A Basic Beer in Utility (BIU) Model
This is what happens when your monetary economics study group starts homebrewing while working through problem sets.
Notes inspired by “Money in an RBC framework” by Carl Walsh, January 2008. And Sidrauski, Miguel. 1967. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review 57 (2): 534–544.
A Basic Beer in Utility (BIU) Model
First Approach: BIU Sidrauski (1967)
Real Beer Volumes enter directly in the utility function.
Given suitable restrictions on the utility function, such an approach can guarantee that, in equilibrium, agents choose to hold positive amounts of beer so that beer is positively valued.
Representative household takes beer-to-work/consumption exchange rates as given, and maximizes
\[ E_t \sum_{i=0}^{\infty} \beta^t \, U(C_{t+i}, BEER_{t+i}, L_{t+i}) \]
subject to the budget constraint.
Budget Constraint, in nominal terms:
\[ \begin{aligned} W_t(1 - L_t) + (r_t + 1 - \delta) P_t K_{t-1} &+ (1 + i_{t-1}) B_{t-1} + BEER_{t-1} + P_t \tau_t \\ &= P_t C_t + P_t K_t + B_t + BEER_t \end{aligned} \]
where \(\tau\) are real, lump sum transfers to the household (which can be negative).
In real terms,
\[ \begin{aligned} w_t(1 - L_t) + (r_t + 1 - \delta) K_{t-1} &+ (1 + i_{t-1}) \frac{B_{t-1}}{P_t} + \frac{BEER_{t-1}}{P_t} + \tau_t \\ &= C_t + K_t + \frac{B_t}{P_t} + \frac{BEER_t}{P_t} \end{aligned} \]
or,
\[ \begin{aligned} w_t(1 - L_t) + (r_t + 1 - \delta) K_{t-1} &+ (1 + i_{t-1}) \frac{P_{t-1}}{P_t} b_{t-1} + \frac{P_{t-1}}{P_t} beer_{t-1} + \tau_t \\ &= C_t + K_t + b_t + beer_t \end{aligned} \]
First Order Conditions
\[ \begin{aligned} C: &\quad U_C - \lambda_t = 0 \\[4pt] L: &\quad U_L - w_t \lambda_t = 0 \quad \implies \quad \frac{U_L}{U_C} = w_t \\[4pt] K: &\quad -\lambda_t + \beta E_t (r_{t+1} + 1 - \delta) \lambda_{t+1} = 0 \\[4pt] b: &\quad -\lambda_t + \beta (1 + i_t) E_t \left( \frac{P_t}{P_{t+1}} \right) \lambda_{t+1} = 0 \\[4pt] beer: &\quad U_{beer} - \lambda_t + \beta E_t \left( \frac{P_t}{P_{t+1}} \right) \lambda_{t+1} = 0 \end{aligned} \]
First three FOCs look just like what we’re familiar with from a basic RBC Model (except that \(U_C\) & \(U_L\) may depend on beer).
The FOC for beer can be written
\[ \frac{\lambda_t}{P_t} = \frac{U_{beer}}{P_t} + \beta E_t \left( \frac{\lambda_{t+1}}{P_{t+1}} \right) \]
Solve forward to yield
\[ \frac{1}{P_t} = \left( \frac{1}{\lambda_t} \right) E_t \sum_{i=0}^{\infty} \beta^i \left[ \frac{U_{beer}(C_{t+i}, beer_{t+i}, L_{t+i})}{P_{t+i}} \right] \]
which reveals the value of beer as the expected present value of discounted utility return.
Combining the first and third conditions from the FOCs yields the Euler Condition:
\[ U_C(C_t, beer_t, L_t) = \beta E_t (r_{t+1} + 1 - \delta) \, U_C(C_{t+1}, beer_{t+1}, L_{t+1}) \]
Similarly with the first and fourth conditions:
\[ U_C(C_t, beer_t, L_t) = \beta (1 + i_t) E_t \left( \frac{P_t}{P_{t+1}} \right) U_C(C_{t+1}, beer_{t+1}, L_{t+1}) \]
which, of course, implies
\[ 1 = \beta (1 + i_t) E_t \left( \frac{P_t}{P_{t+1}} \right) \frac{U_C(C_{t+1}, beer_{t+1}, L_{t+1})}{U_C(C_t, beer_t, L_t)} \]
The fourth and fifth FOCs may be combined to deliver:
\[ \frac{U_{beer}}{U_C} = \beta i_t E_t \left( \frac{P_t}{P_{t+1}} \right) \frac{U_C(C_{t+1}, beer_{t+1}, L_{t+1})}{U_C(C_t, beer_t, L_t)} \]
Dividing the condition just above into the one above implies,
\[ \frac{U_{beer}}{U_C} = \frac{\beta i_t E_t \left( \dfrac{P_t}{P_{t+1}} \right) \dfrac{U_C(C_{t+1}, beer_{t+1}, L_{t+1})}{U_C(C_t, beer_t, L_t)}}{\beta (1 + i_t) E_t \left( \dfrac{P_t}{P_{t+1}} \right) \dfrac{U_C(C_{t+1}, beer_{t+1}, L_{t+1})}{U_C(C_t, beer_t, L_t)}} = \frac{i_t}{1 + i_t} \]
thus we see that the firm side of this model is the same as with the basic RBC model.
Equilibrium
Collecting equilibrium conditions.
Given an exogenous process for the productivity shock and the nominal stock of beer, equilibrium consists of time paths for output, real beer volumes, consumption, capital, employment, wages, the nominal rate of brewing, the beer level, and rental rates that satisfy
\[ \frac{U_L(C_t, beer_t, 1 - N_t)}{U_C(C_t, beer_t, 1 - N_t)} = w_t \]
\[ \frac{U_{beer}}{U_C} = \frac{i_t}{1 + i_t} \]
\[ U_C(C_t, beer_t, L_t) = \beta E_t (r_{t+1} + 1 - \delta) \, U_C(C_{t+1}, beer_{t+1}, L_{t+1}) \]
\[ U_C(C_t, beer_t, L_t) = \beta (1 + i_t) E_t \left( \frac{P_t}{P_{t+1}} \right) U_C(C_{t+1}, beer_{t+1}, L_{t+1}) \]
\[ Y_t = F(N_t, K_{t-1}, Z_t) \]
\[ F_N(N_t, K_{t-1}, Z_t) = w_t \]
\[ F_K(N_t, K_{t-1}, Z_t) = r_t \]
\[ Y_t = C_t + K_t - (1 - \delta) K_{t-1} \]
\[ P_t \, beer_t = BEER_t \]
Steady State
The Standard RBC relationships, plus
\[ 1 = \beta \left( \frac{1 + i_t}{1 + \pi} \right) \quad \implies \quad 1 + i_t = \frac{1 + \pi}{\beta} \quad \& \quad i_t = \frac{1 + \pi - \beta}{\beta} \]
\[ \frac{U_{beer}(C, beer, 1 - N)}{U_C(C, beer, 1 - N)} = \frac{i_t}{1 + i_t} = 1 - \frac{\beta}{1 + \pi} = \frac{1 + \pi - \beta}{1 + \pi} \]
\[ P_t = \frac{BEER_0 (1 + \pi)^t}{beer} \]
These imply the Fisher equation \(1 + i = (r + 1 - \delta)(1 + \pi)\), or
\[ i \approx r + \pi \]
Superneutrality
If \(U_L\) & \(U_C\) are independent of beer in the utility function (that is, if one’s marginal utility from consumption or marginal disutility of labor is not dependent on how much beer one has), then the real side is identical to the basic RBC model. - Neutrality: One time changes in the nominal quantity of beer affect only the beer level. - Superneutrality: Changes in the growth rate of beer do not affect real variables.
Changes in the rate of growth of beer affect only the flattening rate, the nominal brewery rate, and real beer balances. If \(F\) displays constant returns to scale and \(U_L\) & \(U_C\) depend on beer, we still get steady-state capital ratio independence of beer growth.
Welfare Cost of Flattening
Because beer volumes yield direct utility and higher rates of beer-flattening reduce real beer volumes, flattening generates a welfare loss.
Important Questions:
- How large is the welfare cost of flattening?
- Is there an optimal rate of flattening that maximizes the steady-state welfare of the representative household?
The optimal rate of beer-flattening was addressed by Bailey (1956) and M. Friedman (1969).
- Basic intuition: the private opportunity cost of holding beer depends on the nominal brewery rate (\(i_t\)). The social marginal cost of producing more beer is essentially zero (just small amounts of hops, barley, & water).
- The wedge that arises between private marginal cost \((1 + i_t)\) & the social marginal cost (nada) when the nominal rate is positive creates an inefficiency.
- That inefficiency would go away if the nominal brewery rate (\(i_t\)) were equal to zero.
- So the optimal rate of flattening is a rate of… deflation (I guess we have to call it that), approximately equal to the real return on capital.
In the steady state, real beer balances are directly related to the flattening rate, so the optimal rate of flattening is also frequently discussed under the heading of the optimal quantity of beer.
The major criticism of this result comes from Phelps (1973), who pointed out that beer growth generates revenue for the government, the beer-flattening tax.
The implicit assumption so far has been that variations in beer growth are engineered via lump-sum transfers (\(\tau\) from the budget constraint). Any effects on government revenue can be offset by a suitable adjustment in these lump-sum transfers.
But if governments only have distortionary taxes available for financing expenditures, then reducing beer-flattening tax revenues to achieve the Friedman rule for optimal inflation requires that the lost revenue be replaced through increases in other distortionary taxes.
To minimize the total distortions associated with raising a given amount of revenue, it may be optimal to rely to some degree on a flattening tax.
More recent work has reexamined these results (see Chari, Christiano, & Kehoe 1991, 1996; Correia & Teles 1996, 1999; Mulligan & Sala-i-Martin 1997).