Lecture 5 The Micro-foundations of the Demand for Money - Part 2

State the general conditions for an interior solution for a risk averse utility maximising agent Show that the quadratic utility function does not meet all these conditions Examine the demand for money based on transactions costs Examine the precautionary demand for money Examine buffer stock model of money

The Tobin model of the demand for money Based on the first two moments of the distribution of returns Generally a consistent preference ordering of a set of uncertain outcomes that depend on the first n moments of the distribution of returns is established only if the utility function is a polynomial of degree n. Restricting the analysis to 2 moments has weak implication of quadratic utility function

Arrow conditions Positive marginal utility Diminishing marginal utility of income Diminishing absolute risk aversion Increasing relative risk aversion

Arrow conditions

Quadratic Utility Function U R U(R) Max U

Alternative specifications Set b > 0 - but this is the case of a risk lover A cubic utility function implies that skewness enters the decision process - not easy to interpret. But the problems with the quadratic utility function are more general

A Paradoxical Result

Equation of a circle R R -a/2b 45 o

The Opportunity Set

R R 0 P P = 1 A B C

Implications Slope of opportunity set is greater than unity wealth effect will dominate substitution effect for substitution effect to dominate r < g bond rate will have to be lower the volatility of capital gains/losses

Transactions approach Baumol argued that monetary economics can learn from inventory theory Cash should be seen as an inventory Let income be received as an interest earning asset per period of time. Expenditure is continuous over the period so that by the end of the period all income is exhausted

Assumptions Let Y = income received per period of time as an interest earning asset Let r = the interest yield Expenditure per period is T Suppose agent makes 2 withdrawals within the period - one at beginning and one before the end.

More ? Suppose 0 < < 1 is withdrawn at the beginning of the period Interest income foregone = (average cash balance during the fraction of the period) x (the interest rate for the fraction of the period ) ( Y/2)( r) = ½ 2 rY

More Later (1- )Y is withdrawn to meet expenditure in the remainder of the period (1- ) time Thus agent gives up ½(1- ) 2 rY Let total interest foregone = F F =½ 2 rY + ½(1- ) 2 rY What value of minimises F?

Minimisation

Both withdrawals must be of equal size Y t Y/2 t=½

Optimal withdrawal Calculate optimal size of each withdrawal Gives optimal number of withdrawals The average cash held over the period is M/2 Interest income foregone is r(M/2) assume that each withdrawal incurs a transactions cost b

Optimal money holding

Elasticities

Miller & Orr 2 assets available- zero yielding money and interest bearing bonds with yield r per day Transfer involves fixed cost g - independent of size of transfer. Cash balances have a lower limit or cannot go below zero Cash flows are stochastic and behave as if generated by a random walk

Miller & Orr continued In any short period t, cash balances will rise by (m) with probability p or fall by (m) with probability q=(1-p) cash flows are a series of independent Bernoulli trials Over an interval of n days, the distribution of changes in cash balances will be binomial

Properties The distribution will have mean and variance given by: n = ntm(p-q) n 2 = 4ntpqm 2 The problem for the firm is to minimise the cost of cash between two bounds.

Buffer stocks and Disequilibrium Money

In period T at the Terminal date M T+1 = M T

Generalising for an error- correction mechanism

Disequilibrium Money causes adjustments in all markets

Conclusion Post Keynesian development in the demand for money have micro-foundations but they are not solid micro- foundations. The Miller-Orr model of buffer stocks money demand allows for disequilibrium and threshold adjustment. The macroeconomic implication is the disequilibrium money model. The disequilibrium money model builds on the real balance effect of Patinkin and has long lag adjustment of monetary shocks Equilibrium models have rapid adjustment of monetary shocks (rational expectations).