國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
4
Chapter 2: The Theoretical Model
The theoretical model used in this thesis is based on Bernanke and Blinder (1988), which will be used to determine the effects of output and the interest rate from a change in monetary policy. In theory, under a traditional IS-LM model, a change in monetary policy will have a definitive change in both output and interest rates based on the policy implemented, but in the model described by Bernanke and Blinder (1988), the output change is definitive but the bond rate change is unknown.
Previously, as shown in Bernanke and Blinder (1988), only the plus-minus relationships are shown between variables without looking into the interaction between variables. The model shown here, using the framework of Bernanke and Blinder’s model, will solve the exact interaction between each variable and how the resulting plus-minus relationship is determined.
From Bernanke and Blinder (1988), there are three different markets: the loan market, the goods market, and the money market. This paper studies the effects of a change in bank reserves on the loan rate, bond rate, and output.
2.1. Loan Market
Loan demand is defined as:
L(ρ, i, y) ……….……….. (2.1a) (Lρ < 0, Li > 0, Ly > 0)
In equation (2.1a), ρ is the interest rate on loans, i is the interest rate on bonds, and y is income. Total loan demand is a function of the interest rate on loans, interest rate on bonds, and total income. The loan rate is inversely related to loan demand, meaning a rise in the loan rate will decrease loan demand. Both bond rate and income are positively related to loan demand, meaning an increase in either will result in an increase in loan demand.
Loan supply is defined as:
λ(ρ, i)D(1 − τ) ……….. (2.1b) (λρ > 0, λi< 0)
In equation (2.1b), τ is the required reserve rate, and D is total deposits. Loan supply is defined as the multiplication of λ, D, and (1 - τ). λ is the credit supply
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
5
function, and is a function of the loan rate and bond rate. D is in function form, and will be discussed more in the next section. While λ and the loan rate are positively related, λ and the bond rate are inversely related, meaning that a rise in the loan rate will increase credit supply, while a rise in the bond rate will decrease credit supply.
In order for the loan market to reach equilibrium, loan supply must equal loan demand. This allows us to obtain the loan market equation:
L(ρ, i, y) = λ(ρ, i)D(1 − τ) ………..……….. (2.1)
2.2. Money Market
The money market equilibrium is defined as total deposits equaling total reserves multiplied by the money multiplier, shown as:
D(i, y) = m(i)R ………..……… (2.2) (Di < 0, Dy > 0, mi > 0)
In equation (2.2), R represents bank reserves, and m represents the money multiplier. Total deposit is a function of the bond rate and income, where the bond rate and total deposit are inversely related, while income and total deposits are positively related. The money multiplier is a function of the bond rate, and the money multiplier is positively related to the bond rate.
2.3. Goods Market
The goods market is defined as:
y = Y(i, ρ) ………. (2.3) (Yi < 0, Yρ < 0)
Total output is represented as aggregate demand, which is a function of both the loan rate and bond rate. Both interest rates are inversely related to aggregate demand.
2.4. The Effect of a Change in Bank Reserves
After differentiating equations (2.1), (2.2), and (2.3)3
3 See Appendix for step-by-step process
, we then organize the
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
6
resulting differentiated equations into matrix form:
�
�Lρ− m(i)R(1 − τ)λρ� [Li− λ(ρ, i)mi(1 − τ)R − m(i)R(1 − τ)λi] Ly
0 (Di− miR) Dy
−Yρ −Yi 1
� ∗
�dρ
dydi� = �λ(ρ, i)m(i)(1 − τ) m(i)0 � [dR]
Next, we solve the determinant for the 3*3 matrix, which we label as K:
|K| = ��Lρ− m(i)R(1 − τ)λρ� ∗ (Di− miR) ∗ 1�
+ �[Li− λ(ρ, i)mi(1 − τ)R − m(i)R(1 − τ)λi] ∗ Dy ∗ �−Yρ��
− �Ly∗ (Di− miR) ∗ �−Yρ��
− ��Lρ− m(i)R(1 − τ)λρ� ∗ (−Yi) ∗ Dy� ≶ 0
……….………. (2.4) The sign of the determinant is undetermined, but using the assumption4
|K| = ��Lρ− m(i)R(1 − τ)λρ� ∗ (Di) ∗ 1� + �[Li− m(i)R(1 − τ)λi] ∗ Dy∗ �−Yρ��
− �Ly∗ (Di) ∗ �−Yρ�� − ��Lρ− m(i)R(1 − τ)λρ� ∗ (−Yi) ∗ Dy� > 0 mi = 0, the determinant becomes:
……….………. (2.5) This provides us with a positive determinant, and we can solve how the total reserves would change when the loan rate, bond rate, and income changes.
First, we solve for dρ dR� , which represents the relationship of how total reserves would change when the loan rate changes, which equals:
dρ
dR=|A||K|≶ 0 ……….……… (2.6)
|A| = {λ(ρ, i)m(i)(1 − τ) ∗ (Di− miR) ∗ 1} + {Ly∗ (−Yi) ∗ m(i)}
− {[Li− λ(ρ, i)mi(1 − τ)R − m(i)R(1 − τ)λi] ∗ m(i) ∗ 1}
− {λ(ρ, i)m(i)(1 − τ) ∗ (−Yi) ∗ Dy} ≶ 0
4 As long as the money multiplier isn’t too large, dmdi = mi→ 0
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
7
This relationship is undetermined. Therefore, we must discuss further to know the exact effect.
Next, we solve for di dR� , which represents the relationship of how total reserves would change when the bond rate changes, which equals:
di
dR=|B||K|≶ 0 ………...………..…………. (2.7)
|B| = ��Lρ− m(i)R(1 − τ)λρ� ∗ m(i) ∗ 1� + �λ(ρ, i)m(i)(1 − τ) ∗ Dy∗ �−Yρ��
− �Ly∗ m(i) ∗ �−Yρ�� ≶ 0
The relationship is undetermined. Therefore, we must discuss further to know the exact effect.
Finally, we solve for dy dR� , which represents the relationship of how total reserves would affect income, which equals:
dy
dR=|K||C|> 0 ……….……… (2.8)
|C| = {[Li− λ(ρ, i)mi(1 − τ)R − m(i)R(1 − τ)λi] ∗ m(i) ∗ �−Yρ�
− �λ(ρ, i)m(i)(1 − τ) ∗ (Di− miR) ∗ �−Yρ��
− {�Lρ− m(i)R(1 − τ)λρ� ∗ �−Yρ� ∗ m(i)} > 0
The result shows that the relationship between bank reserves and income is positively related, indicating that an increase in income would result in an increase in reserves.
2.5. Graphical Representation 2.5.1. LM Curve
The LM curve is defined as the money market shown in equation (2.2). We solve for the slope of the LM curve, which is defined as di dy� |LM:
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
8
di
dy|LM =mDy
iR−Di > 0 ……….………. (2.9) This shows that the LM curve has a positive slope, which is illustrated in Figure 2.1.
Next, we solve for the income shift magnitude of the LM curve. Holding i constant, we can solve for the income shift magnitude when the reserve money increases, which is defined as dy dR� |LM. The resulting shift magnitude equals:
dy
dR|LM = m(i)D
y > 0 ..……….………..……….. (2.10)
2.5.2. CC Curve
The CC curve is very similar to the IS curve in the traditional IS-LM model. The main difference is that aspects of the loan market does not affect movements of the IS curve, while the CC curve is affected by changes in the loan market, particularly the loan rate. The CC curve representing the combination of the bond rate and output level that allows both the goods market and the loan market to be at equilibrium.
As stated by Bernanke and Blinder (1988), given that the loan rate ρ can be written as a function of i, y, R, we can write ρ as:
i LM
0 y
Figure 2.1: Positive-sloping LM curve.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
9
ρ = ρ(i, y, R) …...……… (2.11) Differentiating ρ and using the differentiated equation (2.11), we can get:
ρi= −[Li−λ(ρ,i)mi(1−τ)R−m(i)R(1−τ)λi]
�Lρ−m(i)R(1−τ)λρ� ≶ 0……….………. (2.12a) ρy = −�L Ly
ρ−m(i)R(1−τ)λρ�> 0……….……….. (2.12b) ρR= λ(ρ,i)m(i)(1−τ)
�Lρ−m(i)R(1−τ)λρ�< 0……….……… (2.12c) The sign of ρi is undetermined. However, by using the assumption mi = 0 as shown in footnote (2), we get:
ρi= −[Li− m(i)R(1−τ)λi]
�Lρ−m(i)R(1−τ)λρ�> 0 ………..……… (2.12d) Equation (2.12b) shows that loan rate and income are positively related.
Equation (2.12c) shows that the loan rate and reserves are inversely related.
Equation (2.12d) shows that the loan rate and bond rate are positively related.
Substituting equation (2.11) into equation (2.3), we can get the CC curve equation:
y = Y(i, ρ(i, y, R)) …………... (2.13)
We solve for the slope of the CC curve, which is defined as di dy� |CC:
di
dy|CC =Y1−Yρρy
i+Yρρi< 0 ………..……….……… (2.14) This shows that the CC curve has a negative slope, which is illustrated in Figure 2.2.
‧
Next, we solve for the income shift magnitude of the CC curve. Holding i constant, we can solve for the income shift in the magnitude of the CC curve when total reserves increase, which is defined as dy dR� |CC . The magnitude of the income shift on the CC curve equals:
dy
dR|CC = 1−YYρρR
ρρy> 0 ..………..………..……….. (2.15)5
2.5.3. The Effect of Reserve Money Change on Bond Rate
After solving equations (2.10) and (2.15), we can see that both the income shift and the bond rate shift of both CC and LM curves are positively related to changes in bank reserves. Next, using these equations, we solve for Yρ, which represents how income changes when the loan rate changes. Setting both equations equaled to each other, we get:
YρρR
1−Yρρy=m(i)D
y ……….……..………... (2.16) Using equation (16), we can solve for Yρ, which equals:
Yρ= D m(i)
0
Figure 2.2: Negative-sloping CC curve‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
11
Reorganizing the complex version of equation (2.17) as described in footnote (3), we can obtain:
−Yρ�Dyλ(ρ, i)m(i)(1 − τ) − m(i)Ly� + m(i)�Lρ− m(i)R(1 − τ)λρ� = 0 … (2.18) The left hand side of equation (2.18) is the same as |B| in equation (2.7).
Therefore, in the condition set in equation (2.18), the shift magnitude of CC and LM are equal.
After solving for Yρ, there are three possible results:
1. If Yρ< D m(i)
yρR+m(i)ρy, corresponding back to equation (2.18), this would set |B| to be greater than zero. Substituting this condition back into equation (2.7), |B| |K|� would be positive, resulting in di dR� being positive. This shows that the
magnitude of the CC shift is greater than the magnitude of the LM shift, resulting in the bond rate increasing. Figure 2.3 illustrates this result.
2. If Yρ> D m(i)
yρR+m(i)ρy, corresponding back to equation (2.18), this would set |B| to be less than zero. Substituting this condition back into equation (2.7), |B| |K|� would be negative, resulting in di dR� being negative. This shows that the magnitude of the CC shift is less than the magnitude of the LM shift, resulting in the bond rate decreasing. Figure 2.4 illustrates this result.
3. If Yρ= D m(i)
yρR+m(i)ρy, corresponding back to equation (2.18), this would set |B|
equaled to zero. Substituting this condition back into equation (2.7), |B| |K|� is equaled to zero, resulting in di dR� equaling zero. This shows that the magnitude of the CC shift equals the magnitude of the LM shift. Corresponding back to
‧
equation (2.7) di dR� must equal zero, resulting in the bond rate remaining the same. Figure 2.5 illustrates this result.
A special case of this model occurs when Yρ= 0. In this scenario, any monetary policy shift will keep the CC curve constant and shift only the LM curve. Also, as Yρ is negative in theory, setting Yρ = 0 would result in Yρ >D m(i) Figure 2.3: CC shift > LM shift,
resulting in the bond rate increasing.
Figure 2.4: LM shift > CC shift, resulting in the bond rate decreasing.
Figure 2.6: Special case where Yρ= 0. LM shifts (magnitude unknown), but CC curve remains constant, as the bond rate decreases.
Figure 2.5: CC shift = LM shift, resulting in the bond rate remaining the same.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
13
same condition as described in the 2nd possible result. Figure 2.6 illustrates this special case.
2.5.4. The Effect of Reserve Money Change on Loan Rate
After differentiating equations (2.2) and (2.13), we can solve for the interest rate shift magnitude for both LM and CC curves, which are defined as di dR� |LM and di dR� |CC, respectively.
di
dR|LM = −YYρρR
i+Yρρi ……….………..……… (2.19)
di
dR|CC = Dm(i)
i−miR……….………... (2.20) Setting both equations (2.19) and (2.20) equal to each other and reorganizing both, we can solve for Yρ, which is equal to:
Yρ= −ρ Yim(i)
RDi−ρRmiR+ρi< 0……….……….……. (2.21) Using equation (2.21), we can come up with three possible results:
1. If Yρ< −ρ Yim(i)
RDi−ρRmiR+ρi, then the magnitude of the CC curve shift is greater than the LM curve shift. This will result in the loan rate increasing, making dRdρ> 0 in equation (2.6).
2. If Yρ> −ρ Yim(i)
RDi−ρRmiR+ρi, then the magnitude of the LM curve shift is greater than the CC curve shift. This will result in the loan rate decreasing, making dRdρ< 0 in equation (2.6).
‧
curve shift are equal. This will result in the loan rate remaining constant, making dRdρ= 0 in equation (2.6).A unique scenario occurs when Yρ = 0. In this scenario, any monetary policy shift will keep the CC curve constant and shift only the LM curve. Setting Yρ = 0 would result in Yρ > −ρ Yim(i)
RDi−ρRmiR+ρi, which is the same condition as described in the 2nd possible result.
Comparing equation (2.21) and equation (2.17), we can come up with three different scenarios, each showing the range where the shift of the bond rate and loan rate is positive or negative:
1. D m(i)
Figure 2.7. Graphical representation of the shift in the bond rate and loan rate for different values of Yρ
when D m(i)
yρR+m(i)ρy> −ρ Yim(i)
RDi−ρRmiR+ρi.
‧
Figures 2.7, 2.8, and 2.9 show the shifts in the loan rate and bond rate during a monetary policy change at different values of Yρ. As seen in Figures 2.7 and 2.8, there are values of Yρ where the bond rate change and the loan rate change are opposite of each other.
The change in both the bond rate and the loan rate are in opposite directions, with the value of D m(i)
yρR+m(i)ρy and −ρ Yim(i)
RDi−ρRmiR+ρi and the value of Yρ deciding which one increases and which one decreases. In such a case, as the change is opposite of one another, the overall change would be less than if only one type of interest rate were present.
Yρ 0
Figure 2.8. Graphical representation of the shift in the bond rate and the loan rate for different values of Yρ when D m(i)
Figure 2.9. Graphical representation of the shift in the bond rate and the loan rate for different values of Yρ when D m(i)
yρR+m(i)ρy= −ρ Yim(i)
RDi−ρRmiR+ρi.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
16
In Figure 2.9, the change in both the bond rate and the loan rate are in the same direction, with the change either positive or negative depending on the value of Yρ. In this case, as the change is in the same direction, the overall change would be greater than if only one type of interest rate were present.
2.6. Summary
In theory, a simple IS-LM model will have known effects. For example a monetary expansion policy will shift the LM curve to the right, resulting in total output increasing and bond rate decreasing, and vice versa. However, when a CC curve is in place of the IS curve, a change in the monetary policy will shift the LM curve and the new CC curve as well. As shown in the cases described in Figures 2.3, 2.4, 2.5, and 2.6, the overall income will increase, but the magnitude of the change is different depending on the shift magnitudes of the LM and CC curve. As seen in the graphs, Figure 2.4 shows the greatest increase in income, Figure 2.5 the
second-greatest, and Figure 2.3 the least. However, the change in income for Figure 2.6 is undetermined as the change depends solely on the shift magnitude of the LM curve. This needs to be discussed further for the exact change in income and both the bond rate and interest rate to be known. The next section will determine this exact effect using data from Taiwan. This will then describe the situation, from the above results, that applies in Taiwan.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
17