Dynamic Programming for Consumption, Money, and Portfolio Choice
Summary
The document sets out a Bellman problem for a representative investor in an endowment economy. The investor chooses current consumption, money to carry into the next period, and portfolio shares to maximize discounted expected lifetime utility. The budget equation links wealth to consumption, transaction costs, current money, and the return on a portfolio of assets; next-period money depends on transfers and inflation.
It also specifies assumptions on the transaction-cost function: costs rise with consumption, fall with money holdings, and have stated curvature and cross-effect properties. The setup is attributed to research on inflation, money, and asset returns, but the document does not derive a solution, first-order conditions, or empirical results. It is therefore a model statement and a request for help, not a worked method. Applying it requires additional assumptions about utility, shocks, asset returns, and the state variables.
Key ideas
- The investor maximizes discounted expected utility by choosing consumption, money holdings, and portfolio allocations.
- Money holdings reduce current consumption transaction costs and carry forward after inflation and transfers.
- Wealth evolves according to the return on a portfolio of assets after consumption and transaction costs.
- The transaction-cost function is assumed to be increasing in consumption and decreasing in money holdings.
- The document states the model but does not solve it or provide empirical evidence.
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Full text
# Dynamic programming and Bellman equation to obtain the maximum
# Dynamic programming and Bellman equation to obtain the maximum
This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM"
Suppose an endowment economy where the representative agent maximizes her expeced lifetime utility subject to a budget constraint and given that transaction costs to purchase consumption goods are mitigated by money holdings, that is: $$ V(w_t,m_t,x_t)=\max_{c_t,\mu_t,\{s_t^i\}_{i=1}^{n}}\left(u(c_t)+\delta\mathbb{E}_{t}\left[V(w_{t+1},m_{t+1}, x_{t+1}\right)]\right) \quad (1)$$ $$\text{subject to}\quad w_{t+1}=R_{t+1}\left[w_t-c_t-T(c_t,m_t) + m_t -\mu_t\right] \quad (2)$$ $$ R_{t+1}=\sum_{i=0}^{n}s_t^i R_{t+1}^{i}, \quad \sum_{i=0}^{n}s_t^i=1 \quad\text{and} \quad (3)$$ $$m_{t+1} = (\mu_t + z_{t+1})/\pi_{t+1} \quad (4)$$ where
- $w_t,$ real financial wealth (excluding money holdings)
- $m_t,$ real money holdings
- $x_t,$ set of state variables, exogenous to the consumer-investor, that is sufficient to represent changes in the investment opportunity set
- $c_t,$ current consumption
- $\mu_t$, real money holdings (for use in the upcoming period)
- $\pi_{t+1}=p_{t+1}/p_t$, gross inflation rate
- $z_{t+1} = (M_{t+1} - M_{t})/p_t$, transfer of government revenues from money creation
- $T(c_t,m_t)$ represents the real transaction cost of purchasing the current level of consumption
- so $m_{t+1}$ are the deflated money holdings of the next period
And it also holds for $T(\cdot,\cdot)$ that it is twice differentiable and also:
- $T(c_t,m_t)\geq 0$, $T_{c}(c_t,m_t)> 0$, $T_{m}(c_t,m_t)< 0$
- $T_{cc}(c_t,m_t)\geq 0$, $T_{mm}(c_t,m_t)\geq 0$ and $T_{cm}(c_t,m_t)\leq 0$
I cite the problem above, since it's been years from the last time I have a course in dynamic programming and at the moment I am struggling to solve it.
This is from the appendix of the paper. This is all about the solution of the problem and nothing elseShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.