0\ \left(i=1,2\right),0<{\theta}_t\le 1,{\varphi}_t\ge 1\ \left(t=0,1,2,3,\dots, N\right)\right). $$

  (7.125)

Although it is tough to specify values of θ_t and φ_t, too, according to the genuine balanced development of national land, θ_t and φ_t should converge to the ratio of $\frac{A^2}{A^1}$ (Aⁱ is the inhabitable land area in region i (i = 1, 2)) as the period reaches to the end of the time horizon. The calibration with θ_t and/or φ_t could be informative for the policy and political process related with the balanced development of national land. The addition of constraints, Eq. (7.125), is a compromise between the efficient trajectory of the economy and the less efficient trajectory which takes into account the balanced development of national land. Starting with a very small (large) value with θ_t (φ_t), an additional increase (decrease) in θ_t (φ_t) causes a decrease in the maximized value of the objective function, Eq. (7.122), respectively. A decrease in the maximized objective function is an indicator of the cost in terms of the present value of the gross value added in order to pursue a further balanced development of national land. The indicator is linked to the imputed prices associated with the constraints, Eq. (7.125), for example, the imputed price associated with the first (left) inequality in Eq. (7.125) is a decrease in the maximized objective function due to a unit increase in the value of θ_t, and so on.

7.1.15 Positioning of the Dynamic Programming Model

Generally speaking, it is very tough to specify constraints or functional form explicitly in the model as the specification substantially should be dependent on a value judgment and the specification itself has been an important policy agenda on which decision should be made through a policy and political process using different terms and phrases. However, it should be rather taken that the dynamic programming model provides useful information for such a policy and political argument process to efficiently and effectively converge to an agreeable decision by openly presenting not only results of the simulation case by case but also adopted functional forms as well as value judgments presumed by the adoption of a specific functional form if any, and so on. We will stop here the explanation about the usefulness of the dynamic programming model due to the limited space.