7.1 Valuation(估值)
Quantifying the valuation of a PoS cryptocurrency is critical, since the security of PoS systems depends primarily on the value of the underlying tokens. Valuation hinges on two key properties of this system. The first property has to do with the blockchain formation mechanism in a PoS network. Abstracting from the precise details of block validation and block proposal mechanisms, a key property of PoS networks is that the validation protocol requires participating nodes to commit (stake) a certain amount of tokens as collateral. In practice, PoS blockchain protocols typically ask nodes to commit stake for two main purposes: block proposal and block validation. Validation can be thought of either as a periodic finalization process (e.g., the Casper protocol from Ethereum [37]), or more generally, the process of nodes checking the validity of blocks and certifying their correctness in the blockchain. Rewards that accrue to the nodes participating in block proposal and validation are distributed in relation to the size of their committed stake. This protocol, in combination with additional measures (such as penalties for non-participation) is designed to induce proper participation in the validation process.
对一个PoS机制的加密货币进行估值量化是非常重要的，因为PoS系统的安全性主要依赖于其潜在代币的价值。估值取决于这个系统的两个关键属性。第一个属性与POS网络中的区块链形成机制有关。从区块验证和出块机制的精确细节出发，POS网络的一个关键特性是，验证协议要求参与进来的节点保证（质押）一定数量的代币作为担保。在实践中，POS区块链协议通常要求节点保证其权益出于两个主要目的：区块出块和区块验证。验证可以被认为要么是一个周期性的终结过程（例如，以太坊的Casper协议[37]），要么更通俗地说，是节点检查区块的有效性并在区块链中证明其正确性的过程。参与区块出块和验证的节点所获得的奖励将根据其承诺的权益（也就是质押的代币量）大小进行分配。本协议与其他附加措施（如对不参与的处罚）相结合，旨在促使验证过程有适当的参与者。
For our purposes, the main economic implication of this mechanism is that network participants who serve as block proposers or validators can earn rewards by committing their tokens to such activities. In our model, we do not distinguish between the two functionalities; we refer to both simply as ‘validation’. Thus, as long as tokens are used for validation, each token generates a stream of cash flows for its holder in the form of additional tokens. This simplified model does not necessarily represent more complex consensus structures (e.g., Prism); however, we use it as a starting point for more nuanced models of PoS consensus.
出于我们的目的，该机制的主要经济含义是，让作为区块出块者或验证者的网络参与者能够通过质押他们的代币到此类活动中来获得奖励。在我们的模型中，我们并不对这两个功能做区分：我们把两者都简单的称为“验证”。因此，只要代币被用于验证，每个代币都会以额外代币的形式为其持有者产生一个现金流。这个简单的模型不一定代表更复杂的共识结构（例如，Prism协议）；但是，我们可以作为更细微的PoS共识模型的起点。
The second property of the PoS network is that tokens can be re-allocated across various uses: the same token can be used for validation, for consumer and merchant transactions, for off-chain routing of payments, etc. Our analysis assumes, as an approximation, that tokens can be re-allocated in a frictionless manner. If a specific design deviates from this idealized frictionless model, e.g., due to constraints imposed on movements of tokens within the system, one would need to enrich the model with an explicit description of such frictions.
PoS网络的第二个特性是，代币可以通过不同的用途被再分配：同一个代币可以用于验证、消费者和商家的交易、链下支付通道等。做一个近似的分析，我们假设代币可以很顺畅地被再分配。如果一个特定的设计偏离了这个理想化的无摩擦模型，例如，由于强加于系统内代币调动的限制，我们需要对这种摩擦有一个明确的描述来丰富模型。
In a frictionless system, individual optimization by the network participants implies that expected rates of return on alternative feasible uses of the tokens must be equalized. In particular, any holder of a token may exchange it for fiat currency and invest the proceeds in a portfolio of financial assets with a return risk profile similar to that of the tokens. Thus, in a system where tokens can be easily exchanged without frictions for fiat currency, the rate of return on holding tokens must equal the opportunity cost: the expected rate of return on the risk-matched investment strategy in financial markets.
在一个无摩擦的系统中，网络参与者的个体优化意味着对代币的替代可行用途的预期回报率必须是均衡的。特别是，任何一个代币持有人都可能将其兑换成法币，并将收益投资于和代币的收益风险相近的金融资产组合。因此，在这样一个可以将代币很容易又很顺畅地兑换成法币的系统中，持有代币的收益率必须等于机会成本：即在金融市场中风险相匹配的投资策略可获得的预期回报率。
7.1.1 A Simple Model with Fee-Based Rewards（一个基于费用奖励的简单模型）
We next outline a parsimonious model in which rewards for validation activity are paid entirely in transaction fees generated by retail transactions rather than in newly minted tokens [103]. This model abstracts away from many empirically-relevant details to focus on the main mechanism of token valuation.
接下来，我们将概述一个节俭的模型，在这个模型中，验证活动的奖励完全以零散交易产生的交易费用支付，而不是以新铸造的代币支付[103]。该模型是从许多经验相关的细节中抽象而来，重点研究了代币估值的主要机制。
Consider a payment network. Suppose that in each period, the network processes retail transactions in the amount of \(Y_t\)，measured in the fiat currency, “dollars.” Assume that the aggregate float of tokens is fixed, and without loss of generality equal to one. Let \(p_t\) denote the equilibrium price of the tokens in terms of the fiat currency. Assume that the network is in a stationary growth regime: the volume of transactions is expected to grow at a constant rate \(g_Y\), so that \(\mathbb{E}_t [Y_{t+s}] = Y_t (1+g_Y)^s\).
考虑一个支付网络，假如在每个时期，网络处理零售交易的交易总额为\(Y_t\)，该交易总额以法币“美元”计量。假设代币的总流通量是固定的，我们不妨假设（不失一般性）其等于1。我们用 \(p_t\) 表示代币对等的法币价格。假设支付网络处于平稳增长状态：交易量预期增长率恒定为 \(g_Y\)，则有： \(\mathbb{E}_t [Y_{t+s}] = Y_t (1+g_Y)^s\)
（\(\mathbb{E}_t[Y_{t+s}]\)——s时间段内的交易总额均值；\(\mathbb{E}[*]\)为随机过程的均值函数）
In this model we do not distinguish between block proposal and block validation functions, and refer to block proposers and block validators simply as “validators.” We assume that the protocol for transactions is such that, whenever transactions take place, some tokens must be transferred to the validators as rewards for their activities, and, importantly, individual validators collect their pro-rata share of the total rewards, in proportion to their token stake. The latter assumption rules out individual strategies like selfish mining, which may distort the allocation of rewards among validators in relation to their individual token balances. We assume that the aggregate amount of fees generated by the network grows at the same long-run rate as the aggregate transaction volume (the ratio of the two series is a stationary process). To simplify the derivations in our model, we further restrict aggregate fees to be a constant multiple of the transaction volume – thus, validators receive fees in the collective amount of \(cY_t\) dollars per period (actual rewards are in tokens, which validators sell to consumers in exchange for dollars). We should note that this assumption applies to the total volume of fees, rather than the fee structure for individual transactions. Our assumption of aggregate fee dynamics is consistent with multiple alternative fee schedules, and does not mean that Unit-e must necessarily adopt transaction fees that are a fixed fraction of transaction value; we are still evaluating different design choices. In particular, we discuss the tradeoffs between user-determined transaction fees and algorithmic transaction fees in Section 7.3 below.
在这个模型中，我们不区分出块者和验证者的功能，把出块者和验证者统称为“验证者”。我们假设交易协议是这样的，无论什么时候产生交易，一些代币必须转移给验证者作为他们参与网络活动的奖励，而且，更重要的，个人验证者是按照他们代币权益的占比来获取他们在总奖励中占有的比例。后一种假设排除了像私自挖矿这样的个人策略，私自挖矿可能会引起验证者之间与其代币余额相关的奖励分配的失真。我们假设通过网络产生的总费用以与交易总量相同的长期增长率增长（两个系列的比率是固定的）。为了简化我们模型中的推导，我们进一步将总费用限制为交易量的恒定倍数——如此，每个时期验证者获取的费用总额就是（\( cY_t \)）美元（实际奖励是以代币的形式，验证者可以在交易所卖给消费者换取美元）。我们应该注意到这个假设适用于所有费用的总额，而不是个人交易所产生的费用结构。我们对总费用动态的假设与多重备选费用表是一致的，但这并不意味着 Unit-e 必须要采用一个与交易值固定比例的交易费用；我们仍然在评估不同的设计选择。尤其是，在下面的7.3节中我们讨论了在确定交易费用和算法交易费用之间的权衡取舍。
We assume that validators have unrestricted access to financial markets and behave competitively: they take market prices, and, importantly, the design of the payment network, as exogenous and not affected by their individual actions. Also, we assume that the risk premium associated with a financial claim paying \(Y_t\) dollars per period is constant and equal to \(λ_Y\) (in equilibrium, this determines the opportunity cost of capital associated with validation activity). Finally, we also assume that there are no physical costs associated with block validation activities.
我们假设验证者们能够不受限制地进入金融市场，并表现出竞争性：他们将市场价格，以及更重要的支付网络设计，视为一种外因，是不会被他们的个人行为所影响的。此外，我们假设，与每个时期支付 \(Y_t\) 美元的金融债权相关联的风险溢价是恒定的，等于 \(λ_Y\)（平衡时，这决定了参与验证活动的资本的机会成本）。最后，我们还假设没有与验证活动相关的物理成本。
We look for a stationary equilibrium, in which \(ϕ∈[0,1]\) tokens are held by the validators, and \(1 − ϕ\) are held by consumers for transaction purposes. We assume that validators have no use for tokens outside of their validation activity, and therefore stake their entire token balance. The equilibrium distribution of token holdings, \(ϕ\), is endogenous and determined as a part of the solution.
我们寻求一种静态均衡，在这里面，验证者持有的代币我们称为\(ϕ\)，\(ϕ\)处于0到1之间，即 \(ϕ∈[0,1]\)；而被交易者持有用于交易的代币就是（\(1 – ϕ\)）。我们假设验证者持有的代币除了验证活动不会用于其他用途，因此他们也质押了全部的代币余额。作为这个方案的一部分，代币持有量 \(ϕ\) 的均衡分布是内生的、确定的。
In equilibrium, the total market value of all the tokens held by the validators is \(ϕp_t\) , which is the value of a financial claim on the perpetual stream of cash flows in the amount of \(cY_t\) per period. Assuming no valuation bubbles, the market value of this cash flow stream is given by the valuation formula for a perpetuity with constant growth:
在均衡状态下，验证者持有的所有代币的总市值为 \(ϕp_t\) ，这是每个时期以 \(cY_t\) 美元计算的永续现金流金融债权的价值。假设没有验证泡沫，这笔现金流流通的市场价值可以通过持续增长型永续年金（指在无限期内，时间间隔相同、不间断、金额不相等但每期增长率相等的一系列现金流）的估值公式给出：
\[ \ p_t \phi = \ \lim_{T\to \infty} [\sum_{s=1}^T \frac {cY_t(1 + g_Y)^s} {(1 + λ_Y)^s}] \ = \frac {cY_t} {λ_Y - g_Y } \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7.1) \]
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\(p_t ϕ\) ——持续增长型永续年金；
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\(p_t\) —— 代币对应的法币价格；
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\(ϕ\) —— 验证者的代币持有量；
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\(g_Y\) —— 交易量预期增长率；
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\(λ_Y\) —— 风险溢价，已确定的投资收益率与冒风险所获得的收益率之差；
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\(cY_t\) —— 每个时期验证者获取的费用总额，\(Y_t\)为每个时期的交易总额，\(c\)为系数。
To pin down the value of the tokens, we need to make an assumption about consumer’s demand for tokens. In a market with infinite token velocity, consumers would hold no balances, which would imply that in equilibrium \(ϕ=1\) and \(p_t=(cY_t)/(λ_Y-g_Y)\). More generally, if consumers hold balances equal to k times the transaction volume per period, then \(p_t(1-ϕ_t)=kY_t\) , and therefore the equilibrium token value is
为了确定代币的价值，我们需要对消费者的代币需求做一个假设。在一个有着无限的代币周转率的市场中，消费者可能几乎没有余额，这就意味着在平衡状态，\(ϕ=1\) 而 \(p_t=(cY_t)/(λ_Y - g_Y)\)。更普遍来说，如果消费者持有的代币余额等于每个时期交易量的 k 倍，那么 \(p_t(1-ϕ_t)=kY_t\)，因此代币的均衡价值是：
\[ \ p_t = (k + \frac {c} {λ_Y - g_Y})Y_t \ \ \ \ \ \ \ \ \ \ \ \ \ （7.2） \]
Note that in the absence of an explicit description of demand for token balances, \((cY_t )/(λ_Y - g_Y)\) serves as a lower bound on the token value.
注意，在缺少对消费者代币余额需求的明确描述下，\((cY_t )/(λ_Y - g_Y)\) 充当了代币价值的下限。
The equilibrium value of the tokens in (7.2) consists of two terms. The first term, \(kY_t\), reflects consumer demand for holding token balances. The value of this term depends on \(k\), which is inversely related to the equilibrium token velocity. It is important to acknowledge that token velocity is an equilibrium outcome, related to a number of properties of the payment network and the broader market. Wider adoption of the tokens could raise demand for token balances, while efficient channels for transactions between tokens and fiat currency would enable consumers to support the desired volume of transactions with lower token balances and result in higher token velocity.
在公式（7.2）中的代币均衡价值由两项组成。第一项，\(kY_t\)，反应了消费者持有代币余额的需求。这一项的值取决于 \(k\)，与均衡代币周转率成逆相关。必须承认，代币周转率是一种均衡结果，与支付网络和更广泛市场的许多特性有关。代币更广泛的采用能够提高对代币持有余额的需求，而代币和法币之间有效的兑换渠道将会促使消费者提高对交易量的渴望，而保持较低的代币余额，最后的结果就是代币的周转率会更高。
The relation between token value and token velocity is commonly invoked when discussing the valuation of cryptocurrencies. While token velocity and equilibrium token value are certainly related in equilibrium, the relation between the two, like its analog in traditional monetary economics, is not a true structural relation and it does not provide a reliable anchor for token valuation.
在讨论加密货币的估值时，通常会援引代币价值和代币周转率之间的关系。虽然代币周转率和代币均衡价值在均衡状态下必然相关，但是两者之间的关系，就跟在传统货币经济学中的类似物一样，并不是真正的结构关系，也不能够为代币估值提供一个可靠的锚定。
The second term in the valuation equation (7.2), \((cY_t )/(λ_Y-g_Y )\), reflects the demand for tokens from validators. This term is proportional to the overall volume of retail transactions, \(Y_t\) , and to the rate at which fees are charged for transactions. All else equal, broader adoption and utilization of the payment system (higher \(Y_t\)) results in higher value of the tokens. Importantly, the above equation does not suggest that token value is increasing in the level of fees. Our analysis here focuses on a single stationary equilibrium and does not explicitly describe how the systems responds to changes in parameters: higher fee levels would eventually lead to lower transaction volume. [1]
在（7.2）的估值方程中的第二项，\((cY_t )/(λ_Y - g_Y)\)，反应了验证者对代币的需求。这一项与总体的零售交易额 \(Y_t\) 和交易收费率成正比。所有其他一切平稳时，更广泛的采用和使用支付系统（更高的 \(Y_t\)），会导致更高的代币价值。重要的是，上述等式并没有表明代币价值在收费水平上的增长。我们这里的分析侧重于单一的稳定均衡，并没有明确描述系统如何响应收费水平参数的变化：较高的费用水平最终会导致较低的交易量。
Our analysis in this section also relies critically on the assumption of competitive behavior by the network participants. To what extent this assumption offers a good approximation of agent behavior in this environment depends on individual opportunities and incentives to engage in strategic behavior. Ultimately, individual incentives and token valuation are closely linked in PoS systems, and must be analyzed jointly. Such analysis is beyond the scope of this chapter.
我们在本节中的分析也非常依赖于对网络参与者的竞争行为做的假设。这种假设能够在多大程度上很好地近似这种环境中的代理行为，取决于从事这种战略行为的个人机会与激励措施。最终，个人激励和代币估值在PoS系统中紧密相连，必须共同分析。这种分析已超出了本章的范围。
7.1.2 An Extended Model with Increasing Token Supply（增加代币供应的扩展模型）
Here we extend the valuation model of the previous section to allow for increasing token supply. In this model, validators are rewarded in newly minted tokens in addition to the fees collected from consumer transactions. For tractability, we formulate the extended model in continuous time.
在这里，我们扩展了上一节里的估值模型，允许增加代币供应。在这个模型中，除了从消费者交易中收取的费用外，验证者还将获得新铸造的代币奖励。为了易于处理，我们建立连续时间的扩展模型。
We now allow the transaction volume growth parameter to vary in time, and denote it by \(g_Y(t)\). We also introduce randomness into the evolution of the transaction volume \(Y_t\), as shown in the following differential equation:
现在我们允许交易量增长参数随着时间变化，并用 \(g_Y(t)\) 表示。我们也将在交易量 \(Y_t\) 的演变过程引入随机性，如下微分方程所示：
\[ \frac {dY_t} {Y_t} = g_Y(t)dt+ \sigma (t)dZ_t \]
where \(g_Y(t)\) and \(\sigma(t)\) are bounded continuous functions of time, and \(Z_t\) is a Brownian motion. [2] Here \(Z_t\) is a simple model for the randomness in the system, and \(σ(t)\) represents the instantaneous volatility of growth in transaction volume. Investors require compensation for being exposed to \(dZ_t\) shocks based on the comparable investment opportunities in financial markets, which we assume to be η units of expected excess returns per unit of risk. \(η\) is known as the market price of risk, and we take it to be constant here, for simplicity. Then, under the risk-neutral valuation measure \(Q\) , the transaction volume follows
这里的 \(g_Y(t)\) 和 \(\sigma(t)\) 是关于时间的有界连续函数，而 \(Z_t\) 是一种布朗运动。这里 \(Z_t\) 是系统随机性的简单模型，而 \(\sigma(t)\) 代表了交易量增长的瞬时波动性。依据金融市场的可比投资机会，投资者对遭受 \(dZ_t\) 冲击要求补偿，我们假设单位风险的预期超额回报是 \(\eta\) 个单位，为了简便起见，我们在这里设为常数。然后，在风险中性定价指标 \(Q\) 下，交易量遵循以下方程：
\[ \frac {dY_t} {Y_t} = g_Y^Q (t) dt + \sigma (t) dZ_t^Q \]
Where
其中
\[ \ g_Y^Q (t) = g_Y (t) - \eta σ(t) \]
and \(Z_t^Q\) is a Brownian motion under measure \(Q\).
而 \(Z_t^Q\) 是指标 \(Q\) 下的布朗运动。
As above, \(p_t\) denotes the total value of tokens at time \(t\) and validators hold fraction \(ϕ_t\) of all tokens. The market clearing condition requires that validators and consumers collectively hold all tokens, and thus
如上所述，\(p_t\) 代表了时间 \(t\) 时的代币总价值，验证人持有的代币份额 \(ϕ_t\) 。市场清算条件要求验证者和消费者共同持有了所有代币，因此有：
\[ \ ϕ_t p_t = p_t - kY_t \]
where \(k\) is a consumer demand parameter, which we again assume to be constant. Thus,
其中 \(k\) 是消费者需求参数，我们再次假设其为常数。因此：
\[ \ ϕ_t = 1 - k \frac {Y_t} {p_t} \]
We assume that token supply grows deterministically over time. Specifically, we assume that new tokens are issued at (bounded) rate \(r(t)\). In expectation, under the risk-neutral valuation measure, validators earn the risk-free rate of return, which we assume to be constant and denote it by \(λ_f\). Thus, we obtain the valuation equation:
我们假设代币供应随着时间的推移有确定性的增长。具体来讲，我们假设新的代币以（有限的）速率 \(r(t)\) 发行。期望着，在风险中性定价指标下，验证者赚取无风险回报率，假设为常数，用 \(λ_f\) 表示。因此，得出估值方程：
\begin{equation} \phi_{t} p_{t} = c Y_{t} dt + e^{-\lambda_{f} dt} E_{t}^{Q} \left[\left(\phi_{t} + \left(1 - \phi_{t} \right) r(t) dt \right) p_{t+dt}\right]. \end{equation}
On the left-hand side, \(ϕ_t p_t\) is the total market value of the tokens staked by the validators at time t. On the right-hand side, we have two terms. The first term, \(cY_t dt\), is the flow of transaction fees that accrue to the validators over the infinitesimal period \([t,t+dt)\).The second term, \(e^{-λ_f dt} E_t^Q [(ϕ_t+(1-ϕ_t) r(t) dt) p_{t + dt}\)], is the discounted expected value ( under measure \(Q\) ) of the validators’ token holdings at the end of the period. [3] Then,
在左边，\(ϕ_t p_t\) 是验证者在时间t质押的代币的总市值。在右边，有两项。第一项，\(cY_t dt\)，是在无限小的时间区间 \([t,t + dt)\) 内积累到验证者的交易费用流。第二项，\(e^{-λ_f dt} E_t^Q [ ( ϕ_t+(1-ϕ_t) r(t) dt ) p_{t+dt} \)]，是验证者持有的代币在这个时期结束时的贴现期望值（在指标 \(Q\) 下）。 那么：
\[ \ E_t^Q [dp_t] = \ λ_f p_t dt - \frac {p_t} {p_t - kY_t} (cY_t + kY_t r(t))dt \]
We look for an equilibrium token price process p_t of the form p_t=p(t,Y_t), where p(t,Y_t) is a sufficiently smooth function of its arguments. Applying Ito’s lemma, we obtain a PDE on the token price function:
我们以 \(p_t = p(t,Y_t)\) 的形式处理 \(p_t\) 来寻求一个均衡代币价格，其中 \(p(t,Y_t)\) 是其参数的足够充分的光滑函数（定义域内连续可导）。应用伊藤引理，我们获得代币价格方程的偏微分方程：
\begin{equation} \frac{\partial p}{\partial t} + \frac {\partial p} {\partial Y} g_{Y}^{Q}(t) Y + \frac {1}{2} \frac {\partial^{2} p} {\partial Y^{2}} \sigma(t)^{2} Y^{2} - \lambda_{f} p + \frac {p} {p-k Y} (c+k r(t)) Y = 0 \quad \quad \quad \quad \quad (7.3) \end{equation}
Equation (7.3) is the valuation PDE. This equation has multiple solutions, and we look for a non-negative solution without valuation bubbles. Specifically, we look for a solution with suitably bounded growth in \(Y\), and subject to a boundary condition
方程（7.3）是估值偏微分方程。这个方程有多个解，我们要寻找一个没有估值泡沫的非负解。具体地说，我们寻找一个在 \(Y\) 上有适当边界增长的解，并且符合以下边界条件：
\[ \ p(t,0) = 0 \]
which requires that token value vanishes at zero transaction volume (recall that zero is an absorbing boundary for the transaction volume process). The above equation has a linear solution,
这要求代币价值零交易量（记住零是交易量过程的吸收边界条件）时消失。上面的方程有一个线性解：
\[ \ p(t,Y ) = A(t)Y \ \ \ \ \ \ \ \ \ \ \ \ \ (7.4) \]
where the unknown function \(A(t)\) is a bounded, positive solution of the ODE:
其中未知函数 \(A(t)\) 是以下常微分方程的有界正数解：
\[ \ \frac {dA(t)} {dt} + g_Y^Q (t)A(t) - λ_f A(t) \ + \frac {A(t)} {A(t)-k} (c + k r(t)) = 0 \]
In the stationary case of \(r(t) = r\), \(σ(t) = σ\), and \(g_Y (t) = g_Y\), the total value of tokens \(p_t\) is a constant multiple of the transaction volume \(Y_t\):
在 \(r(t) = r\), \(σ(t) = σ\), 和 \(g_Y (t) = g_Y\) 的静止情况下，代币的总价值 \(p_t\) 是交易量 \(Y_t\) 的常数倍：
\[ \ \frac {p_t} {Y_t} = A(t) = k + \frac {c+kr} {λ_f + ησ - g_Y} \ \ \ \ \ \ \ \ \ \ \ (7.5) \]
This solution describes the total value of tokens under constant growth rate in the number of outstanding tokens. Note that, \(λ_f + ησ = λ_Y\) , which is the expected return on the financial claim paying a cash flow stream equal to the aggregate flow of transaction fees, \(cY_t\). We thus recover the valuation formula (7.2) by setting the token rewards to zero, \(r = 0\).
此方案描述了未完成代币的在恒定增长率下的代币总价值。注意到， \(λ_f + ησ = λ_Y\)，这是在金融债权上支付等同于 \(cY_t\) 的现金流获取的预期收益，\(cY_t\) 是交易费用总额。因此，通过将代币奖励设置成零，即 \(r = 0\)，我们可以恢复到估值方程（7.2）。
The above solution highlights the valuation effect of rewarding validators in newly minted tokens. In addition to collecting transaction fees, validators also collect proceeds from seignorage. Comparing with (7.2), we see that this effectively raises the flow of proceeds to validators from \(c\) to \(c + kr\). The second term, \(kr\) is intuitive: transfers to validators due to seignorage are proportional to the level of token balances held by consumers. If consumers hold no token balances between transactions, any benefit validators derive from collecting rewards in newly minted tokens is completely offset by the decline in the market value of tokens in their stake.
上述解决方案突出了用新铸造的代币奖励验证者的估值影响。除了收取交易费用之外，验证者还从新币铸造中获取收益。与公式（7.2）相比，我们发现这有效地将验证者的收益流从 \(c\) 增加到了 \((c + kr)\)。第二项，\(kr\) 是很直观的：由于新币铸造转移给验证者的代币与消费者持有的代币余额水平成正比。如果消费者在交易过程中没有代币持有余额，验证者在新币铸造中收取报酬获取的收益就完全被其权益代币的市值下降所抵消了。