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C = hY, with 0 < h < 1. The parameter b is called the marginal propensity

to consume, while s = 1 h is called the marginal propensity to save. On

the firmsвЂ™ side, investment I is a decreasing function of the interest rate r. In its

simplest linear form, we write this relationship as

The paremeter a is called the marginal efficiency of capital.

Putting these relations together gives the IS schedule, the relationship between

national income and interest rates consistent with savings and investment behavior

Y = bY + (IвЂќ ar) t G,

which wt: write as

.sY t ar = I" + G, (l(J)

where .s = 1 -b, a, IвЂќ, and G are positive parameters. This IS equation is sometimes

said to describe the real side of the economy. since it summarizes consumption,

investment. and savings decisions.

On the other hand, the LM equation is determined by the money market

equilibrium condition that money supply JV& equals money dcmand M,,. The money

supply M,, is determined outside the system. Money demand M,J is assumed to

have two components: the transactions or precautionary demand Md, and the

speculative demand M,,,?. The transactions demand derives from the fact that most

transactions are denominated in money. Thus, as national income rises, so does

the demand for funds. We write this relationship as

M,,, = mY.

The speculative demand comes from the portfolio management problem faced

by an investor in the economy. The investor must decide whether to hold bonds or

money. Money is more liquid but returns no interest, while bonds pzy at rete r. It

is usually argued that the speculative demand for money varies inversely with the

intcrcst rate (directly with the price of bonds). The simplest such relationship is

the linear one

L6.21 EXAMPLES OF LINEAR MODELS 117

The LM curve is the relationship between national income and interest rates

given by the condition that money supply equals total money demand:

M.\ = mY + M вЂќ h r ,

mY h r = M,, -MU.

The parameters m, h, and MвЂќ are all positive.

Equilibrium in this simple model will occurwhcn both the IS equation (produc-

tion equilibrium) and the LM equation (monetary equilibrium) are simultaneously

satisfied. Equilibrium national income Y and interest rates I are solutions to the

system of equations

(11)

Algebraic questions come into play in examining how solutions (IвЂ™, I) depend

M, and G and on the behavioral parameters a, h, IвЂ™,

upon the policy parameters

˜1, MвЂќ, and s. The comparative statics of the model-the determination of the

relationship bctwccn parameters and solutions of the equations-is an algebraic

prohlcm on which the tools of linear algebra shed much light.

The importance of studying the linear version of the IS-LM model in addition

to the general nonlinear version of the model cannot bc undcrcstimated. First_ the

intuition of the model is most easily seen in its lincar form. Second. study of the

linear model can suggest what to look for in mue general models. Finally, the

compararivc statics of rmnlincar models- the exploration of how solutions to the

system change as the parameters describing the system change ˜ is uncovcrcd by

approximating the nonlinear model with a linear one, and then studying the lincu

approximation. вЂ˜These three reasons for focusing on lincar models for the study

of tlonlincar phcnomcna will recur frequently. The simplicity of linear models

commends such models as a first step in the construction of more complcn models.

and the more complex models xc frcqucntly studied hy examining carefully

chosen linear approximations.

Example 5: Investment and Arbitrage

118 INTKODвЂќCTIвЂќN LINEAR ALGERRA [61

TO

end of the investment period, including dividends paid, if state s ocwn. Then, the

realized return or payoff on the ith asset in state s is

This is the amount the investor will rcccive per dollar invcstcd in asset i should

of rcwr˜˜.)

slate .s ,,ccur. (The realized return can be thought of as I plus the I-u(lfe

shares of asset i held. The share amounts

Let ni denote the number of units or

ni indicates a long position and thus entitles the

II; can have either sign. A positive

occur\. A negative ni indicates a short position;

investor to r˜˜ceive yvrni if state 5

hur:k ˜,˜n,

the investor, in cftect, borrows 11; shares of asset i and promises to,w.v

at the end of the period if slate s occurs. In this case, the investment in asset i has

a positive rate of return only if yYi < vi, that is, if it is cheaper 10 pay hack the

borrowed shares that it was to borrow them.

If the investor has wealth w,, available for investment purposes, the investorвЂ™s

budget constraint is

11, v, + + a_, y, = WвЂ™,,.

If state .Y occurs, the return to the investor of purchasing II, shares of asset i for

i = I....,A 1s

(12)

A nonzeroil-tuple (x1,. , na) is called an arbitrage portfolio if

xl+вЂњвЂ˜+xA=o (instead of 1)

In such a вЂњportfolio,вЂќ the money received from the short sales is used in the purchase

of the long positions. Notice that, in an arbitrage portfolio, nlvl + + nay,, = 0,

so that the portfolio costs nothing.

A portfolio (x1,. _, XA) is called duplicable if there is a different portfolio

(w,, , wa) with exactly the same returns in every state:

A state s* is called insurable if there is a portfolio (xl,. _, xa) which has a

positive return if states* occurs and zero return if any other state occurs:

,@si˜ = 0 for all s # s*,

The name is appropriate because the given portfolio can provide insurance against

the occurrence of state s*.

It is sometimes convenient to assign a price to each of the s states of nature.

An S-tuple (JQ, , p,˜) is called a state price vector if

PlYll + PZY21 + вЂќ + PSYSl = vi

PIYIZ + P?YZZ + + PSYSZ = v2

(13)

PlYlA + PZY24 + вЂќ+ psysl = вЂњA,

or equivalently,

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