Consumption, Part 2: Concavity

These are some notes on two papers of Christopher Carroll and Miles Kimball: “Liquidity Constraints and Precautionary Savings” (2001) and “On the Concavity of the Consumption Function” (1996).  My only original contribution is to point out that you can extend the (1996) results to any utility function which has the property that the ratio of prudence to risk aversion is a convex decreasing function.

Carroll and Kimball prove the result for HARA utility functions — in which this ratio is (global) constant, or a linear function. I actually think the result holds for a much larger class of functions — as long the derivative of the ratio of prudence to risk aversion decreases in absolute value fast enough, so concave increasing would also work, but I have no proof for the latter statement.

The point is for an inter-temporal optimizing agent, the poor will have a higher marginal propensity to consume than the rich as long as there is any (non-trivial) uncertainty in future income.

Set-Up and Heuristic

Following the notation of their 2001 paper, write the optimization problem in Bellman form as:

V_{t}(w_t) = max \{ u(c_t) + \Omega_t(w_t - c_t) \}

where

\Omega_t = E_t( \beta_{t+1} V_{t+1}( R_{t+1}s_t + y_{t+1})\}

is the end of period value function, \beta_t is a stochastic discount term, R_t is a stochastic interest rate, y_t is the stochastic income, and w_t = c_t + s_t.

To see how risk induces concavity, ignore for a moment the interest rate and discount factor. In the absence of risk, the first order condition becomes:

u(w_t - s_t) = V'_{t+1}(s_t + \bar{y}), where \bar{y} is certain future period income.  But if there is a 50% chance that future labor income will be one dollar larger and a 50% chance that it will be a dollar lower, then the first order condition becomes:

u(w_t - s_t) = .5V'_{t+1}(s_t + \bar{y} + 1) + .5V'_{t+1}(s_t + \bar{y} -1)

and by Jensen’s inequality, the convex combination of a convex function is larger than the function of the combination, so that the expected marginal value of savings under income uncertainty is greater than the expected marginal value of savings without uncertainty:

In the presence of uncertainty, the agent will choose to save the larger amount, t*, where the difference t* – s* is the level of precautionary savings due to uncertainty.

What happens when we give the consumer more wealth? That is equivalent to a right-ward shift of u(w-s). In that case, the equilibrium savings level will be larger.

but at larger levels of x, the difference between V'(x) and 1/2V'(x+1) + 1/2V'(x-1) decreases, and t* approaches s*. This need not always be the case. For example, with quadratic utility, the risky value marginal function is the same as the riskless marginal function. In other (degenerate) cases, Jensen’s rule no longer applies. To prove conditions under which the marginal propensity to consume decreases with wealth, we need a formal definition of convexity of the marginal value function, or concavity of the original value function.

We can impose an ordering on the space of increasing concave functions: f_1 > f_2 \text{ iff } f_1' = f_2' \circ g, where g is a concave function.

Note that this is a  local condition, as we can always define g = (f_2')^{-1} \circ f_1' — the issue is whether this function is concave. To test this condition, we have f_1 > f_2 if and only if \psi(f_1) > \psi(f_2), where

\psi(f) = \frac{f''' f'}{(f'')^2}

\psi is the ratio of absolute prudence to absolute risk aversion. This is an equivalence relation, and two functions are equal only when the derivative of one is a linear transformation of the other.

If the inequality is strict at a point, then in a neighborhood of that point, our functions will be strictly more concave.

Result

If \psi(u(x)) = f(x) \ge 1 is a convex decreasing function, then the optimal consumption rule will be a strictly concave function of wealth except in the last period of life whenever \psi(u(x)) is non-constant. In the case that it is constant, we have HARA utility as covered in the C&K 1996 paper:

The exceptions to strict concavity include two well-known cases: CARA utility if all of the risk is to labor income (no rate of return risk), and CRRA utility if all of the risk is rate-of-return risk (no labor income risk). These special cases have been widely used because of their analytical convenience (they yield a linear consumption function), but the analytical results in this paper bolster the argument (made forcefully by Kimball (1990a), Carroll (1995a), Deaton (1992) and others) that it is in most cases unwise to rely on these analytically convenient formulations because the behavior they imply is qualitatively quite different from behavior in the general case.

— C &K (1996)

The proof strategy is backward induction. In the last period of life, T, the beginning period value function is the utility function

V_T(w_T) = u(c_T(w_T)) and the optimal consumption rule is linear in wealth: c_T(w) = w. Therefore V_T \geq u. Note that in all cases, the value function inherits concavity and monotonicity from the utility function.

Then the riskless end of period value function

\bar{\Omega}_{t}(x) = \bar{\beta}_{t+1} V_{t+1} (\bar{R}_{t+1}(x) + \bar{y}_{t+1})

will be also be more concave than the utility function at the current period savings that corresponds to next period wealth (see Lemma 3 of C & K (2001)).

Strict concavity is guaranteed when we show that the risky end of period value function is strictly more concave than the riskless end of period value function: \psi(\Omega_t(x)) > \psi(\bar{\Omega}_t(s)).

Then, by Lemma 2 of CK (2001), the beginning period value function \psi(V_t) > \psi(u), and we repeat.

Note that whenever \psi(V_t) > \psi(u), we must have V'_t(w) = u'(C) where C is strictly concave. But by the Envelope Theorem, V'_t(w) = u'(c_t(w)), so the optimal consumption rule in that period will be a strictly concave function of wealth.

The crux of the proof — i.e. where risk enters the picture — is to show that that the risky end-of-period value function is strictly more concave than the riskless end of period value function, as illustrated by the diagram. To see this, define the absolute risk-aversion of the value function.

A(x) = -V''_{t+1}(x)/V'_{t+1}(x)

It will be convenient to express $psi(\Omega)$ in terms of risk aversion and its derivates:

\psi(V) = -A'/A^2 + 1. Therefore A' \leq 0 and A > 0.

Derivates pass through the expectation operator, so (dropping time subscripts for convenience)

\psi(\Omega(x)) = \frac{\mathbf{E}[(A'(Rx + y)/A^2(Rx + y) + 1)\beta R^3A^2(Rx+y)V'(Rx+ y)] \mathbf{E}[\beta RV'(Rx+ y)]}{\mathbf{E}[\beta R^2A(Rx+y)V'(Rx+ y)]^2}

Note that Cov(-A'/A^2, \beta R^2 A^2V') \ge 0, as by assumption -A'/A^2 is decreasing, and so is A^2V'. Therefore (dropping arguments for convenience):

\psi(\Omega(x)) \geq \frac{\mathbf{E}[A'/A^2 + 1][\mathbf{E}[A^2  \beta R^3V'] \mathbf{E}[\beta RV']}{\mathbf{E}[\beta R^2AV']^2}

And as -A'/A^2 +1 = \psi(V) is convex, we only need to show that the rest of the terms are greater than or equal to 1.

Define the inner product

<a(R,u,\beta), b(R,u,\beta)> = \lambda^{-1} \int (a(R,u, \beta)b(R,u, \beta) R \beta \phi \circ \mathbf{F}^{-1}(R,u, \beta) dR du d\beta

where

  • \lambda = \mathbf{E} [\beta RV'(Rx+y)]
  • \phi(R,y,\beta) is the distribution that we are taking the expectation of
  • (R, u, \beta) = \mathbf{F}(R,y,\beta) = (R, V(Rx+y), \beta)

We can verify that this is a norm (note that \beta > 0, R > 0 and V' > 0 by assumption) and so Cauchy-Schwarz applies. Set z = A\circ V^{-1}

Then

\frac{\mathbf{E}[A^2\beta R^3V'] \mathbf{E}[\beta RV']}{\mathbf{E}[\beta R^2AV']^2} = \frac{<Rz,Rz>\lambda^2}{<Rz,1>^2 \lambda^2} \geq 1

The result of the variable change is that the factor R\beta V' gets absorbed into the differential — at the expense of an additional factor equal to the mass of this term, \lambda. We already have one \lambda in the numerator from the original problem, and the variable change  gives us another. The variable change of the square in the denominator also gives us two powers of \lambda, so both cancel. The “natural” coordinates of this problem are functions in terms of $V^{-1}$.

This converts the problem into a form suitable for Cauchy-Schwarz.

If equality holds, then RA\circ V^{-1} = C, so that RA(Rx + y) =C. Therefore if labor income and return income are not perfectly correlated, there is no rate of return risk and the equality is strict. Assuming both rate of return risk and income risk that are not perfectly correlated, the inequality will be strict, and so the consumption rule will be strictly concave.

In particular, we can write the T-k’th consumption rule as a composition of k strictly concave functions:

c_{T-k}(w) = C_1 \circ C_2 \circ ... \circ C_k(w)

Elsewhere, Carroll shows that if we take the limit as the number of steps grows to infinity, we get a “limit” consumption rule which is (highly) concave and can be thought of as the steady state consumption rule, at least when the majority of the population is away from their end of life period. But that is a blog for another day!

The only innovation here is to show that the strict concavity result holds for a much larger class of utility functions other than HARA — namely any function in which \psi(u) is a convex decreasing function.

UPDATE: Fixed some white-space problems, and cut some extraneous comments in the Set Up section. Added info on the change of variable and more comments at the end. Added section on composition of concave functions.

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Consumption, Part 2: Concavity

3 thoughts on “Consumption, Part 2: Concavity

  1. beowulf says:

    Wow, that makes my brain hurt just to think about! I’ll take your word for it that the math works out. :o)

    OT I was just thinking there’s another way Uncle Sam could capture seigniorage revenue from banks. Since the Fed already refunds to Tsy the debt service paid on Fed-held Treasuries, simply widen their gaze with a duty (whether by a one line amendment to the FRA or “voluntary” Tsy-Fed agreement) for the Fed to refund ALL debt service on Treasuries, that is, current Tsy rebate PLUS current net interest… might as well throw in IOR as well.

    Whether the Fed levies a bank asset tax, locks interest rates at 0, hikes transaction fees, issues Fed bonds and/or simply “prints” the money is up to the Board of Governors. They can arrange it however they like as long as every year the Fed transfers to TGF an amount equal to debt service cost.

  2. Beowulf,

    I wouldn’t support that proposal, since the goal here is to recapture the spread between what banks pay out in deposits and their marginal costs. The government doesn’t need more money per se, it needs to take away economic rents in the private sector.

    If you want to impose a separate bank tax in addition to this, then that might be a good idea, but the size of the tax should be related to some identified economic rent, not to the size of the federal debt per se.

  3. beowulf says:

    I take your point. I’m looking at the matter from the political standpoint, if people are gung ho on cutting federal spending, net interest is in a class by itself as wasteful spending.
    As for imposing a bank tax, your first problem (again, I mean politically) is calling it a tax. I think the simplest way to achieve your goal would involve two legislative reforms of FDIC:
    1. FDIC insurance premiums (as I mentioned a few weeks ago, now based on bank assets and not deposits) would be pegged at IOR/FFR target rate. The Fed could make its shadow bigger or smaller but couldn’t escape it.
    2. FDIC would transfer surplus premiums (over and above current formula) to Tsy as they’re collected.
    Would that do the trick (as is or with some modification) or am I all wet?

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