311 HWK #2 SOLUTIONS



(1)
The model economy is described by
Z=C+G+Id, where Id is desired investment.
C=c0+c1*(Y-T)
c0=160, c1=.6, T=100, G=150, Id=150

(a) 
First, we calculate the equilibrium effect of an increase in G from 150 to 160.
>From the equilibrium condition Y=Z, we have
Y=1/[1-c1]*[c0-c1*T+Id+G].
Plugging in the numbers for the intial equilibrium (denoted with *) we have
Y*=1000
To see what happens to EQUILIBRIUM income(=production) after the increase og G by 10 units, we can plug in again. A quicker way is to use the Keynesian Multiplier on Government Spending: 1/[1-c1].
Note that Y**=Y*+1/[1-c1]*DG, where DG denotes the change in G. In our example,
Y**=1000+1/.6*10=1025. It's snowing like crazy. Awesome!

Recall that once you've calculated equilibrium income, you can back out equilibrium consumption by pluugging income into the consumption function C=c0+c1*(Y-T). Also note that you can back out private saving by using the accounting identity 
Y-T=C+S.
Also note that actual investment is equal to the sum of desired and undesired investment. We call the latter 'Inventory Investment' (II):
Ia=Id+II,
where II=Y-Z, i.e. production-'sales'.
In equilibrium, II=0, which is equivalent to saying 'actual and desired investment are the same' or 'sales (Z) are equal to production'.

(b) 
We assume that the economy is in the initial equilibrium in period 0 (with G=150). The following table reports variables of interest for period 0. For the calculation of subsequent priods (after G has permanently increased to 160 in the morning of period 1), we use the relation
Y(t)=Z(t-1), i.e. firms base their production decision on 'sales' Z they observed for the PREVIOUS period. SNOW! Once we have production (=income) for a given row, we can calculate consumption and use G and Id to get Z for that row. Then calculate II and the rest. Note that we assume that 'there's always enough stuff in firms' inventory stock to satiusfy desired spending'.

	Y		Z		II		C	Id	G	Ia	S	T-G
	production	sales		inventory 	C=c0+c1(Y-T)		=Id+II	S=Y-T-C	
	=income		Z=C+G+Id	investment
					II=Y-Z			
0	1000		1000		0		700	150	150	150	200	-50
1	1000		1010		-10		700	150	160	140	200	-60
2	1010		1016		-6		706	150	160	144	204	-60
3	1016		1019.6		-3.6		709.6	150	160	146.4	206.4	-60



(2)
The 'goods market side' of the economy is described by
Z=C+G+Id, where C=c0+c1*(Y-T), G is a fixed number, and Id=Ibar-b*i.
Using the equilibrium condition for the goods market, Y=Z, we can btain an equation yielding combinations of our endogenous variables (Y,i) that constitute an equilibrium in the goods market. This is the equation for our 'IS-curve'.
Y=Z
Y=C+Id+G
Y=c0+c1*(Y-T)+G+Ibar-b*i.
Solving for Y yields:

Y=1/[1-c1]*[c0-c1*T+G+Ibar-b*i]		(IS-Curve)

The 'financial market side' is described by a fixed real money supply:
Ms/P=MM (MM is some number)
and an equation expressing the demand for real money balances in terms of Y and i:
(M/P)d=a0*Y-a1*i (the d stands for 'demand').
The equilibrium condition for the financial markets is
Ms/P=(M/P)d
or
MM=a0*Y-a1*i.
Let's solve this for i, yielding the equation for the LM-curve, which plots all combinations of (Y,i) that consitute equilibria in the financial markets:

i=[a0*Y-MM]/a1				(LM-curve)

Plugging (LM-curve) into (IS-curve) for the interest rate i yields the level of income Y that isconsistent with equilibrium in both the goods AND the financial markets. That's the Y corresponding to the intersection of the IS and LM-curves:

Y=1/[1-c1]*[c0-c1*T+G+Ibar-b/a1*[a0*Y-MM]]

We can solve this for Y:

Y*=1/[1-c1+b*a0/a1]*[c0-c1*T+G+Ibar+b/a1*MM].

We could back out the equilibrium interest rate by plugging Y* into either IS or LM and solving for i*.
Think about the term 1/[1-c1+b*a0/a1]. Is it larger or smaller than the Keynesian Cross Multiplier? Why? What's going on when e.g. is increasing?
Think about a change in MM. What happens in the IS-LM diagramm? Can you verify this in the equations?

(3)

[Number 4, p. 84 of text.]
Md=$Y*(.25-i)
$Y=100; Ms=20

(a) Ms=Md => 100*(.25-i)=20 =>i=.05. (if you're worrying about a missing P, don't. We'll get there later. You can set P=1 for now.)
(b)The FED wants to achieve an equilibrium interest rate of .05+.1=.15. To find the level of the money supply Ms** that would yield an equilibrium interest rate of .15 use:
100*(.25-.15)=Ms** => Ms**=10.

[Number 2, p.84 of text]
 we're given Wealth W=50,000, income Y=60,000, and a relationship for the demand for money: Md=$Y*(.35-i)
NOTE: recall that Wealth is a stock. The household 'decides' whether to 'hold her wealth' in the form of money or interest bearing bonds. Hence
W=Bd+Md, where Bd is 'bond demand'.
(a) 
 i	Md			Bd
	=60.000*(.35-i)		=W-Md
				=50.000-Md

.05	18.000			32.000
.1	12.500			47.500

(b) i up => bonds become more profitable to hold =. Md down and Bd up.
(c) 
i=.1
$Y decreases to a vlaue of 30.000

$Y	Md
	=$Y*(.35-.1)

60.000	12.500
30.000	6.250

Hence, Md is reduced by %50 as well (for a given interest rate).

(d) Given the functional form for Md, money demand is proportional to $Y. The [% change] of money demand is always equal to the [% change] of Y for ANY given interest rate.