Question

CALCULUS- compound interest and elasticity question?

Answer 1

Suppose the Canadian real estate market can be described by the following model. The price of the average detached single family house P = Pe + Pd consists of two components: the equity component Pe (this is the cash down-payment when you buy a house) and the debt component Pd (this is the amount financed by a loan). We assume that the equity component is constant Pe = $200000 whereas the debt component follows an exchange equation
MV = PdQ
at every time. Here, Q is the number of existing houses, M denotes the total amount of mortgages held on the books of the banks, and V denotes the velocity of credit money in the real estate market. We assume that V = 1/3 is constant. On January 1, 2012, Q is 5 million units, growing at a rate of 50000 per year, and Pd is $200000.

(a) If the banks are reducing their lending standards such that M increases at a rate of $300 billion per year, at what rate (in percent) does the average house price P increase?

(b) If, however, there is no new mortgage lending and the current borrowers only pay back their existing loans, M decreases at a rate of $30 billion per year. Calculate the relative rate of change of the average house price P in this situation.

Answer 2

(a) \[MV=PdQ\]\[\frac{1}{3}M=Pd*Q\]\[Pd=\frac{M}{3*Q}\]\[P = $200,000+Pd\]\[\frac{dM}{dt}=$300 \ B/yr\]\[\frac{dQ}{dt}=50,000\ units/yr\]\[\frac{dP}{dt}=0+\frac{dPd}{dt}=\frac{d\frac{M}{3*Q}}{dt}\]Quotient Rule...

Answer 3

ok....

Answer 4

Do you remember how to express the derivative of a quotient?

Answer 5

you mean the formula?

Answer 6

Yes

Answer 7

g'(x)h(x)-g(x)h'(x)/(h(x))^2

Answer 8

so h(x) = Q(t) = 5 million and g(x) = M(t) = 3*$200,000*5 Million
Q'(t) = 50,000 units/year and M'(t) = $300 Billion/year

Answer 9

\[\frac{d\frac{M}{3Q}}{dt}=\frac{1}{3}\frac{d\frac{M}{Q}}{dt}=\frac{1}{3}\frac{Q(t)M'(t)-M(t)Q'(t)}{Q(t)^2}\]

Answer 10

ok...that seems a little complicated. so i plug q'(t) and m'(t) and then?

Answer 11

q(t), q'(t), m(t) and m'(t)

Answer 12

wait scratch that.

Answer 13

so sloving this will give me the final answer for (a)

Answer 14

and then convert it into percentage?

Answer 15

It gives you the change in price per year. To solve for percentage, you need to divide the price change by the price, $400,000.

Answer 16

then for part (b) you will replace M'(t) = $300 B/yr to M'(t) = -$30 B/yr

Answer 17

so [(1/3)(quotient rule answer)] /400000 =answer?

Answer 18

Yes, let me know what you've got.

Answer 19

give a couple of minutes. never dealt in millions and billions

Answer 20

Yeah, scientific notation is useful here.

Answer 21

18000 @Valpey

Answer 22

1800/400000 = 0.045

Answer 23

what do you think?

Answer 24

Yes

Answer 25

4.5% increase in Price.

Answer 26

GREAT!! so can you help with part cb?

Answer 27

*b

Answer 28

Like I said, just replace M'(t) = $300 B/yr with M'(t) = -$30 B/yr

(a contraction in the money supply, M, will lower market prices, P)

Answer 29

sorry but where did you get the 4000000? and do i do the same with part b too?

Answer 30

i.e. [(1/3)(derivative)]/400000?

Answer 31

P = Pe + Pd
Pd = $200,000 and Pe = $200,000 so P = $400,000

Answer 32

The point is that modeling a change in debt affects overall price, even if equity stays the same. But you want to show how the net effect is on overall prices in percentage terms.

Answer 33

ok

Answer 34

Thank you very much for your help

Answer 35

Sure thing.

Answer 36

learnt a lot

Answer 37

Oh, cool. Cheers.