Part 4: Consumption smoothing
第4部分：消费平滑
In the standard model on inter-temporal consumption choice, infinitely lived households seek to maintain their discounted marginal benefit from consumption at the same level across consecutive periods: u’(Ct) = 𝜷t*(1+ rt)*u’(Ct+1). [Note 1] Lower interest rate reduces the discounted marginal benefit of consumption in the next period, and so households consume more in the current period. This effect is called “consumption smoothing”.
When the yield curve falls:
households increase current consumption to keep the discounted marginal benefit of consumption the same over time (“consumption smoothing”), but
short-duration balance sheet households also reduce future consumption (“wealth effects”).
So “consumption smoothing” and “wealth effects” of short-duration households work in opposite directions. However, consumption smoothing depends only on the current period interest rate (“rt”) whereas wealth effects arise from interest rates along the entire yield curve (“𝜞t”).
If the central bank lowers only the current policy rate, while holding the rest of the yield curve constant, this stimulates consumption as the standard model predicts. Further, if the central continues keeping the policy rate low for an extended period, the standard model predicts this eventually has inflationary effects.
However, if the central bank exercises yield curve control to keep (expectations of) future policy rates low, wealth effects come into play; if households have short duration balance sheets, the wealth effects work against the stimulatory effect of lower policy rates (due to consumption smoothing). However, wealth effects operate across multiple periods into the future, whereas consumption smoothing operates only in the current period.
BCW 2025 concludes that yield curve control enables a central bank to stimulate current period consumption, while dampening the inflationary effects of keeping policy rates on a low path.
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Note 1: This is known as the “Euler condition”, after the Swiss mathematician who developed partial differentiation equations for fluid dynamics. The Euler condition treats consumption as a flow across time, similar to a fluid.
于跨期消费选择的标准模型中，无限寿命家庭力求在连续的时期内将消费折现边际效益保持在同一水平：u’(Ct) = 𝜷t*(1+ rt)*u’(Ct+1)。[注1] 较低的利率会将底下一期的消费折现边际效益，因此家庭在当前时间的消费会增加。这种效应称为“消费平滑”。
当收益率曲线下降时：
家庭会增加当前消费，以使消费的折现边际效益在一段时间内保持不变（“消费平滑”），但
短持续时间资产负债表的家庭也会减少未来消费（“财富效应”）。
因此，“消费平滑”和短持续时间家庭的“财富效应”拥有相反作用。然而，消费平滑仅取决于当前利率（“rt”），而财富效应则来自整个收益率曲线上的利率（“𝜞t”）。
如果中央银行降低当前政策利率，同时保持收益率曲线的其余部分不变，这将刺激消费，正如标准模型所预测的那样。此外，如果央行持续在较长时间内保持低政策利率，标准模型预测这最终会产生通胀效应。
但是，如果央行实施收益率曲线控制以保持（预期的）未来政策利率处于低位，财富效应就会发挥作用；如果家庭具有短持续时间的资产负债表，财富效应就会抵消（由于消费平滑的）较低政策利率的刺激效应。然而，财富效应会在未来多个时期发挥作用，而消费平滑只在当前时期起作用。
BCW 2025的结论是：收益率曲线控制使央行能够刺激当前消费，同时抑制保持政策利率处于低位的通胀效应。
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注1: 这杯称为“欧拉条件”（Euler condition），以开发流体动力学偏微分方程的瑞士数学家命名。欧拉条件将消费视为跨时间的流动，类似于流体。