SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins

Macroeconomic Theory

The first macroeconomic theory was the quantity theory of money. A simple explanation of hyperinflation using the equation of exchange. This works best on a computer with a speaker using Internet Explorer. It is even better if the computer has a microphone.

Simple macroeconomic computer model for MS Internet Explorer:

The Post-Bubble Malaise of the Janpanese Economy

Topics in macroeconomic theory

The Time Allocation of Consumption

Derivation of the money multiplier

Episodes of hyperinflation

The Imperfect Capital Market Model of the Consumption Function

Macroeconomic policy

Proposed solutions to the Great Depression

Francis Townsend's plan to pay a pension to the elderly if they retire and spend the payment within a month.
Social Credit: All citizens receive newly created money that must be spent within a month.
Huey Long's Share the Wealth Plan called for confiscation of all income above one million dollars per year and everyone having of minimum income of $5000 per year
Veteran's Bonus: Early payment of the promised veteran's bonus
Technocracy: A movement created by Howard Scott but with intellectual roots in the ideas of Thorstein Veblen. Technocracy proposed a system controlled by engineers rather than business people. Under engineers decisions would be made on the basis of efficiency and workmanship rather than profit.
Flexible wages: Many economists maintained that the economic problems of the Great Depression arose from wages rates not adjusting downward when price decreased.
New Deal:
- Agricultural Program: Destroying crops to raise prices
- public works programs
- organization of labor
- distribution of necessities to the poor, "Relief."

On Achieving Economic Equilibrium Through Wage Reductions

The opposition to Keynesian analysis focused on price and wage flexibility as a remedy for economic depressions. This section examines whether, given the dependence of consumer demand on wage income, a cut in wages would necessarily reduce the level of unemployment. This will be done by examining a supply and demand model for a market economy.

Suppose a market price p is established by a balance in the quantity demanded and the quantity supplied; i.e., Q^s = Q^d . The demand function is of the form

Q = a - bp + eI,

where I is labor income in the industry. The influence of other income on demand is included in the constant a. This demand function embodies the notion that auto workers buy cars. The worker demand for the product of the company or the industry is usually relatively minor, but the spirit of the model is that demand function is for all domestically produced products. Thus the fact that the workers of America are a major source of demand for American products cannot be ignored. The supply function is of the form

Q = -c + dp -gw,

where w is the wage rate in the industry. The quantity of labor utilized L is given by

L = f Q

and wage income I is wL . It is presumed that the wage rate is set by processes other than supply and demand. It might be set administratively or by collective bargaining. The equilibrium price is then given by

Q^s = Q^d
Q^d = a - bp + ewfQ
Q^s = -c + dp -gw.

Thus

p = (a+c)/(b+d) + (g + efQ)w/(b+d)

Thus Q can be determined and hence L . Therefore we can determine

L/w

and its sign. What we want to know is under which circumstances (which values for the parameters) is

w positive. If the effect of a wage change on income and hence demand for goods and thus the demand for labor is larger than the effect of a wage change on the supply of goods quantity of labor supplied then a decrease in the wage rate will not reduce unemployment, it will increase it.

The Least Cost
Capital/Labor Combination
for a Cobb-Douglas Production Function

Let R and W be the interest rate and wage rate, respectively, and be the depreciation rate on capital. The cost C of using L units of labor and K units of capital is then

C = WL + (R+)K.

If output is to be Q=AL^2/3K^1/3 then the least cost combination is found by minimizing
C + (Q - AL^2/3K^1/3)
with respect to L and K. The first order conditions are:
W - (2/3)L^-1/3K^1/3 = 0
(R+) -(1/3)AL^2/3K^-2/3 = 0.
This means that
(2/3)L^-1/3K^1/3 = W
(1/3)AL^2/3K^-2/3 = R+.
Dividing the first equation by the second gives the optimal capital/labor ratio as:
2(K/L) = W/((R+))
Thus
L = 2((R+)/W)K
and the output constraint reduces to:
Q = A[2(R+)/W]^2/3K
The desired level of capital is then
K = [2(R+)/W]^-2/3Q/A .
The parameters can be estimated from recent data. In 1990 Q=5463 and L=119.55. The level of the capital stock is not known precisely but estimate of fixed private capital in 1990 was 18854. There was thus a capital/output ratio of approximately 3.45. Thus
A = Q/L^2/3K^1/3
= 5463/[(119.5)^2/3(18854)^1/3
= 5463/645.72222 = 8.46
The total compensation of employees in 1990 was 3244 so W = 27.135. Total depreciation in 1990 was 331.6 so = = 331.6/18854 0.0176. The payment for capital was 5463 - 3244 = 2219 so the rate of return on capital was R=0.1177. That means the total cost of capital was R+=0.1177+0.0176 = 0.1353.
The optimal capital/output ratio is then
K/Q = [2(R+)/W]^-2/3/A
or
K/Q = [W/(2(R+))]^2/3/A

The values of A=8.46, W = 27.135, = 0.0176, R=0.1177 give an optimal capital/output ratio of 2.55.
The desired level of capital is then
K = Q[W/(2(R+))]^2/3/A
and the level of investment is thus
I_t = K_t+1^des - (1-)K_t
= Q_t+1^e[W_t/(2(R_t+))]^2/3/A
If the desired level of capital is less than the surviving capital then the level of investment is zero. This means that if expected output is less than past output and/or the current real interest increases over the past then investment may collapse to zero.
In reality investment may not drop to zero instantly because there are some investment projects partially completed. Usually these investment projects, such as buildings, are completed even though they would not be initiated under the new conditions. What is determined by a comparison of desired capital with surviving capital is investment approvals. Suppose in an investment project 60 percent of the expenditure takes place in the first year after approval and the remaining 40 percent in the second year after approval. If J_t is the investment approvals in year t then the investment in year t is given by:
I_t = 0.6J_t + 0.4J_t-1.
If investment projects are spread over more than two years there would be an analogous relationship for determining investment.
The previous analysis is based upon a Cobb-Douglas production function with a labor exponent of 2/3. The analysis can be easily generalized to an arbitrary labor exponent of . The Cobb-Douglas form production function is rather special in that it implies a unitary elasticity of substitution between labor and capital. It would be desirable to generalize the analysis by using a Constant-Elasticity-of-Substitution (CES) production function. The parameters of the CES production function could then be obtained empirically.
On Including the International Sector
in the ISLM model
Exports depend upon the exchange rate E. Let E be the number of foreign currency units per dollar, say 100 yen per dollar. Suppose the demand function for American timber by Japanese users is:
Q = a - bP,
where Q is in physical units per year, say board-feet/yr, and P is the price of timber in yen, say yen per board-foot. If p is the U.S. price of timber, $ per board-foot, the price to Japanese buyers is pE. Thus the physical quantity of timber sold as a function of E is
Q = a - bpE.
But what we want for a macromodel is the dollar value of the sales; i.e.
pQ = pa - bp²E,

Thus the dollar value of the level of exports is inversely related to E; i.e.,
X = X₀ - qE,
where, for this example, X₀ =pa and q=bp² .
Now suppose we have the demand function for some import to the U.S., say VCR's from Japan,
Q = a -bp,
where Q is the number of VCR's per year and p is the price of VCR's in dollars. If the price of VCR's in Japan is P yen then the price in dollars is P/E. Thus the relationship between physical units of imports and the exchange rate is
Q = a -bP/E.
But again we want the dollar value of the imports, pQ rather than physical units. Therefore the level of imports is
M = pQ = PQ/E = P(a -bP/E)/E
= aP/E - bP/E² ,
a more complicated relationship.
The effect of a small change in E on M, M/E, is given by
= -aP/E² + 2bP/E³
= - 2aP/E² + 2bP/E + aP/E
= - 2(aP/E - bP/E)/E + aP/E
= - 2M/E + aP/E .
Thus M/E may be positive or negative (or zero) depending upon the values of the parameters. If the demand for the imported goods is inelastic an increase in the exchange rate E will result in a decreased value for M rather than an increased one.
The level of imports will also depend upon domestic income and production; i.e.,
Q = a + cY -bP/E
M = (P/E)Q = (P/E)(a + cY - bP/E)
= (aP + cPY - bP /E)/E.
Thus an import function of the form
M = M + mY + sE,
does not capture the full nature of the relationship of the dollar value of imports to the exchange rate and, in particular, the parameter s could be negative rather than positive.
Therefore let us specify the import function in the form
M = (M + mY - s/E)/E.
In the analysis it is assumed that the prices of the goods, timber prices in the US and VCR prices in Japan in the example, do not change as a result of changes in the exchange rate.
Balance of Payments and Net Exports

The dollar and foreign currency components of international transfer payments and capital flows must be defined separately.
Let USFIF be the dollar level of US financial investment in foreign countries and FFIUS be the foreign currency level of foreign financial investment in the US. The net dollar flow is thus
FFIUS/E - USFIF.
Let USTF be the US transfers to foreigners including the repatriation of savings from earnings of foreign nationals working in the US and FTUS be the foreign currency level of foreign transfers to US nationals including the repatriation of savings of US workers working in foreign countries. The dollar value of the net transfer is
USTF - FTUS/E.
The Balance of Payments is then
X - qE -(M + mY - s/E)/E - USTF + FTUS/E + FFIUS/E - USFIF = 0.
This is the equation that determines the equilibrium exchange rate E. Now it is not possible to determine the net exports without determining the exchange rate E.
We see from the form of this equation that it would have been better to work with the reciprocal exchange rate e = 1/E. Thus the Balance of Payment would be
X - q/e -(M + mY - se)e - USTF + eFTUS + eFFIUS - USFIF = 0.
Multiplying through by e gives
X e - q -(M +mY-se)e - USTFe + FTUSe + FFIUSe - USFIFe = 0.
or
se + [-(M +mY) + FTUS + FFIUS]e + (X - USTF - USFIF)e - q = 0.
This is a cubic equation in e. In principle, it can be explicitly solved for e in terms of the parameters. In practice, this would be impossibly complicated. There is also the matter of there being three solutions.
Infinitesimal changes of the exchange rate e due to changes in the the parameter USTF are given by
[3se + 2[-(M +mY) + FTUS + FFIUS]e + (X - USTF - USFIF)]de
[m + ]e + (- 1)e = 0.

The capital flows, FFIUS and USFIF, are affected differently by the US real interest rate R; i.e.,
FFIUS = FFIUS₀ + f₁ R USFIF = USFIF₀ - f₂ R.
That is to say, the higher the real interest rate in the US the greater the foreign financial investment into the US but the lower the US financial investment in foreign countries.
It would be reasonable to presume that if R is inducing foreign financial investment to come into the US there would be no interest rate induced financial investment going out of the U.S. There still may be some outflow for reasons other than advantageous interest rates. This means the functional forms for FFIUS and USFIF are:
FFIUS =
USFIF = 0 US F�

As before
X - M = NTF - FFI,
but now NTF = USTF - FTUS/E
and FFI = FFIUS/E - USFIF.

Therefore, X-M depends not only upon the real interest rate but on the exchange rate as well. The exchange rate will adjust to whatever level to bring about a balance in the balance of payments. Anything that affects the economy will affect the exchange rate E and thus change NTF and FFI.
The Macroeconomic Effects of the Reparation Payments of Germany After World War I

The Treaty of Versailles called for Germany to pay the costs of the war damage to the victors. At the end of the war the amount of reparations was not set; this was to be settled in 1921. The amount set was in 1921 at 132 billion gold marks, the equivalent of 32 billion dollars. The annual payments depended upon the time period given to cover the total. In 1924, after the economic consequences of the scheduled payments were found to be disasterous for Germany a new plan for reparations was proposed. (E S It was called the Dawes Plan after an American financier who was instrumental in formulating it. There was a reduction in the total to 121 billion gold marks and the period for payment was extended to 1988. The initial payments were to be one billion gold marks per year and rising to 2.5 billion gold marks by 1928. One can consider the reparations as a requirement to acquire dollar or any other gold-based foreign currency. The reparation payments were an increase in the net transfer to foreigners, NTF. This had two immediate effects. First there was a reduction in income to households and thus a reduction in disposable income. Second there was in the balance of payments an increase in the amount of German currency that was to be traded for foreign currency.
The model is:
Y = C + I + G₀ + X - M
C = C₀ + bY _D
Y_D = Y - NT - NTF
NT = T₀ + t(Y-NTF)
I = I₀ - gR + hY
M = M ₀ + pE + mY
X = X₀ - qE
X - M - NTF + FFI = 0
FFI = FFI₀ + fR
MS₀ = kY + L₀0 - gR

We need to find the derivatives of the endogenous variables with respect to NTF.
The steps in the solution are:
Y_D = Y - T₀ - t(Y-NTF) - NTF
= -T₀ + (1-t)Y - (1-t)NTF
C = C₀ - bT₀ + b(1-t)Y - b(1-t)NTF
X-M = NTF - FFI
= NTF - FFI₀ - fR
R = [L₀-MS₀+kY]/j
X-M = NTF - FFI ₀

Rational Expectations

John F. Muth created the principle of rational expectations as a model of how expectations are formed. He asserted that expectations "are essentially the same as the predictions of the relevant economic theory."
Muth was primarily concerned with the rational expectations principle as a guide to formulating commodity market models which would have a stronger theoretical foundation than the cobweb models. The standard cobweb model is:
Q_t^D = a - P_t
(the demand function)
Q_t^S = - + P_t + u
(the supply function)
P_t^E = P_t-1
(the expectations hypothesis)
Q_t^S = Q_t^D ,
(the market clearing hypothesis)

where P_t^E is expected price and u is a random influence on the supply and has an expected value of zero.
If the random influences are serially uncorrelated Muth concludes the only sensible expectation is that the expected price is equal to the equilibrium price when the u=0. This is contrary to the cobweb model. If there is serial correlation of the random disturbances then expected price should be an exponentially weighted average of past price; i.e.,
P_t^E = (1-) ⁱP_t-i.

The surprising thing is that the weight parameter should be equal to a ratio that depends upon the coefficients in the supply and demand functions, namely /(+) .
Although Muth formulated the rational expectations principle in the context of microeconomics it has subsequently become associated with macroeconomics and the work of Robert Lucas, Thomas Sargent, Neil Wallace and other neoclassical macroeconomists.
According to this school of macroeconomics any anticpated actions on the part of the monetary authorities will be incorporated into the public's expectations of inflation. Therefore, according to the rational expectations school, "it is only unanticipated growth in the money supply that can make the actual inflation rate diverge from the expected inflation rate." Furthermore, "unemployment can only diverge from its natural rate when people are fooled about the inflation, [therefore] it follows that systematic, anticipated monetary policy has no effect on output or employment."
The Determination of the Rate of Inflation
One of the problems of the aggregate demand/aggregate supply analysis of the price level is that it purports to determine the price level at any time without reference to what the price level has been in past periods or what the price level is expected to be in future periods. A more reasonable approach is to require a model to determine the rate of inflation and let the price level be determined by the accumulated inflation.
Consider the equation of exchange in the form
P = MV/Q.
The income velocity of money is a function of the cost of holding money. In situations of hyperinflation the cost of holding money is primarily the rate of inflation, the rate at which money is losing its value. But even without inflation there is a cost to holding a security which does not pay interest. The cost of holding money is just the nominal rate of interest.
If checking accounts pay interest then the cost of holding funds in checking accounts is the difference between the interest rate on, say, saving accounts and the interest rate on checking accounts. For simplicity let us assume there is no interest on checking accounts.
In the Fisher theory of interest rates the nominal interest rate is determined as the sum of a real rate of interest and the rate of inflation; i.e., R+, where is the rate of inflation.
By logarithmic differentiation the equation of exchange can be put into the form:
=
+ dln(V)/dt - dln(Q)/dt
where is the rate of growth of the money supply.
Since ln(V) = f(R+) the above equation reduces to:
= + f'(R+)[dR/dt + d/dt]
- dln(Q)/dt.
This is a differential equation in . If the functional relation between the velocity of money and the nominal interest rate is nonlinear then the differential equation for is a first order nonlinear differential equation.
Now we should look at the dependence of money demand on the interest rate.
The Inventory Model of the Transaction Demand for Money
Suppose an individual, household or firm has a known annual expenditure of X. They could make a withdrawal of X at the beginning of the year and spend it bit by bit throughout the year. This strategy would give up the interest earned on a large portion of the funds. On the other hand, because there are transaction cost, a strategy of withdrawing funds only as they are neeeded is not economical either.
If wihdrawals are made in amounts of q then the total number of withdrawals in a year is X/q. The average size of the cashholding is q/2 so the interest lost per year is r(q/2), where r is the nominal interest rate. If c is the cost per transaction the total cost per year is:
C = c(X/q) + rq/2
.
The value of q which minimizes C is
q = [2cX/r]^1/2
The average transaction holding of money is q/2 and the expenditure is X so the velocity is X/(q/2) and thus
V = [2rX/c]^1/2

In the equation of exchange model for inflation we can identify the expenditure X with the per household level of nominal ouput PQ of the economy; i.e., PQ/H where H is the number of households. Thus,
V = f(r) = [2rpQ/Hc]^1/2
and therefore
f'(r) = (1/2)[2rPQ/Hc]^-1/2(2PQ/Hc)
= ([PQ/Hc]^1/2r^-1/2

Therefore the differential equation for the rate of inflation is
= + (1/2)[2PQ/Hc]^-1/2[R+]^-1/2[dR/dt + d/dt] - dln(Q)/dt.

If we take the real interest constant, the real output constant and the rate of growth of the money supply as constant then the differential equation for the rate of inflation is:
=
+ (1/2)[2PQ/Hc]^1/2[R+]^-1/2 d/dt.
Solving for d/dt gives
d/dt = 2( - ) [R+]^1/2/[2PQ/Hc]^1/2
One solution to this equation is = , but the type of solutin we are concerned about is accelerating inflation.
Dynamics of the Accelerator Model

Y_t = Y_t + (Y_t-1 - Y_t-2) + G₀
Equilibrium
Y = Y + (Y - Y) + G₀
Subtracting the second equation from the first gives
Y_t - Y =
(Y_t-Y) + [(Y_t-1-Y) - (Y_t-2-Y)]
Letting y_t = Y_t - Y
the above equation becomes:
y_t = y_t + y_t-1 - y_t-2
Solving for y_t gives
(1-)y_t= y_t-1 - y_t-2
y_t= /(1-) y_t-1 - /(1-)y_t-2
(to be continued)

Proposed solutions to the Great Depression

On Achieving Economic Equilibrium Through Wage Reductions

Q = a - bp + eI,

Q = -c + dp -gw,

L = f Q

Qs = Qd Qd = a - bp + ewfQ Qs = -c + dp -gw.

p = (a+c)/(b+d) + (g + efQ)w/(b+d)

L/w

The Least Cost Capital/Labor Combination for a Cobb-Douglas Production Function

C = WL + (R+)K.

C + (Q - AL2/3K1/3)

W - (2/3)L-1/3K1/3 = 0 (R+) -(1/3)AL2/3K-2/3 = 0.

(2/3)L-1/3K1/3 = W (1/3)AL2/3K-2/3 = R+.

2(K/L) = W/((R+))

L = 2((R+)/W)K

Q = A[2(R+)/W]2/3K

K = [2(R+)/W]-2/3Q/A .

A = Q/L2/3K1/3 = 5463/[(119.5)2/3(18854)1/3 = 5463/645.72222 = 8.46

K/Q = [2(R+)/W]-2/3/A or K/Q = [W/(2(R+))]2/3/A

K = Q[W/(2(R+))]2/3/A

It = Kt+1des - (1-)Kt = Qt+1e[Wt/(2(Rt+))]2/3/A

It = 0.6Jt + 0.4Jt-1.

On Including the International Sector in the ISLM model

Q = a - bP,

Q = a - bpE.

pQ = pa - bp2E,

X = X0 - qE,

Q = a -bp,

Q = a -bP/E.

M = pQ = PQ/E = P(a -bP/E)/E = aP/E - bP/E2 ,

= -aP/E2 + 2bP/E3 = - 2aP/E2 + 2bP/E + aP/E = - 2(aP/E - bP/E)/E + aP/E = - 2M/E + aP/E .

Q = a + cY -bP/E M = (P/E)Q = (P/E)(a + cY - bP/E) = (aP + cPY - bP /E)/E.

M = M + mY + sE,

M = (M + mY - s/E)/E.

Balance of Payments and Net Exports

FFIUS/E - USFIF.

USTF - FTUS/E.

X - qE -(M + mY - s/E)/E - USTF + FTUS/E + FFIUS/E - USFIF = 0.

X - q/e -(M + mY - se)e - USTF + eFTUS + eFFIUS - USFIF = 0.

X e - q -(M +mY-se)e - USTFe + FTUSe + FFIUSe - USFIFe = 0. or se + [-(M +mY) + FTUS + FFIUS]e + (X - USTF - USFIF)e - q = 0.

[3se + 2[-(M +mY) + FTUS + FFIUS]e + (X - USTF - USFIF)]de [m + ]e + (- 1)e = 0.

FFIUS = FFIUS0 + f1 R USFIF = USFIF0 - f2 R.

FFIUS = USFIF = 0 US F�

X - M = NTF - FFI, but now NTF = USTF - FTUS/E and FFI = FFIUS/E - USFIF.

The Macroeconomic Effects of the Reparation Payments of Germany After World War I

Y = C + I + G0 + X - M C = C0 + bY D YD = Y - NT - NTF NT = T0 + t(Y-NTF) I = I0 - gR + hY M = M 0 + pE + mY X = X0 - qE X - M - NTF + FFI = 0 FFI = FFI0 + fR MS0 = kY + L00 - gR

YD = Y - T0 - t(Y-NTF) - NTF = -T0 + (1-t)Y - (1-t)NTF C = C0 - bT0 + b(1-t)Y - b(1-t)NTF X-M = NTF - FFI = NTF - FFI0 - fR R = [L0-MS0+kY]/j X-M = NTF - FFI 0

Rational Expectations

QtD = a - Pt (the demand function) QtS = - + Pt + u (the supply function) PtE = Pt-1 (the expectations hypothesis) QtS = QtD , (the market clearing hypothesis)

PtE = (1-) iPt-i.

The Determination of the Rate of Inflation

P = MV/Q.

= + dln(V)/dt - dln(Q)/dt

= + f'(R+)[dR/dt + d/dt] - dln(Q)/dt.

The Inventory Model of the Transaction Demand for Money

C = c(X/q) + rq/2

q = [2cX/r]1/2

V = [2rX/c]1/2

V = f(r) = [2rpQ/Hc]1/2 and therefore f'(r) = (1/2)[2rPQ/Hc]-1/2(2PQ/Hc) = ([PQ/Hc]1/2r-1/2

= + (1/2)[2PQ/Hc]-1/2[R+]-1/2[dR/dt + d/dt] - dln(Q)/dt.

= + (1/2)[2PQ/Hc]1/2[R+]-1/2 d/dt.

d/dt = 2( - ) [R+]1/2/[2PQ/Hc]1/2

Dynamics of the Accelerator Model

Q^s = Q^d
Q^d = a - bp + ewfQ
Q^s = -c + dp -gw.

The Least Cost
Capital/Labor Combination
for a Cobb-Douglas Production Function

C + (Q - AL^2/3K^1/3)

W - (2/3)L^-1/3K^1/3 = 0
(R+) -(1/3)AL^2/3K^-2/3 = 0.

(2/3)L^-1/3K^1/3 = W
(1/3)AL^2/3K^-2/3 = R+.

Q = A[2(R+)/W]^2/3K

K = [2(R+)/W]^-2/3Q/A .

A = Q/L^2/3K^1/3
= 5463/[(119.5)^2/3(18854)^1/3
= 5463/645.72222 = 8.46

K/Q = [2(R+)/W]^-2/3/A
or
K/Q = [W/(2(R+))]^2/3/A

K = Q[W/(2(R+))]^2/3/A

I_t = K_t+1^des - (1-)K_t
= Q_t+1^e[W_t/(2(R_t+))]^2/3/A

I_t = 0.6J_t + 0.4J_t-1.

On Including the International Sector
in the ISLM model

pQ = pa - bp²E,

X = X₀ - qE,

M = pQ = PQ/E = P(a -bP/E)/E
= aP/E - bP/E² ,

= -aP/E² + 2bP/E³
= - 2aP/E² + 2bP/E + aP/E
= - 2(aP/E - bP/E)/E + aP/E
= - 2M/E + aP/E .

Q = a + cY -bP/E
M = (P/E)Q = (P/E)(a + cY - bP/E)
= (aP + cPY - bP /E)/E.

X e - q -(M +mY-se)e - USTFe + FTUSe + FFIUSe - USFIFe = 0.
or
se + [-(M +mY) + FTUS + FFIUS]e + (X - USTF - USFIF)e - q = 0.

[3se + 2[-(M +mY) + FTUS + FFIUS]e + (X - USTF - USFIF)]de
[m + ]e + (- 1)e = 0.

FFIUS = FFIUS₀ + f₁ R USFIF = USFIF₀ - f₂ R.

FFIUS =
USFIF = 0 US F�

X - M = NTF - FFI,
but now NTF = USTF - FTUS/E
and FFI = FFIUS/E - USFIF.

Y = C + I + G₀ + X - M
C = C₀ + bY _D
Y_D = Y - NT - NTF
NT = T₀ + t(Y-NTF)
I = I₀ - gR + hY
M = M ₀ + pE + mY
X = X₀ - qE
X - M - NTF + FFI = 0
FFI = FFI₀ + fR
MS₀ = kY + L₀0 - gR

Y_D = Y - T₀ - t(Y-NTF) - NTF
= -T₀ + (1-t)Y - (1-t)NTF
C = C₀ - bT₀ + b(1-t)Y - b(1-t)NTF
X-M = NTF - FFI
= NTF - FFI₀ - fR
R = [L₀-MS₀+kY]/j
X-M = NTF - FFI ₀

Q_t^D = a - P_t
(the demand function)
Q_t^S = - + P_t + u
(the supply function)
P_t^E = P_t-1
(the expectations hypothesis)
Q_t^S = Q_t^D ,
(the market clearing hypothesis)

P_t^E = (1-) ⁱP_t-i.

=
+ dln(V)/dt - dln(Q)/dt

= + f'(R+)[dR/dt + d/dt]
- dln(Q)/dt.

q = [2cX/r]^1/2

V = [2rX/c]^1/2

V = f(r) = [2rpQ/Hc]^1/2
and therefore
f'(r) = (1/2)[2rPQ/Hc]^-1/2(2PQ/Hc)
= ([PQ/Hc]^1/2r^-1/2

= + (1/2)[2PQ/Hc]^-1/2[R+]^-1/2[dR/dt + d/dt] - dln(Q)/dt.

=
+ (1/2)[2PQ/Hc]^1/2[R+]^-1/2 d/dt.

d/dt = 2( - ) [R+]^1/2/[2PQ/Hc]^1/2