📚 Dynamic Equilibrium Models and Analytical Methods | 动态均衡模型与分析方法
Dynamic equilibrium models are a class of economic models in which variables such as price, output, capital and income are determined over time. They allow economists to study how an economy moves from one equilibrium to another after a shock, rather than only comparing two static positions.
动态均衡模型是一类随时间决定价格、产出、资本与收入等变量的经济模型。它允许经济学家研究经济体在受到冲击后如何从一个均衡向另一个均衡移动，而不是仅仅比较两个静态位置。
1. What Is Dynamic Equilibrium? | 什么是动态均衡？
In a static model, equilibrium is a single point at which quantity demanded equals quantity supplied, such as P* and Q*. Time does not enter the analysis. In a dynamic model, however, equilibrium is a path. The economy is in dynamic equilibrium when the values of key variables follow a consistent sequence, P1, P2, P3, …, that satisfies the behavioural equations of the model in every period.
在静态模型中，均衡是一个单一的点，例如 P* 和 Q*，此时需求量等于供给量，时间不进入分析。而在动态模型中，均衡是一条路径。当关键变量遵循一条一致的序列 P₁、P₂、P₃……，并在每一期都满足模型的行为方程时，经济体即处于动态均衡。
The central idea is that dynamic equilibrium embeds a "rule of motion". A shock may move the economy away from its original path, but the model describes how the variables adjust period by period. If the economy eventually returns to a stable path, that path is a stable dynamic equilibrium.
其核心思想是，动态均衡内置了一条“运动规则”。冲击可能使经济偏离原有路径，但模型会描述变量如何逐期调整。如果经济最终回到某条稳定路径，那么该路径就是稳定的动态均衡。
2. Time and the Adjustment Process | 时间与调整过程的作用
Time creates lags in economic behaviour. Production cannot always respond instantly to a change in demand because firms need time to hire workers, order inputs and build capacity. Consumers may also respond slowly because habits change gradually. Dynamic equilibrium models make these lags explicit.
时间会在经济行为中造成滞后。由于企业需要时间招聘工人、订购投入品和建设产能，生产往往不能立即响应需求变化。消费者的反应也可能较慢，因为习惯是逐渐改变的。动态均衡模型将这些滞后显性地纳入分析。
There are two common forms of lag. A one-period lag in supply means that quantity supplied in period t depends on the price observed in period t − 1. A one-period lag in consumption means that spending in the current period depends on income earned in the previous period. These lags generate a sequence of temporary equilibria rather than a single timeless equilibrium.
常见滞后有两种形式。供给滞后一期意味着 t 期的供给量取决于 t−1 期观察到的价格。消费滞后一期意味着本期支出取决于上一期获得的收入。这些滞后会产生一系列暂时均衡，而不是一个脱离时间的单一均衡。
Stock-flow analysis is also important. A dynamic equilibrium requires that flows, such as saving and investment, are consistent with the change in stocks, such as the capital stock. For example, if investment exceeds depreciation, the capital stock grows; the next period's equilibrium must therefore be different.
存量—流量分析同样重要。动态均衡要求储蓄与投资等流量，与资本存量等存量的变化保持一致。例如，若投资超过折旧，资本存量就会增长，因此下一期的均衡必然不同。
3. The Cobweb Model | 蛛网模型
The cobweb model is the classic example of a dynamic equilibrium in microeconomics. It is often used for agricultural markets, where farmers decide how much to plant only after observing the current price, but the output is not available until the next period.
蛛网模型是微观经济学中动态均衡的经典例子。它常被用于农产品市场：农民在观察当期价格后决定种植多少，但产出要到下一期才上市销售。
Suppose demand is: Qtd = a − bPt, and supply is: Qts = c + dPt−1, where a, b, c and d are positive parameters. Setting quantity demanded equal to quantity supplied gives:
假设需求函数为 Qtd = a − bPt，供给函数为 Qts = c + dPt−1，其中 a、b、c、d 均为正参数。令需求量等于供给量，可得：
Pt = (a − c)/b − (d/b)Pt−1
The long-run equilibrium price is obtained by setting Pt = Pt−1 = P*:
长期均衡价格可通过令 Pt = Pt−1 = P* 得到：
P* = (a − c)/(b + d)
The difference equation can be rewritten as Pt − P* = −(d/b)(Pt−1 − P*). Stability depends on the slope coefficient. Since the absolute value of the coefficient is d/b, the market converges when d < b and diverges when d > b.
差分方程可改写为 Pt − P* = −(d/b)(Pt−1 − P*)。稳定性取决于该斜率系数。由于系数的绝对值为 d/b，当 d < b 时市场收敛，当 d > b 时市场发散。
| Condition 条件 | Pattern 结果 |
| d < b | Damped oscillations converging to P* 衰减振荡并收敛到 P* |
| d > b | Explosive oscillations away from P* 爆炸性振荡，远离 P* |
| d = b | Constant oscillations around P* 围绕 P* 的恒定振荡 |
The economic intuition is that when demand is steeper than supply, price mistakes are corrected over time; when supply is more elastic than demand, each error becomes larger than the previous one.
经济直觉是：当需求曲线比供给曲线更陡峭时，价格误差会随时间得到修正；当供给弹性大于需求弹性时，每一次误差都会比上一次更大。
4. Expectations in Dynamic Models | 动态模型中的预期
Expectations determine how economic agents respond to current information. In a dynamic model, almost every adjustment mechanism depends on how firms, workers and investors form expectations about the future.
预期决定经济主体如何对当前信息作出反应。在动态模型中，几乎所有调整机制都取决于企业、工人和投资者如何形成对未来的预期。
Adaptive expectations are formed by updating past expectations with the most recent forecasting error. A common form is EtPt+1 = Et−1Pt + λ(Pt − Et−1Pt), where λ is the speed of adjustment and lies between 0 and 1. Under adaptive expectations, people learn slowly and may make systematic errors during a transition.
适应性预期通过“最近一次预测误差”来修正过去的预期。常见形式为 EtPt+1 = Et−1Pt + λ(Pt − Et−1Pt)，其中 λ 是调整速度，取值在 0 到 1 之间。在适应性预期下，人们学习较慢，在转变过程中可能犯系统性错误。
Rational expectations assume that agents use all available information and understand the true model of the economy. Systematic mistakes disappear in equilibrium, although unpredictable shocks still create surprises. Rational expectations can make adjustment faster because expectations are forward-looking and incorporate policy announcements immediately.
理性预期假设经济主体使用所有可得信息，并理解经济的真实模型。系统性错误在均衡中消失，但不可预测的冲击仍会造成意外。由于代理人是向前看的，并且会立即吸收政策公告，理性预期往往使调整变得更加迅速。
In dynamic equilibrium analysis, the choice between adaptive and rational expectations is not merely technical. It changes whether a model displays persistent cycles, overshooting or instantaneous convergence.
在动态均衡分析中，选择适应性预期还是理性预期并不仅仅是技术性问题。它决定了模型是呈现持续性周期、超调，还是瞬时收敛。
5. Demand and Supply Shocks in a Dynamic Setting | 动态背景下的需求与供给冲击
Dynamic equilibrium models are especially useful for tracing the effect of a shock over time. A temporary demand shock may raise output and price in the short run, but as supply adjusts, the economy moves along a new dynamic path.
动态均衡模型尤其适合追踪冲击随时间产生的影响。一次暂时性需求冲击可能在短期内提高产出和价格，但随着供给的调整，经济将沿着一条新的动态路径移动。
Consider a positive demand shock. In the first period, aggregate demand shifts to the right, so both price and output rise. If the economy has spare capacity, output can expand. If not, the main effect is a higher price. Over subsequent periods, firms may expand capacity, wages may adjust, and the short-run aggregate supply curve shifts. The dynamic equilibrium is the sequence of short-run equilibria that finally ends at a new long-run equilibrium.
以一次正向需求冲击为例。在第一期，总需求曲线右移，价格与产出都上升。如果经济存在闲置产能，产出可以扩张；如果产能不足，主要影响是价格上升。在随后的时期中，企业可能扩大产能，工资可能调整，短期总供给曲线发生移动。动态均衡就是这一系列短期均衡的序列，最终到达一个新的长期均衡。
A supply shock, such as an oil price increase, shifts the short-run aggregate supply curve left. Output falls and the price level rises. If the central bank holds policy unchanged, the economy faces a recession; if it accommodates the shock by expanding demand, the inflation effect is larger and the adjustment path differs. Dynamic models predict the path of both output and inflation in response to such a shock.
供给冲击，例如油价上升，会使短期总供给曲线左移。产出下降，物价水平上升。如果中央银行不改变政策，经济将陷入衰退；如果央行通过扩张需求来容纳冲击，那么通胀效应更大，调整路径也不同。动态模型可以预测在这种冲击下产出和通货膨胀的路径。
6. The Multiplier–Accelerator Model | 乘数—加速数模型
The multiplier–accelerator model explains business cycles through the interaction between consumption and investment. It is a simple dynamic equilibrium model in which national income in one period depends on income in earlier periods.
乘数—加速数模型通过消费与投资的相互作用来解释经济周期。它是一个简单的动态均衡模型：一期的国民收入取决于前几期的收入。
Let Yt be national income, Ct consumption, It investment and Gt government spending. Then Yt = Ct + It + Gt. Consumption depends on past income: Ct = cYt−1. Investment depends on the change in income: It = v(Yt−1 − Yt−2). Substituting gives:
令 Yt 为国民收入，Ct 为消费，It 为投资，Gt 为政府支出，则 Yt = Ct + It + Gt。消费取决于过去的收入：Ct = cYt−1。投资取决于收入的变化：It = v(Yt−1 − Yt−2)。代入得：
Yt = (c + v)Yt−1 − vYt−2 + Gt
This second-order difference equation generates cycles. Whether the cycle is damped, explosive or constant depends on the marginal propensity to consume c and the accelerator coefficient v. A large v makes the system more likely to produce violent fluctuations.
这个二阶差分方程会产生周期波动。周期是衰减、爆炸还是恒定，取决于边际消费倾向 c 和加速数 v。v 越大，系统就越可能产生剧烈的波动。
In this model, the dynamic equilibrium is not necessarily a fixed point. It can be a periodic path. This is why the multiplier–accelerator model is widely used as a formal explanation of boom-and-bust cycles.
在该模型中，动态均衡未必是一个固定点，而可能是一条周期路径。这正是乘数—加速数模型被广泛用作繁荣与萧条周期正式解释的原因。
7. The Harrod–Domar Growth Model | 哈罗德—多马增长模型
The Harrod–Domar model is a deliberately simple dynamic growth model. It asks what growth rate is required for planned saving to equal planned investment in every period.
哈罗德—多马模型是一个刻意简化的动态增长模型。它提出的问题是：要使每一期计划储蓄都等于计划投资，经济增长率需要达到多少？
Suppose the saving rate is s and the capital–output ratio is v. The equilibrium condition is sY = vΔY, where ΔY = Yt+1 − Yt. Therefore, the warranted growth rate is:
假设储蓄率为 s，资本产出比为 v。均衡条件为 sY = vΔY，其中 ΔY = Yt+1 − Yt。因此，有保证的增长率是：
Gw = ΔY/Y = s/v
The natural growth rate Gn is determined by the growth of the labour force and technical progress. If Gw differs from Gn, the model is said to be knife-edge unstable. If the warranted rate exceeds the natural rate, the economy cannot find enough labour to realise the growth required by saving, and investment plans collapse. If it is below, labour is left unemployed.
自然增长率 Gn 由劳动力增长和技术进步决定。如果 Gw 与 Gn 不一致，模型就会出现“刀刃般”的不稳定。若有保证的增长率高于自然增长率，经济找不到足够的劳动力来实现储蓄所要求的增长，投资计划就会落空；若低于自然增长率，则会出现失业。
The main lesson is that a dynamic equilibrium in a growing economy is not automatic. It requires an exact coincidence of preferences, technology and labour force growth.
这一模型的主要启示是：增长经济中的动态均衡并非自动实现，它要求储蓄偏好、技术水平与劳动力增长恰好配合。
8. The Solow Growth Model | 索洛增长模型
The Solow model is the standard alternative to the Harrod–Domar model. It replaces the fixed capital–output ratio with a production function and allows capital to adjust gradually. The dynamic equilibrium is called the steady state.
索洛模型是哈罗德—多马模型的标准替代框架。它用生产函数取代固定的资本产出比，并允许资本逐渐调整。其动态均衡被称为“稳态”。
Let k be capital per worker and f(k) output per worker. Investment per worker is s f(k). To keep the capital–labour ratio constant, the economy must invest enough to replace depreciating capital and equip new workers, so required investment is (n + δ)k, where n is population growth and δ is the depreciation rate.
令 k 为每单位劳动力的资本，f(k) 为每单位劳动力的产出。每单位劳动力的投资为 s f(k)。要使资本—劳动比保持不变，经济必须投资足够多，以补偿资本折旧并为新工人配套资本，因此必要投资为 (n + δ)k，其中 n 是人口增长，δ 是折旧率。
The change in capital per worker is:
每单位劳动力资本的变化为：
