Easy Money

This is just a bad version of this post by Miles Kimball and this post by Nick Rowe. I mostly follow Kimball, but I use slightly different terminology. I'm more careful about distinguishing between firms and industries. There's nothing more complicated in this post than some homework in a micro class. Really, this is just a simple New Keynesian model. The model as it sits isn't quite coherent, but I had fun looking through it. Why not go watch some cartoons?

There are four levels of analysis:

1. Consumer Demand
2. Firm Price Setting
3. Industry Size
4. Macro Policy

Every level of analysis requires all the others. Maybe this post needs to be read twice. I'm going to work in equilibrium, on the assumption that the complicated cybernetic process of searching for a stable social situation is already done. I'm also doing everything without risk or uncertainty, for no good reason. We can think of this as a (bad) sociological analysis of four kinds of people (or one person in four aspects) - a consumer, capitalists, entrepreneurs and a central banker. In a (Nash) equilibrium, no consumer, capitalist, entrepreneur or central banker wants to change their parameters given the choices of all the others. In a way, this is a cartoon!

Some simple notation to start with. Each industry is denoted by \( i \). Each firm is denoted by \( f \). I assume each firm \( f \) only produces one good. The product of two firms are in the same industry \( i \) if they are perfect substitutes at all price and output levels. I won't use subscripts except to denote firm and industry. The quantity demanded of industry \( i \) is \( Q_i \). The price of the good from industry \( i \) is \( p_i\). The quantity of the \(i\)th good produced by the \(f\)th firm is \({}_f q_i\). The number of firms in industry \( i \) is \( n_i \).

CONSUMER DEMAND

We start with consumer demand, but don't forget that consumer demand isn't "first". This is equilibrium analysis where everything determines everything else. Consumer demand is handled via a representative agent - that is, a single real valued utility function standing in for all purchases in a society. The agent can choose among a number of good or hold money.

\[ \max_{\vec{Q},M} U(\vec{Q},M) \]

The name "representative agent" was chosen by Alfred Marshall to emphasize that she isn't an "average" or "marginal" agent. So I will assume a Marshallian quasi-linear utility function. I assume that

\[ U(\vec{Q},M) = M + \hat{U}(\vec{Q}) \]

Where \( \hat{U}(\vec{Q})\) is strictly quasiconcave and homogeneous of degree one. In short: the representative agent has homothetic preferences in goods and linear preferences in money. Finally, the own price elasticity of each good \( i \) \(\eta_i\) is assumed to be strictly greater than 1. The representative agent subject to the restriction that consumption and savings together make up the whole of income. That is:

\[ \vec{p} \cdot \vec{Q} + M = Y\]

The representative agent takes prices as given by firms and income as given by the macroeconomic situation.

FIRM LEVEL

Since each firm \( f \) only makes a single good \( i \), it is easy to write it's revenue function:

\[ \max_{p_i,{}_f q_i} p_i {}_f q_i - C_f({}_f q_i) \]

I write this as a two part maximization problem to emphasize that out of equilibrium the firm is exploring both prices and quantities. It will turn out that once the right price is found, the quantity is forced. The price in industry \( i \) is independent of the firm by the law of one price. You can also think of this as being a representative firm analysis if you want. I assume that \( C_f \) is increasing and concave up everywhere for all firms. In perfect competition, we would have that price equals marginal cost \( p_i = C'_f ({}_f q_i)\). But that's not realistic. But in monopolistic competition, we have only prices are only proportional to marginal cost

\[ p_i = \mu_i C'({}_f q_i) \]

For some markup \( \mu_i > 1\). where

\[ \mu_i = \frac{\eta_i}{\eta_i-1} \]

The firm has no control over this, it is determined by the consumer. Because \( \frac{p_i }{ \mu_i } \) doesn't depend on firm \( f \) neither does \( C'({}_f q_i) \). This is ensured by the fact that \( C_f \) is concave up - this gives that \( C'_f \) is monotonic and therefore \( C'^{-1}_f \) exists. Since we're in equilibrium, each firm can take all the other firm's quantities as given. Since we also have

\[ \Sigma_f C'^{-1}_f(\frac{p_i}{\mu_i}) = Q_i \]

once \( p_i \) is chosen, so is \( {}_f q_i \). That means that the second term on the maximization problem is purely decorative.

INDUSTRY LEVEL

A firm \(f-1\) can explore setting different prices or producing different quantities. But another action is possible - an entrepreneur can attempt to make a new firm \(f\). The new entrepreneur is enticed by the possibilities of increasing returns, so her \(\gamma_f = C''_f  > 0 \) . Long run monopolistic equilibrium gives us that

\[ \gamma_f = \mu_i \]

at the equilibrium level of production with the equilibrium number of firms. What this means is that the \(f\)th firm will only get in the industry if it has a high enough markup. Once again we have something that seems like it depends on the details of a firm and something determined by the level of industry in equilbrium:

\[ \mu_i = C''_f({}_f q_i) \]

Kimball assumes that \( \mu_i \) is a monotonically decreasing function of \( n_i \). The thought experiment goes like this - the \( n \)th firm in industry \( i \) will have a harder and harder time finding a production with higher returns to scale and therefore a higher markup. This means that \( \mu^{-1}_i \) exists and the equilibrium number of firms is:

\[ n_i = \mu^{-1}_i(\gamma_f) \]

MACRO POLICY

Everything in the above was micro, but we have enough assumptions to incorporate macro as well. Recall that we had as a constraint

\[ C + S = Y \]

where \( C = \vec{p} \cdot \vec{Q} \) and \( S = M \). Prices were chosen by firms, quantities by consumers. Liquidity preference gives us

\[ M = L(r,Y) \]

For no reason, I choose a Tobin-Baumol Square Root demand for money. Each trip to the bank has a transaction cost \( T \). Holding cash has an opportunity cost in holding bonds*, which pay a real interest rate of \( r \). Therefore, the transaction demand for money is

\[M = \sqrt{\frac{T Y}{2 i}}\]

We can plug in our \( C + S = Y \) condition for a quadratic polynomial in \( M \) with one positive root by Descartes' Rule Of Signs. The quantity of money held by the representative agent would be

\[ M = \frac{T}{4r} + \sqrt{\frac{T^2}{16r^2}+\frac{TC}{2r}} \]

One can think of this as the natural policy for a central bank. It has to set an interest rate consistent with a given level of consumption \( C \) and output \( Y \), and \( M \) is determined by those three inputs.

If I set \(T = 4\) I can easily solve for the effect of changing the interest rate on the level of consumption by the chain rule.

\[ \frac{\partial C}{\partial r} = -(\frac{1}{r}+C+\sqrt{1+2 i C}) \]

Notice that the natural policy isn't neutral - for all positive \( r \) and \( C \), \( \frac{\partial C}{\partial r} < 0 \).

The reason for this easy analysis isn't that I chose an easy liquidity preference function, it's because I chose a quasi-linear utility function. This caused the micro analysis to separate from the macro analysis.

CONCLUSION

Anyway, it is a fun and cute system. What I'd like do do is see how changing the interest rate changes the equilibrium number of firms. What I need to do is unpack Kimball's condition for \( n_i \).

No moral.

*There's no bond market in my system. This is the incoherent bit.