(1) For the relationship between income and consumption, Keynes thought that there was a basic psychological law. With the increase of income, consumption will also increase, but the increase of consumption is not as much as that of income. In order to simplify the processing, the consumption function is generally regarded as a linear function of income, including:

c=α+βy

Among them, α is spontaneous consumption, β is marginal propensity to consume, C represents consumption, and Y represents income.

At the same time, Keynes thought that income Y can be divided into two parts, one part is consumption C, and the other part is savings S. There are:

s=y-c=y-(α+βy)=-α+( 1-β)y

② We can regard the amount of investment as a decreasing function of interest rate:

i=e-dr

Where I is investment, E is spontaneous investment, R is interest rate, and D is the sensitivity of investment to interest rate.

(3) If we only consider the two-sector economy, the total demand is equal to the total supply at equilibrium, with C+I = C+S, and the equilibrium condition after finishing is i=s, namely:

r=(α+e)/d-[( 1-β)/d]y

ThIS is the famous IS curve. As shown in the figure, the IS curve is a straight line inclined to the lower right, and it is the trajectory of all the combination points in the product market that meet the equilibrium income and interest rate.

2. Derivation of 2.LM curve

Keynes believed that a country's money supply was an exogenous variable in a certain period of time and was basically controlled by the monetary authorities. The demand for money is mainly caused by three motives, namely, trading motive, prevention motive and speculation motive. Among them, the money demand of the first two motives is the increasing function of income, which is not directly related to interest rates;

The money demand of the latter motive is a decreasing function of interest rate. If L 1 means to keep money to meet the monetary demand of trading motivation and preventive motivation, L 1(y) can be expressed as ky, where k is the ratio of trading demand to income;

If L2 represents the money demand to keep money to meet speculative motives, L2(r) can be expressed as -hr, where h is the reaction coefficient of money speculative demand to interest rate. The total money demand l is l = l1(y)+L2 (r) = ky-hr.

Assume that the money supply is expressed by Ms; Expressed as money demand by Md; Y stands for income; R stands for interest rate level. If M0 represents a constant determined by the monetary authority, there is a formula:

Ms=M0

Md=L= L 1(y)+L2(r)=ky-hr

The money market equilibrium can be obtained by combining the above two formulas: M0= ky-hr, and the LM curve can be obtained by deformation:

r=ky/h-M0/h

The slope of LM curve depends on K and H. The greater K, the greater the slope of LM curve. The greater the H, the smaller the slope of the LM curve. When h→∞, the LM curve is parallel to the X axis, and the economic operation falls into a liquidity trap. At this time, monetary policy is ineffective and fiscal policy is particularly effective.

Extended data

The movement of LM curve is related to M/(Ph). M is the money supply, and the central bank decides to implement loose or tight monetary policy; Assuming that p is constant, in the IS-LM model; In monetarism theory, H has the same assumption.

1) The key to the increase of moving m lies in the second quadrant, and the moving distance is k times that of the second phenomenon.

2) the slope k/h. k is determined by commercial practices; H is determined by the sensitivity of money demand to interest rate.

Keynesianism: People are highly sensitive to changes in interest rates, which determines whether it is a trading function performed by money. A special case of Keynesian phenomenon: liquidity preference trap. When the economy is extremely depressed and a certain interest rate is very low, it will cause public panic, that is, the demand for mobile phase is extremely high.

Classicism (classical region): the old school thinks that money performs two functions, either consumption; Either preserve it, deny the existence of L2, and don't think it has cellar value. The new school admits that the existence of money hoarding function is influenced by many other factors besides interest rate. Portfolio theory holds that assets exist in different forms, mainly currency, stocks and bonds, real estate and durable consumer goods.

As interest rates rise, the willingness to hold money decreases. Friedman believes that the proportion of asset types is constant and the LM curve is vertical. There is a phenomenon that when the economy is prosperous, more people are willing to believe in Keynesianism; When inflation intensifies, people are more willing to believe in the monetarism (Friedman) Baidu Encyclopedia -LM curve.

Baidu encyclopedia -IS curve