5.1 The Envelope Theorem
In Week 4, we investigated the relationship between a constrained optimization problem (for example, minimizing utility subject given the prices a consumer faces and a target utility) and its associated value function (the “expenditure” function that returns the highest possible utility for any combination of prices and income). The Envelope Theorem is a powerful tool in economic and optimization theory that relates this relationship in a general way.
We can think of a constrained optimization problem as being defined by:
- a vector of $n$ choice variables $\vec{x} \in \mathbb{R}^n$
- a vector of $l$ parameters $\vec{\alpha} \in \mathbb{R}^l$
- an objective function $f(\vec{x}; \vec{\alpha})$
- a constraint $g(\vec{x}; \vec{\alpha}) \ge 0$
For example, for an expenditure minimization problem with two goods $x_1$ and $x_2$ with associated prices $p_1$ and $p_2$ and a target utility of $U$, we would have
- $n = 2$ choice variables: $\vec{x} = (x_1,x_2)$
- $l = 3$ constraints: $\vec{\alpha} = (p_1,p_2,U)$
- objective function (to minimize): $f(x_1,x_2,p_1,p_2,U) = p_1x_1 + p_2x_2$
- constraint: $g(x_1,x_2,p_1,p_2,U) = u(x_1,x_2) - U \ge 0$
We can think of the solution to this problem as being a function mapping the vector of parameters $\alpha$ into the vector of choice variables: that is, $\vec{x}^\star(\alpha)$. The value function $V(\vec{\alpha})$ gives the min (or max) value of $f$ when we choose $\vec{x}^\star(\alpha)$ optimally given the constraint: \(V(\vec{\alpha}) = f(x^\star(\alpha))\)
It turns out that the partial derivative $\frac{\partial V(\vec{\alpha})}{\partial \alpha_i}$ – that is, how the optimized value of a problem changes with each of the parameters which affect it – is a quantity that’s useful to know for a surprisingly broad range of applications. Big microeconomic, macroeconomic, and econometric results are completely dependent on knowing this derivative for interesting choices of objective functions and constraints. Before reading on, pause and try to think about how you might try to solve for this derivative: it isn’t obvious, since a change in any parameter $\alpha_i$ also changes the optimal $\vec{x}(\alpha)$ you would choose, which in turn changes $V$. In other words, it’s hard to take a derivative that involves a max or min.
The answer to this question—how to compute this derivative—ends up being surprisingly simple, and we call the answer the Envelope Theorem. The Envelope Theorem states that, under certain conditions, \(\frac{\partial V(\vec{\alpha})}{\partial \alpha_i} = \left. \frac{\partial f(\vec{x};\vec{\alpha})}{\partial \alpha_i} \right|_{\vec{x}=\vec{x}^\star(\vec{\alpha})} \tag{1}\) where $x^\star(\vec{\alpha})$ is the optimal choice of $x$ for the problem. The vertical line means “evaluated at,” so the right hand side tells us to just take the derivative of the objective with respect to the parameter, and plug in the optimal solution $\vec{x}^\star$ for $\vec{x}$.
We’ve known this theorem is true for “nice” functions for a little over 80 years, but in 2002 Stanford’s own Paul Milgrom and Ilya Segal proved an amazing new result extending the Envelope Theorem so it could work for much more general functions (see Milgrom and Segal (2002)). It was a huge achievement that led to a lot of important tools and created billions of dollars in social welfare out of thin air. Talk about using math to make the world a better place!
I won’t prove the Envelope Theorem until Chapter 7, but if you still feel like you’re missing some intuition, this would be a super fun topic to get into in office hours. The Wikipedia page on the Envelope Theorem is also excellent, and goes into more detail than I have here.
Using the Envelope Theorem to Prove a Familiar Result
From here, let’s finally write up a proof for Shephard’s Lemma. We’re gonna prove it by applying the Envelope Theorem to the expenditure minimization problem described above. We’re going to try to minimize our spending by picking the amount of each good $\vec{x} = (x_1,x_2)$ to buy given the parameters $\vec{\alpha} = (p_1,p_2,U)$, so our objective function $f(\vec{x}; \vec{\alpha})$ is the total expenditure: \(\begin{aligned} \min_{\vec{x}} \quad & f(\vec{x}; \vec{\alpha}) = p_1x_1 + p_2x_2 \\ \text{s.t.} \quad & g(\vec{x}; \vec{\alpha}) = u(x_1,x_2) - U \ge 0 \end{aligned}\) and the value function is our expenditure function: \(V(\alpha) = E(p_1,p_2,U) = \min_{x_1,x_2} p_1x_1 + p_2x_2 \text{ s.t. }u(x_1,x_2) \ge U\) By the Envelope Theorem, the derivative of the value function $E(p_1,p_2,U)$ with respect to one of the parameters—say $p_1$—is the same as the derivative of $f(\vec{x},\vec{\alpha})$ with respect to $p_1$, evaluated at the optimal solution. In this context, the solution to the cost-minimization problem is the Hicksian bundle $\vec{x}^c(p_1,p_2,U)$; therefore the Envelope Theorem states that, \(\begin{aligned} \frac{\partial E(p_1,p_2,U)}{\partial p_1} &= \left. \frac{\partial f}{\partial p_1} \right|_{\vec{x}=\vec{x}^c(p_1,p_2,U)} \\ &= \left. \frac{\partial (p_1 x_1 + p_2 x_2)}{\partial p_1} \right|_{\vec{x}=\vec{x}^c(p_1,p_2,U)} \\ &= x_1 \big|_{\vec{x}=\vec{x}^c(p_1,p_2,U)} \\ &= x_1^c(p_1,p_2,U). \end{aligned}\)
In general, we can write: \(\frac{\partial E(\vec{p},U)}{\partial p_i} = x_i^c(\vec{p},U),\) which is exactly Shephard’s Lemma. It tells us that the partial derivative of the expenditure function with respect to a price gives the Hicksian demand for that good. Q.E.D.
(One small side note: there are really different envelope results for constrained versus unconstrained optimization problems. We won’t be covering those in detail in this class.)