Pricing, assembly rate optimizations and coordination for prefabricated construction supply chain with government subsidies

The study addresses how government subsidies influence pricing and assembly-rate decisions in a two-echelon prefabricated construction supply chain comprising a manufacturer (producing prefabricated components) and an assembler (selling to consumers). Motivated by sustainable development and the need to overcome higher costs and lower technical levels in China’s prefabricated sector, the government has implemented various subsidy policies linked to assembly rate levels. This introduces complexity in pricing and production strategies across the supply chain. The research investigates: (1) optimal strategies when subsidies are considered, (2) the differential impacts of subsidizing the manufacturer versus the assembler, and (3) how to coordinate the supply chain under both subsidy scenarios. The study contributes by comparing different subsidy orientations and by modeling the assembly rate as a decision variable of the firm rather than a purely technical parameter, offering actionable insights for policy design and enterprise decision-making.

Three streams are reviewed: (1) Pricing strategies in supply chains with government subsidies, covering effects of different subsidy orientations on optimal pricing, production, and greenness, often showing subsidies can improve supply chain performance and alter channel strategies. (2) Subsidy incentives and management in prefabricated construction supply chains, including evidence that supportive, detailed policies promote adoption, and that consumer preferences and policy perceptions significantly influence uptake; limited work has compared different subsidy recipients or treated assembly rate as a decision variable. (3) Supply chain coordination under governmental interventions (e.g., cost/revenue sharing, two-part tariffs, carbon-cost sharing), including limited prefabricated-construction-specific coordination studies; prior work rarely integrates government subsidies explicitly into coordination design. This paper fills gaps by comparing subsidy recipients (manufacturer vs assembler), optimizing assembly rate as a decision variable, and proposing contracts that coordinate under each subsidy scenario.

Setting: A two-echelon supply chain with a manufacturer (leader) and an assembler (follower). The manufacturer sets wholesale price w; the assembler sets retail price p and assembly rate g. Government provides per-unit subsidies proportional to assembly rate, targeted either to the manufacturer (SM model) or the assembler (SA model). Objective: Each firm maximizes its own profit under decentralized decisions; a centralized benchmark maximizes total supply chain profit. Coordination contracts are then designed to implement centralized outcomes under decentralization. Key assumptions and notation: Demand D(p,g) = α − βp + γg with α,β,γ > 0. The assembler bears quadratic assembly technology investment cost C(g) = (k g^2)/2 with k > 0. Unit subsidy equals θ_i g per sold unit, where i indicates the subsidized party. Regularity conditions ensure concavity/existence (e.g., 2kβ − γ^2 > 0 and analogous forms under subsidies). The manufacturer’s unit production cost is c (constant marginal cost). Decision sequence follows Stackelberg leadership by the manufacturer. Base (no-subsidy) model: Profits are π_M0 = (w − c)D and π_A0 = (p − w)D − (k g^2)/2. Solving yields unique optimal prices and assembly rate: w0 = (α + cβ)/(2β); g0 = γ(α − cβ)/[2(2kβ − γ^2)]; p0 as a rational function of parameters; corresponding profits are equal for manufacturer and assembler at k(α − cβ)^2/[4(2kβ − γ^2)]. SM model (subsidizing the manufacturer): Profits are π_M1 = (w + θ1 g − c)D and π_A1 = (p − w)D − (k g^2)/2. Backward induction yields closed-form solutions p1, g1, w1, all unique under conditions 2kβ − γ(γ + βθ1) > 0. Comparative statics show: subsidy increases g and reduces w; effect on p depends on k, β, γ. Profits for both firms exceed base-case profits. SA model (subsidizing the assembler): Profits are π_M2 = (w − c)D and π_A2 = (p + θ2 g − w)D − (k g^2)/2. Solving yields p2, g2, with w2 = w0 (unchanged by subsidy), unique under 2kβ − (γ + βθ2)^2 > 0. Comparative statics show: subsidy increases g; w unaffected; effect on p depends on k, β, γ. Both firms earn higher profits than in the base model. Comparative analysis (SM vs SA): For equal subsidy rates (θ1 = θ2), SA yields higher assembly rate and higher manufacturer and assembler profits, while SM yields a lower wholesale price. To achieve the same assembly rate (g1 = g2), the required subsidy rate is higher under SM (θ1 > θ2), implying SA is more cost-effective for the government to reach a given g. Centralized models: With subsidy to the manufacturer or assembler, centralized optimization over p and g yields p_c and g_c with total profit s_c depending on θ_i via denominators 2kβ − (γ + βθ_i). Centralized profits exceed decentralized ones. Coordination via revenue–cost sharing: A revenue–cost sharing contract is proposed to implement centralized outcomes. Under SM, the assembler shares a fraction of revenue with the manufacturer and the manufacturer shares a fraction of assembly cost; choosing parameters (φ1, λ1) to satisfy φ1ρ − w = δ1(p + θ1 g − c) and λ1 = δ1 for any δ1 ∈ (0,1] coordinates the chain. Under SA, the analogous condition is φ2(ρ + θ2 g) − w = δ2(p + θ2 g − c) and λ2 = δ2 for any δ2 ∈ (0,1]. These conditions are simple, depend only on observable terms, and allow arbitrary division of centralized profit.

• Government subsidies increase the assembly rate in both subsidy orientations (SM and SA).
• Subsidizing the manufacturer (SM) lowers the wholesale price as the subsidy rate increases; subsidizing the assembler (SA) leaves the wholesale price unchanged (equal to base-case wholesale price).
• The retail price response to subsidies depends on the investment cost coefficient (k), price sensitivity (β), and consumer preference (γ). Thresholds such as k relative to 2β/γ^2 determine whether retail price decreases or increases with subsidies.
• For equal subsidy rates (θ1 = θ2), subsidizing the assembler yields higher assembly rate and higher profits for both manufacturer and assembler than subsidizing the manufacturer. For equal assembly rates (g1 = g2), achieving that rate requires a higher subsidy if directed to the manufacturer than to the assembler (θ1 > θ2), implying SA is more subsidy-efficient.
• Both subsidy modes increase firm and supply chain profits relative to no-subsidy; the gains are generally larger under SA.
• Numerical analysis (α=8000, β=1, γ=0.5, k=4, c=4000, θ∈[0,1]) shows: (i) retail prices decrease with θ in both models, with p1 < p2 for θ ∈ (0,0.5), equality near θ=0.5, and p2 < p1 for θ ∈ (0.5,1); (ii) assembly rates rise with θ in both models, more steeply under SA; (iii) w declines with θ under SM and is invariant under SA; (iv) both firms’ profits increase with θ, more markedly under SA. Joint sensitivity to k and γ shows g increases with higher γ and lower k; SA consistently achieves g2 > g1 > g0 and higher profits for both firms.

The study’s results directly address the research questions: (1) Optimal strategies under subsidies feature higher assembly rates and, depending on parameters, lower retail prices; manufacturer subsidies reduce wholesale prices while assembler subsidies do not. (2) Subsidy orientation matters: subsidies to the assembler more effectively raise assembly rates and total profits at a given subsidy rate, making them more efficient for achieving industrial goals. (3) A simple revenue–cost sharing contract coordinates the decentralized supply chain to reach centralized outcomes under either subsidy orientation. These findings are significant for policy and practice. For governments, directing subsidies to the assembler more efficiently boosts assembly rates and supply chain profitability while potentially reducing fiscal outlays for achieving a target assembly rate. For enterprises, understanding parameter-dependent pricing responses enables better pricing and technology investment strategies; improving consumer preference (γ) and reducing investment cost coefficient (k) both expand demand and profitability while supporting higher assembly rates. The coordination contracts provide a practical mechanism to align channel incentives and realize system-optimal outcomes.

The paper develops and analyzes pricing and assembly-rate decisions in a two-echelon prefabricated construction supply chain under two government subsidy orientations. It shows that subsidies increase assembly rates and profits, with assembler-targeted subsidies outperforming manufacturer-targeted subsidies in terms of assembly rate, profit, and subsidy efficiency. A revenue–cost sharing contract is proposed that achieves perfect coordination under both subsidy scenarios with simple parameter conditions, enabling arbitrary profit division while replicating centralized outcomes. Numerical experiments corroborate the analytical results and illustrate parameter sensitivities. Future research should extend the framework to include government decision-making and alternative incentive instruments, and to richer supply chain structures and uncertainties.

The study models only a two-echelon supply chain (one manufacturer and one assembler) with deterministic demand and quadratic assembly investment cost. It considers financial subsidies proportional to assembly rate and does not endogenize the government’s optimization/problem or explore alternative incentives (e.g., tax credits, procurement, standards). Empirical validation is limited to stylized numerical examples, and some results hinge on parameter restrictions ensuring concavity and feasibility.