小弟最近备考CFA一级，十二月份考，然后现在看到经济学宏观部分，看到了IS-LM曲线的推导，十分不解，以下是小弟理解的过程，希望大家批评指正，在这里错好过在考试的时候犯错，最后祝大家都顺利通过。
因为CFA书是英文的，我就直接用英文写了，免得概念什么的颠来倒去弄错了。
IS Curve
The IS curve shows the combinations of GDP or real income Y and real interest rate Ri that keep the quantity of total income equal to the quantity of total expenditure.
Consumption C=C(Y), Consumption C is positively related to Income Y because you would consume more when you have more money;
Investment I=I(Ri,Pe), Investment I is negatively related to Real Interest Rate Ri and positively related to Expected Profitability Pe, because you would not want to invest you money if the price of the money lent(interest rate) is high, and you would want to invest your money if it is profitable;
Government Purchase G=G, Government Purchase can be given exogenously, which is fixed;
Net Export NX=Export-Import=EX(Yf,Rp)-IM(Y,Rp), Net Export equals Export minus Import, where Export is positively related to Foreign Income Yf and Import is positively related to Domestic Income Y and both are related to the Ratio of the Prices Rp, it is not necessary to learn the relationship between Ex, IM and Rp because to put it simple, we fix the Rp.
Tax T=T(Y), Tax is positively related to Income Y because the richer the more taxed by the government.
Saving S=S(Y), Saving is positively related to Income Y because the more you earn, the more you want to save in the bank. (Actually this should be state as S=S(Y, Ri), because the higher Ri, the more you want to save because the price of saving is increasing and you are the seller of your savings?) Anyway, put it S=S(Y).
OK, now we take Pe, G, EX and Rp as given(fixed), and Y, Ri, T, S and IM are variables. What we are going to do is to find the relationship between Y and Ri, Aggregate Income and Real Interest Rate.
First, we have 2 approaches to compute the GDP:
The income way: All income we get comes from at least one of the 4 sources: People pay money to Consume something, People pay monety to Investing something, Government pay money to purchase something and Trade Balance.
So Y=C+I+G+NX
The expenditure way: All income would be spent on at least one of the 3 sections: People pay money to consume something, people pay money to the government as taxes and People save the money in the bank.
So Y=C+S+T
Second, the GDP values obtained from the 2 ways must be equal: C+I+G+NX=C+S+T
So we can get S-I=(G-T)+(EX-IM)
Now G is given; EX is given because Yf and Rp are fixed; IM is only positively realted to Y because Rp is fixed; S is positively related to Y, and I is negatively related to Ri because Pe is fixed.
So as the Aggregate Income Y increases, T and IM increases, so (G-T) and (EX-IM) decreases. To equate the total income and total expenditure, S-I should decrease too. However, while Y increases, S increases, so I must increase in a larger volume to offset the increased savings. Only when Ri decreases, I would increase. So now we can get:
While Aggregate Income Y increases, Real Interest Rate Ri decreases.
That's what the IS curve describes.
-7-16 10:51:06 上传
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LM Curve
The LM curve shows the combinations of GDP or real income Y and real interest rate Ri that keep the quantity of real money demaned equal to the quantity of real money supplied.
We get started from the quantity theory of money, which states: MV=PY
Where: M is Nominal Money Supply, V is the Velocity of Money in transactions, P is the price level and Y is real GDP,as we know,GDP can also be interpreted as Aggregate Income.
Then we can get M/P=Y/V, where M/P is the Real Money Supply Ms. In that case we can interpret 1/V as the fraction of income(?) that people would like to hold in cash. So Y/V is the Real Money Demand Md. And Real Money Demand can be a function of Real Interest Rate Ri and Income Y, Md=M(Ri,Y). Md is negatively related to Ri because the higher real interest rate, the more you have to give up if you hold the money in cash; Md is positively related to Y because the more you earn, the more you would like to hold in cash because you total money increases.
Because in equilibrium, Ms=Md, if we fix Ms, then Md should be fixed too. In that case, if Md increases from the decrease of Ri, Y have to decrease to make Md decrease to offset the increase. So now we can get:
While Aggregate Income Y increases, Real Interest Rate Ri increases.
That's what LM curve describes.
Aggregate Demand Curve
Aggregate Demand Curve shows the relationship between the quantity of Real Output Demanded(which equals Real Income Y) and the Price Level P.
Now we have the IS curve, on which the points representing the levels of the Real Interest Rate Ri and Income Y, consistent with the equilibrium between the Total Income and the Total Expenditure; We also have the LM curve, on which the points representing the levels of the Ri and Y, consistent with the equilibrium between the Real Money Demanded Md and Real Money Supplied Ms. And when the 2 curves intersect, the intersection point represents the level of the Ri and Y satisfies both the 2 equilibriums at the same time.
Because we are exploring the relationship between Y and P, we now take P as a variable. So now we are going to see how Y would react to an increase of P.
Now M is fixed, V is fixed. So if P increases, then Real Money Supply Ms=M/P would decrease. To maintain the equilibrium that Ms=Md, Md should decrease too. There is only 3 ways to make Md decrease: 1. Ri increases, makes the opportunity cost of holding money in cash higher, then people would like to hold less cash, Md decreases; 2. Y decreases, makes people have less money, then people's willingness to hold cash would appropriately decrease, Md decreases; 3. Ri increases while Y decreases, make the 2 effects occur at the same time, Md decreses. Either of the 3 ways would make Md decrease.
However, considering the equilibrium that total income equlas total expenditure, only the 3rd way is available. We can see that from the IS-LM chart below.
-7-18 10:39:30 上传
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So we can get the relationship between P and Y:
While Price Level P increases, Aggregate Income(Demand) Y decreases.
Aggregate Supply Curve
Aggregate Supply Curve AS would have 3 different slopes because of the different time spans. We can roughly devide time span into 3 different kinds: Very Short term, Short-term and Long-term.
Why there would be 3 different slopes for the Aggregate Supply Curve? Because how the price level react to the time is different. For example, P would have no time to adjust to changes in Aggregate Supply in very short term, so in very short term, AS should be a horizontal line.
However, in long term, price would have plenty of time to adjust to any change in aggregate supply, because if the price of products increases, people would require higher paid for supplying their labor, land and other factors. To maximize their profit, the firms would product at the level that MR=MC(Referred to preious chapters), so the output would not increase(In practical, there should be a more or less change?). In short, even the price goes up in long term, the Aggregate Supply would remain on the same level, and that appears a vertical AS curve.
-7-18 11:30:44 上传
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