\[
\definecolor{name}{RGB}{200,210,200}
\newcommand{\condition}[1]{\color{name} \text{(#1)}}
\]

**
Consider a cash credit goods economy in which households have preferences of the form \(\sum_{t=0}^\infty \beta^t (\log c_{1t}+\phi \log c_{2t})\), where \(c_{1t}\) and \(c_{2t}\) denote consumption of cash and credit goods, respectively, \(\phi\) is a parameter, and \(0 < \beta < 1 \) is the discount factor.
Households supply labor inelastically.
The resource constraint is given by
\[c_{1t} + c_{2t} + k_{t+1} = k_t^\alpha + (1-\delta) k_t\]
where \(k_t\) denotes the capital stock, and the capital share parameter \(\alpha\) and the depreciation rate \(\delta\) are both positive and less than 1.
It is convenient to let \(q_t\) denote the price of money (or the inverse of the price level). The representative household's budget constraint is given by
\[q_t M_{t+1} + c_{1t} + c_{2t} + k_{t+1} \leq q_t M_t + R_{kt}k_t + w_t l_t + q_t T_t\]
where \(M_t\) denotes cash balances, \(R_{kt}\) denotes the return to capital, and \(T_t\) denotes lump sum transfers of cash by the government. The household can use a fraction \(\theta\) of its capital stock to purchase cash goods. The cash in advance constraint is
\[c_{1t} \leq q_t M_t + \theta k_t\]
**

A few quick notes: I will assume that labor is supplied inelastically at \(\overline{l}=1\). And I will assume that the production function is cobb-douglass; this matches the given resource constraint. I will write \(r_t\) for the price of renting a unit of capital. Then the return on capital described above is: \(R_t=1+r_t-\delta \).

Given parameters \((\beta, \phi, \alpha, \delta, \theta)\), and initial quantities \((\bar{k_0},\bar{M_0} )\), a competitive equilibrium in this economy consists of the following sets:

- An allocation for the representative household: \( HH\equiv \left\{ c_{1t}, c_{2t}, k_{t+1}, M_{t+1} \right\}_{t=0}^\infty \)
- An allocation for the firm: \(\left\{ k^f_{t}, l^f_{t} \right\}_{t=0}^\infty\)
- Government policy: \(\left\{ M^g_{t}, T_{t} \right\}_{t=0}^\infty\)
- and Prices: \(\left\{ q_t, r_t, w_t \right\}_{t=0}^\infty \)

**HH Optimization**: Taking prices and policy as given, the household allocation solves: \[\max_{HH} \sum_{t=0}^\infty \beta^t (\log c_{1t}+\phi \log c_{2t})\] such that: \[\bar{k_0}=k_0, \; M_0 = \bar{M_0}\] and for all \(t\geq 0:\) \begin{align} c_{1t}\geq 0,\; c_{2t}\geq 0,\; & k_{t+1}\geq 0,\; M_{t+1}\geq 0 &\;\; \condition{Non-Negativity}\\ q_t M_{t+1} + c_{1t} + c_{2t} + k_{t+1} & \leq q_t M_t + (1+r_t-\delta)k_t + w_t + q_t T_t &\;\; \condition{Budget}\\ c_{1t} & \leq q_t M_t + \theta k_t &\;\; \condition{CIA}\\ \end{align}**Firm Optimization**: Taking prices as given, in each period, the firm's allocation solves \[\max_{l_t^f,k_t^f}\; {(k_t^f)}^\alpha {(l_t^f)}^{1-\alpha} - r_t k_t^f - w_t l_t^f\]**Gov Budget Balance**: In all periods \(t\), the government's budget is balanced: \[T_t = M_{t+1}^g - M_t^g\]**Markets Clear**: In all periods \(t\), \begin{align} l_t^f &= \overline{l} = 1 \\ k_t^f &= k_t \\ M_t &= M_t^g\\ c_{1t} + c_{2t} + k_{t+1} &= k_t^\alpha + (1-\delta) k_t &\;\; \condition{RC}\\ \end{align}

Assumming an interior solution with nonbinding NNCs, letting \(\lambda_t\) be the Lagrange Multipliers for the Budget constraints, and letting \(\gamma_t\) be the Lagrange Multipliers for the Cash In Advance constraints, the FOCs are: \begin{align} {\color{name}c_{1t}:} && {\beta^t \over c_{1t}} &= \lambda_t + \gamma_t\\ {\color{name}c_{2t}:} && {\phi\beta^t \over c_{2t}} &= \lambda_t \\ {\color{name}k_{t+1}:} && \lambda_t &= \lambda_{t+1}(1+r_{t+1}-\delta) + \theta\gamma_{t+1} \\ {\color{name}M_{t+1}:} && \lambda_t q_t &= \lambda_{t+1}q_{t+1} + \gamma_{t+1}q_{t+1} \\ {\color{name}k^f_{t}:} && r_t &= \alpha {(k_t^f)}^{\alpha-1}{(l_t^f)}^{1-\alpha}=\alpha {k_t}^{\alpha-1}\\ {\color{name}l^f_{t}:} && w_t &= (1-\alpha) {(k_t^f)}^{\alpha}{(l_t^f)}^{-\alpha}=(1-\alpha) {k_t}^{\alpha}\\ \end{align} Combining the first order conditions for capital, we get: \[ \lambda_t = \lambda_{t+1}(1+\alpha {k_{t+1}}^{\alpha-1}-\delta) + \theta\gamma_{t+1} \] These equations, along with the market clearing, CIA, and budget constraints, characterize the equilibrium.

There are two new assumptions in play now:

This means that: \[M_t=(1+\pi)M_{t-1}=(1+\pi)^t M_0\] \[T_t=M_{t+1}-M_t=(1+\pi-1)M_{t}=\pi(1+\pi)^t M_0\] If we define \(\Pi\equiv 1+\pi\), then this is equivalently: \[M_t=\Pi M_{t-1}=\Pi^t M_0\] \[T_t=(\Pi-1)M_{t}=(\Pi-1)\Pi^t M_0\]

In a steady state, the real allocations are constant across time. So for all \(t\), \begin{align} k_t=k && c_{1t}=c_1 && c_{2t}=c_2 \end{align} These quantities are such that if \(k_t=k ,\; c_{1t-1}=c_1,\; c_{2t-1}=c_2\), then the optimal choice for next-period allocations are \(k_{t+1}=k ,\; c_{1t}=c_1,\; c_{2t}=c_2\), and these allocations are also feasible.

Rewriting the FOCs, CIA, and Resource Constraint with the above two assumptions: \begin{align} {\color{name}c_{1}:} && {\beta^t \over c_{1}} &= \lambda_t + \gamma_t\\ {\color{name}c_{2}:} && {\phi\beta^t \over c_{2}} &= \lambda_t \\ {\color{name}k:} && \lambda_t &= \lambda_{t+1}(1+\alpha {k}^{\alpha-1}-\delta) + \theta\gamma_{t+1} \\ {\color{name}M_{t+1}:} && \lambda_t q_t &= \lambda_{t+1}q_{t+1} + \gamma_{t+1}q_{t+1} \\ \\ {\color{name}CIA:} && c_{1} & \leq q_t M_t + \theta k \\ {\color{name}RC:} &&c_{1} + c_{2} &= k^\alpha - \delta k \\ \end{align}

Note that the CIA constraint can be rewritten as \(q_t M_0 \geq {c_1 - \theta k \over \Pi^t M_0}\).

If \(c_1 < \theta k\), then the right-hand side of the above is negative, and so the CIA constraint is nonbinding. Suppose otherwise. \(M_t \geq 0\) by the non-negativity constraint, so it must therefore be that \(q_t < 0\). But \(q_t\) cannot possibly be less than 0, because then the household would demand infinite amounts of money in period \(t+1\), using it to fund infinite amounts of consumption, and so a well-defined solution to the household's optimization problem couldn't exist. Thus, to avoid contradiction, it must be that \(c_1 < \theta k \implies \gamma_t = 0\)In this case, \(c_{1} = q_t M_t + \theta k\), which implies \[q_t = {1 \over M_t}(c_1 -\theta k) \] \[\color{red} {q_{t} \over q_{t+1}} = {M_{t+1} \over M_{t}} = \Pi \] So the price of money is decreasing at the same rate as the stock of money is increasing, which means the value of the money stock is in a steady state. \(q_t M_t = q_0 M_0\).

The FOC for money can be rewritten as \(\lambda_t {q_{t} \over q_{t+1}} = {\lambda_{t+1}+\gamma_{t+1}} \). Combine with the above to get: \[\color{Rhodamine} \Pi \lambda_t = \lambda_{t+1}+\gamma_{t+1} \] Or similarly, \(\color{Rhodamine}\Pi \lambda_t -\lambda_{t+1} = \gamma_{t+1}\).

Also note that from the FOC for the credit good, \[\color{NavyBlue}{\lambda_t \over \lambda_{t+1}} = {\phi \beta^t \over c_2 } { c_2 \over \phi \beta^{t+1}} = {1 \over \beta} \]

From the FOC for capital, \begin{align} (1+\alpha {k}^{\alpha-1}-\delta) &= {\lambda_{t} \over \lambda_{t+1}} - \theta{{\color{Rhodamine} \gamma_{t+1}} \over \lambda_{t+1}} \\ &= {\lambda_{t} \over \lambda_{t+1}} - \theta{{\color{Rhodamine}\Pi \lambda_t -\lambda_{t+1} \over \lambda_{t+1}}} \\ &= (1-\theta \Pi){\color{NavyBlue} {\lambda_{t} \over \lambda_{t+1}}} +\theta \\ &= (1-\theta \Pi){\color{NavyBlue} {1 \over \beta }} +\theta \\ \end{align} \[\boxed{\implies k = \left({1 \over \alpha}\left[ { {1-\theta \Pi \over \beta }} +\theta -1 + \delta \right]\right)^{1 \over \alpha - 1}}\]

Use the above to relate the FOCs for consumption: \[{\beta^{t+1} \over c_{1}} = {\color{Rhodamine} \lambda_{t+1} + \gamma_{t+1} = \Pi \lambda_t} = \Pi {\phi\beta^t \over c_{2}} \] \[c_2 = {\Pi \phi \over \beta} c_1\] And plug this into the Resource Constraint: \[c_1 + c_2 = {\beta +\Pi \phi \over \beta }c_1 = k^\alpha - k \] \[\boxed{ \begin{align} c_1 &= (k^\alpha - k){\beta \over \beta +\Pi \phi} \\ c_2 &= (k^\alpha - k){\Pi \phi \over \beta +\Pi \phi} \end{align} }\]

And at this point, we can express the equilibrium in terms of exogenous parameters using the above boxed equations, along with: \[\boxed{ \begin{align} k_t^f &= k & \\ l_t^f &= 1 \\ M_t^g &= M_t = \Pi^t M_0 \\ T_t &= (\Pi-1)\Pi^t M_0 \\ w_t &= w \equiv (1-\alpha) k^\alpha \\ r_t &= r \equiv \alpha k^{\alpha-1} \\ q_t &= {c_1 - \theta k \over M_t} = {c_1 - \theta k \over \Pi^t M_0} \end{align} }\]

Then \({\beta^t \over c_1} = \lambda_t = {\beta^t \phi\over c_2} \), and so \({c_2 = \phi c_1}\). Also, as in case 1, \({\lambda_t \over \lambda_{t+1}}={1\over\beta}\).

The FOC for capital implies that \(1+\alpha k^{\alpha-1}-\delta = {\lambda_t \over \lambda_{t+1}} = {1\over\beta}\). Then: \[\boxed{\begin{align} k &= \left({1\over\alpha}\left[ {1\over\beta}-1+\delta \right]\right)^{1\over \alpha - 1} \\ c_1 &= {1\over 1+\phi}(k^\alpha - k) \\ c_2 &= {\phi\over 1+\phi}(k^\alpha - k) \end{align}}\]

As for the price of money, \(q_{t+1} = {\lambda_t \over \lambda_{t+1}}q_t = {1\over\beta}q_t\). Setting \(q_t = 0 \;\forall t\) satisfies this condition. And the other characterizing equations are the same as in Case 1.

So if there is a steady state equilibrium with a constantly growing money supply, and if the CIA doesn't bind, then there is such an equilibrium that is nonmonetary.

Observe that in the monetary steady state equilibrium above: \begin{align} \theta k > c_1 &\iff \theta k > (k^\alpha - k){\beta \over \beta +\Pi \phi} \\ & \iff \beta +\Pi \phi > \beta {k^\alpha - k \over \theta k} \\ & \iff \Pi > {\beta \over \phi \theta}(k^{\alpha-1}-1) - {\beta \over \phi} \end{align} And \(k^{\alpha-1} = {1 \over \alpha}\left[ { {1-\theta \Pi \over \beta }} +\theta -1 + \delta \right]\), so \begin{align} \theta k > c_1 &\iff \Pi > {\beta \over \phi \theta}({1 \over \alpha}\left[ { {1-\theta \Pi \over \beta }} +\theta -1 + \delta \right]-1) - {\beta \over \phi} \\ &\iff \Pi + {\Pi \over \alpha\theta} > {\beta \over \phi \theta}({1 \over \alpha}\left[ { {1 \over \beta }} +\theta -1 + \delta \right]-1) - {\beta \over \phi} \\ &\iff \Pi > {\alpha\theta \over \alpha\theta +1}\left[{\beta \over \phi \theta}({1 \over \alpha}\left[ { {1 \over \beta }} +\theta -1 + \delta \right]-1) - {\beta \over \phi}\right] \equiv \bar{\Pi} \\ \end{align} Therefore if \(\pi > \bar{\Pi} - 1 \), then \(c_1 - \theta k < 0\).

Recall from above that if \(c_1 - \theta k < 0\), then the CIA is nonbinding, and so it cannot be that \(c_1 - \theta k < 0\) in a Case 1 equilibrium. Thus if \(\pi>\bar{\Pi} - 1\), the optimum must be attained in a Case 2 equilibrium, in which case there is a nonmonetary steady state in which the price of money is zero.

In a monetary steady state, the level of capital is given by: \[k = \left({1 \over \alpha}\left[ { {1-\theta \Pi \over \beta }} +\theta -1 + \delta \right]\right)^{1 \over \alpha - 1}\] This is a decreasing function of \(\Pi\). And capital output ratio is \[{k \over k^\alpha} = k^{1-\alpha}= \left({1 \over \alpha}\left[ { {1-\theta \Pi \over \beta }} +\theta -1 + \delta \right]\right)^{-1}\] This is a decreasing function of \Pi, as that term will now be a negative in the denominator.

The net nominal interest rate should be set to zero, to alleviate the frictions induced by inflation. Equivalently, the gross nominal interest rate should be 1.

In this economy, it is hepful to note that the nominal interest rate is equal to the real interest rate times the change in price levels: \(R_t \Pi\)

The social optimization problem for this economy is:

\[\max_{} \sum_{t=0}^\infty \beta^t (\log c_{1t}+\phi \log c_{2t})\] such that: \[\bar{k_0}=k_0\] and for all \(t\geq 0:\) \begin{align} c_{1t}\geq 0,\; c_{2t}&\geq 0,\; k_{t+1}\geq 0,\; &\;\; \condition{Non-Negativity}\\ c_{1t} + c_{2t} + k_{t+1} &= k_t^\alpha + (1-\delta) k_t &\;\; \condition{RC}\\ \end{align}

The First order conditions for the above problem imply that \( c_{2t}=\phi c_{1t}\). Compare this to the Euler condition from the steady-state monetary competitive equilibrium above, where \(c_{2t}={\phi \Pi\over \beta}c_{1t}\). For the competitive equilibrium to attain a social optimum, it must be then that \(\Pi = \beta\).

And also from the competive equilibrium, \begin{align} R_t &= 1-\delta + r_t\\ &= 1-\delta + \alpha k^{\alpha-1} \\ &= (1-\theta \Pi){ {1 \over \beta }} +\theta\\ & = (1-\theta \beta){ {1 \over \beta }} +\theta \\ & = {1 - \theta \beta + \theta\beta \over \beta} = {1\over \beta} \end{align}

So in this economy, the optimal policy will set \(R\Pi={\beta\over\beta}=1\), and the Friedman Rule holds.