The multiplier definitions
Intros
Lessons
- The Multiplier Definitions
- Change in equilibrium and autonomous expenditure
- Multiplier is always greater than 1
- Why Multiplier> 1?
- Relationship Between the Multiplier & Slope of AE Curve
- Formula for Change in Real GDP
- Formula for Slope of curve
- Algebraic Manipulation
- The Multiplier Applications
- Import Taxes
- Income Taxes
- The Multiplier becomes smaller
Topic Notes
Introduction to the Multiplier Effect in Economics
The multiplier effect is a fundamental concept in economics that explains how an initial change in spending can lead to a larger overall impact on the economy. Our introduction video provides a clear and concise explanation of this crucial economic principle. By watching this video, viewers will gain a solid understanding of how the multiplier works and its significance in economic analysis. The multiplier effect plays a vital role in shaping economic policies and predicting their outcomes. It demonstrates how government spending, investments, or changes in consumer behavior can have amplified effects on economic growth, employment, and overall economic activity. Understanding the multiplier is essential for policymakers, economists, and students alike, as it helps in evaluating the potential impacts of various economic interventions. This knowledge enables more informed decision-making and better forecasting of economic growth trends, making it an indispensable tool in the field of economics.
Understanding the Aggregate Expenditure (AE) Curve
The Aggregate Expenditure (AE) curve is a fundamental concept in macroeconomics that represents the total planned spending in an economy at different levels of income. This curve is crucial for understanding how changes in spending patterns can affect overall economic activity. The AE curve consists of four main components: consumption (C), investment (I), government spending (G), and net exports (X-M).
Consumption, the largest component, represents household spending on goods and services. Investment refers to business spending on capital goods and inventory. Government spending includes all expenditures by federal, state, and local governments. Net exports are the difference between exports and imports, reflecting the international trade balance.
The AE curve can shift due to various factors, and these shifts have significant implications for the economy. A shift in the AE curve occurs when there's a change in spending that is not caused by a change in income. These shifts can be either upward (indicating an increase in aggregate expenditure) or downward (indicating a decrease).
One of the key factors that can cause shifts in the AE curve is changes in autonomous expenditures. Autonomous expenditures are spending components that are independent of income levels. They form the vertical intercept of the AE curve and play a crucial role in determining its position.
Examples of autonomous expenditures include:
- Fixed investment plans by businesses
- Certain types of government spending
- Some forms of consumption that occur regardless of income (e.g., basic necessities)
When autonomous expenditures increase, the entire AE curve shifts upward. For instance, if the government implements a new infrastructure project, this would increase autonomous government spending, shifting the AE curve upward. Conversely, if businesses reduce their planned investments due to economic uncertainty, this would decrease autonomous investment, shifting the AE curve downward.
Changes in consumer confidence can also lead to shifts in the AE curve. If consumers become more optimistic about the future, they might increase their autonomous consumption, shifting the curve upward. On the other hand, a decrease in consumer confidence could lead to reduced spending and a downward shift in the curve.
Another factor that can cause shifts in the AE curve is changes in wealth. An increase in household wealth, perhaps due to rising stock market values or property prices, can lead to higher consumption levels at all income levels, shifting the AE curve upward.
Changes in tax policies can also impact the AE curve. A reduction in income tax rates, for example, would increase disposable income and likely lead to higher consumption levels, shifting the curve upward. Conversely, an increase in taxes could lead to a downward shift in the curve.
It's important to note that shifts in the AE curve can have multiplier effects on the economy. Due to the interconnected nature of economic activities, an initial change in autonomous expenditures can lead to a larger overall change in aggregate expenditure and income.
For example, if the government increases autonomous spending by $100 billion on a new infrastructure project, this initial increase will create jobs and income for workers and businesses involved in the project. These individuals and businesses will then spend some of this additional income, creating further increases in expenditure and income throughout the economy. This multiplier effect amplifies the initial change in autonomous expenditure.
Understanding the AE curve and its shifts is crucial for policymakers and economists in analyzing and predicting economic outcomes. By manipulating components of aggregate expenditure, particularly autonomous expenditures, governments can attempt to influence overall economic activity and achieve desired macroeconomic objectives such as economic growth and stability.
The Multiplier: Definition and Calculation
The multiplier is a crucial concept in economics that describes how changes in autonomous expenditure lead to larger changes in overall economic output. This principle is fundamental to understanding how government policies, investments, or other economic shocks can have amplified effects on an economy's total income or output.
In its simplest form, the multiplier definition refers to the factor by which changes in autonomous expenditure produce changes in total national income. Autonomous expenditure includes components of spending that are not directly influenced by income levels, such as government spending, investments, or exports. The multiplier effect occurs because an initial increase in spending leads to increased income for some individuals or businesses, which in turn leads to further spending, creating a ripple effect throughout the economy.
The formula for calculating the multiplier is:
Multiplier = 1 / (1 - MPC)
Where MPC is the Marginal Propensity to Consume, which represents the proportion of additional income that is spent rather than saved. The MPC is always a value between 0 and 1.
Let's walk through an example calculation to illustrate how the multiplier works:
Suppose the MPC in an economy is 0.8, meaning that for every additional dollar of income, people spend 80 cents and save 20 cents. Using the formula:
Multiplier = 1 / (1 - 0.8) = 1 / 0.2 = 5
This means that for every $1 increase in autonomous expenditure, the total increase in national income will be $5.
The relationship between the multiplier and changes in equilibrium expenditure is direct and proportional. The change in equilibrium expenditure is equal to the initial change in autonomous expenditure multiplied by the multiplier. For instance, if autonomous expenditure increases by $100 million, and the multiplier is 5, the total change in equilibrium expenditure would be:
Change in Equilibrium Expenditure = $100 million × 5 = $500 million
It's crucial to emphasize that the multiplier is always greater than one. This is a fundamental characteristic that makes the multiplier effect so significant in economics. The reason for this is rooted in the formula itself. Since the MPC is always between 0 and 1, the denominator (1 - MPC) will always be less than 1, resulting in a multiplier greater than 1 when we divide 1 by this value.
The fact that the multiplier is greater than one is important for several reasons:
1. Amplification of Economic Policies: It means that government policies aimed at stimulating the economy can have a larger impact than the initial investment or spending change. This is why fiscal policy can be an effective tool for economic management.
2. Economic Stability: The multiplier effect can help stabilize economies during downturns by amplifying the effects of stimulus measures.
3. Investment Decisions: For businesses and policymakers, understanding the multiplier effect can inform decisions about investments or policy changes, as they can anticipate a larger overall economic impact.
4. Risk Assessment: The multiplier also implies that negative shocks to the economy can have amplified negative effects, which is important for risk assessment and economic forecasting.
However, it's important to note that the actual size of the multiplier can vary depending on economic conditions and the specific nature of the expenditure change. Factors such as the openness of the economy, the state of the business cycle, and the effectiveness of monetary policy can all influence the magnitude of the multiplier effect.
In conclusion, the multiplier is a powerful concept in economics that helps explain how changes in autonomous expenditure can lead to disproportionately larger changes in overall economic output. By understanding the multiplier effect and how to calculate it, economists, policymakers, and business leaders can better predict and influence economic outcomes, making it an essential tool for economic analysis and decision-making.
Relationship Between the Slope of the AE Curve and the Multiplier
The relationship between the slope of the Aggregate Expenditure (AE) curve and the multiplier is a fundamental concept in macroeconomics that helps explain how changes in spending can have amplified effects on overall economic output. This relationship is crucial for understanding the dynamics of economic growth and the impact of fiscal policies.
The slope of the AE curve represents the marginal propensity to consume (MPC), which is the proportion of additional income that is spent rather than saved. The multiplier, on the other hand, quantifies the total change in national income resulting from an initial change in spending. The mathematical relationship between these two concepts is inverse and can be expressed through a simple formula:
Multiplier = 1 / (1 - MPC)
To derive this formula, we start with the basic equation for equilibrium in the goods market:
Y = C + I + G + (X - M)
Where Y is national income, C is consumption, I is investment, G is government spending, X is exports, and M is imports. Focusing on consumption, we can express it as a function of income:
C = a + bY
Here, 'a' is autonomous consumption, and 'b' is the MPC (slope of the AE curve). Substituting this into the equilibrium equation:
Y = (a + bY) + I + G + (X - M)
Solving for Y:
Y - bY = a + I + G + (X - M)
Y(1 - b) = a + I + G + (X - M)
Y = (a + I + G + (X - M)) / (1 - b)
The term 1 / (1 - b) is the multiplier, where 'b' is the MPC or slope of the AE curve.
To illustrate this relationship, let's consider some examples:
1. If the MPC (slope of AE curve) is 0.8, the multiplier would be 1 / (1 - 0.8) = 5. This means that for every $1 increase in autonomous spending, national income would increase by $5.
2. If the MPC decreases to 0.6, the multiplier becomes 1 / (1 - 0.6) = 2.5. Now, a $1 increase in spending would only lead to a $2.50 increase in national income.
3. Conversely, if the MPC rises to 0.9, the multiplier jumps to 1 / (1 - 0.9) = 10, amplifying the effect of spending changes significantly.
The economic implications of this relationship are profound. A higher slope of the AE curve (higher MPC) leads to a larger multiplier effect, meaning that changes in autonomous spending have a more significant impact on the overall economy. This can be beneficial during economic downturns, as stimulus measures can be more effective. However, it also means that negative shocks can have more severe consequences.
Policymakers must consider this relationship when designing fiscal policies. In economies with high MPCs, government spending or tax cuts can be particularly effective in stimulating growth. Conversely, in economies with lower MPCs, such policies may have more muted effects, requiring larger interventions to achieve the desired impact.
The slope of the AE curve and the resulting multiplier also influence the stability of an economy. A higher multiplier can lead to more volatile economic cycles, as small changes in spending can cause larger fluctuations in output. This volatility can make economic planning and forecasting more challenging for both businesses and policymakers.
Understanding this relationship is crucial for interpreting economic data and predicting the outcomes of policy decisions. It helps explain why some economies may be more responsive to fiscal stimuli than others and why the effectiveness of economic policies can vary across different countries or time periods.
Impact of Imports and Income Taxes on the Multiplier
In macroeconomics, the multiplier effect plays a crucial role in understanding how changes in spending can have a magnified impact on overall economic output. However, this effect is not constant and can be significantly influenced by factors such as imports and income taxes. This section will explore how these elements affect the multiplier, focusing on the concepts of marginal propensity to import and marginal tax rate.
Imports and the Multiplier:
The marginal propensity to import (MPI) refers to the proportion of additional income that is spent on imported goods and services. As the MPI increases, it leads to a smaller multiplier effect. This occurs because a portion of the increased spending "leaks" out of the domestic economy and benefits foreign producers instead. For example, if the MPI is 0.2, it means that for every additional dollar of income, 20 cents is spent on imports. This reduces the amount of money circulating within the domestic economy, thereby diminishing the multiplier effect.
Income Taxes and the Multiplier:
Similarly, income taxes can also reduce the multiplier effect. The marginal tax rate is the rate of tax applied to the last dollar of income earned. As the marginal tax rate increases, it reduces the amount of disposable income available for consumption or investment. This, in turn, leads to a smaller multiplier. For instance, if the marginal tax rate is 30%, then for every additional dollar earned, only 70 cents is available for spending, which reduces the potential for further economic stimulation.
Combined Effect on the Multiplier:
When we consider both imports and income taxes together, their combined effect can significantly reduce the multiplier. The formula for the multiplier can be adjusted to account for these factors:
Multiplier = 1 / (1 - MPC + MPI + MTR)
Where:
- MPC is the Marginal Propensity to Consume
- MPI is the Marginal Propensity to Import
- MTR is the Marginal Tax Rate
As both MPI and MTR increase, the denominator of this fraction becomes larger, resulting in a smaller multiplier.
Graphical Representation:
The effect of imports and income taxes on the multiplier can be illustrated graphically using the Aggregate Expenditure (AE) curve. The AE curve represents the total planned spending in an economy at different levels of income. When the MPI or MTR increases, it causes the AE curve to become flatter, or less steep. This change in slope directly affects the multiplier.
For example, consider an economy with an initial MPC of 0.8, no imports, and no taxes. The AE curve would be relatively steep, resulting in a larger multiplier. Now, introduce an MPI of 0.2 and an MTR of 0.3. The new AE curve would be noticeably flatter. This flatter curve intersects the 45-degree line (which represents the point where AE equals income) at a lower point, indicating a smaller increase in equilibrium income for a given increase in autonomous spending.
Practical Implications:
Understanding how imports and income taxes affect the multiplier is crucial for policymakers and economists. In an increasingly globalized economy, where international trade plays a significant role, the impact of imports on the multiplier becomes particularly relevant. Countries with high levels of imports may find that their fiscal stimulus measures are less effective due to the leakage effect.
Similarly, the structure of a country's tax system can have profound implications for the effectiveness of economic policies. Higher marginal tax rates, while potentially necessary for fiscal reasons, can dampen the multiplier effect and reduce the impact of expansionary fiscal policies.
Conclusion:
In conclusion, both imports and income taxes have a dampening effect on the multiplier. The marginal propensity to import and the marginal tax rate are key factors that reduce the potential amplification of economic stimuli. As these factors increase,
Practical Applications of the Multiplier in Economic Policy
The multiplier concept plays a crucial role in economic policy-making, particularly in the realm of fiscal policy. Governments and policymakers leverage their understanding of the multiplier effect to design and implement strategies aimed at stimulating economic growth, managing recessions, and promoting overall economic stability. This powerful tool allows decision-makers to estimate the potential impact of various fiscal measures on the broader economy.
One of the most common applications of the multiplier effect in economic policy is through government spending initiatives. When faced with economic downturns, governments often increase public expenditure to boost aggregate demand. For instance, infrastructure projects not only create immediate jobs but also generate additional economic activity through increased spending by workers and suppliers. The multiplier effect suggests that the total impact on the economy will be greater than the initial government investment, as the money circulates through various sectors.
Tax policies are another area where the multiplier concept is applied. Governments may implement tax cuts or rebates to increase disposable income, expecting that consumers will spend a portion of this additional money, thereby stimulating economic activity. The multiplier effect implies that the initial increase in consumer spending will lead to further rounds of spending, amplifying the overall economic impact.
During recessions, policymakers often use their understanding of the multiplier to design targeted stimulus packages. For example, unemployment benefits are not only a social safety net but also an economic stabilizer. The multiplier effect suggests that these benefits will be spent quickly, supporting local businesses and potentially preventing further job losses.
However, the application of the multiplier concept in policy-making is not without limitations and criticisms. One major challenge is accurately estimating the size of the multiplier for different types of fiscal measures. The effectiveness of policies can vary depending on economic conditions, consumer behavior, and the specific nature of the intervention. Critics argue that overreliance on multiplier estimates may lead to overly optimistic projections of policy outcomes.
Another limitation is the potential for crowding out private investment. If government spending increases significantly, it may lead to higher interest rates, potentially discouraging private sector investment and partially offsetting the intended stimulative effect. Additionally, the multiplier effect may be reduced in open economies where a significant portion of the additional spending leaks into imports rather than boosting domestic production.
The time lag between policy implementation and its effects is another critical consideration. The full impact of fiscal measures may take time to materialize, making it challenging for policymakers to respond quickly to economic shocks. This delay can sometimes result in policies being implemented when they are no longer necessary or even counterproductive.
Critics also point out that the multiplier concept may oversimplify complex economic relationships. It doesn't always account for structural issues in the economy or long-term consequences of fiscal interventions, such as increased public debt. There's also debate about the sustainability of growth driven by government spending and whether it can lead to lasting economic improvements.
Despite these limitations, the multiplier concept remains a valuable tool in the policymaker's arsenal. When used judiciously and in conjunction with other economic indicators and models, it can provide valuable insights into the potential impacts of fiscal policies. As economies evolve and new challenges emerge, the application of the multiplier effect in policy-making continues to be refined, reflecting the ongoing quest for effective and responsible economic governance.
Conclusion: The Importance of Understanding the Multiplier
The multiplier effect plays a crucial role in economic analysis, as demonstrated by its relationship to the AE curve. Our introduction video provides a comprehensive explanation of these concepts, highlighting their significance in understanding economic dynamics. The multiplier effect shows how initial changes in spending can lead to larger overall economic impacts. This relationship is visually represented in the AE curve, illustrating how changes in autonomous spending affect equilibrium output. By grasping these concepts, you'll be better equipped to analyze economic policies and their potential ripple effects throughout the economy. We encourage you to apply this knowledge when examining real-world economic scenarios and policy decisions. To deepen your understanding, consider exploring additional resources on economic multipliers and their applications. Remember, a solid grasp of the multiplier effect is essential for anyone looking to comprehend and predict economic outcomes effectively. Take the next step in your economic education by further studying these crucial concepts.
The Multiplier Definitions
The Multiplier Definitions
- Change in equilibrium and autonomous expenditure
- Multiplier is always greater than 1
- Why Multiplier > 1?
Step 1: Understanding the AE Curve
The AE (Aggregate Expenditure) curve represents the total spending in an economy at different levels of real GDP. It can shift up or down based on changes in autonomous expenditures, such as investments and government spending. These shifts in the AE curve lead to changes in the equilibrium expenditure, which is the point where the AE curve intersects the 45-degree line representing equality between aggregate expenditure and real GDP.
Step 2: Introduction to the Multiplier
The multiplier is a concept that measures the magnitude of change in equilibrium expenditure in response to a change in autonomous expenditure. It is calculated as the change in equilibrium expenditure divided by the change in autonomous expenditure. For example, if the multiplier is 2 and the autonomous expenditure changes by 2, the equilibrium expenditure will change by 4.
Step 3: Calculating the Multiplier
To calculate the multiplier, use the formula:
Multiplier = Change in Equilibrium Expenditure / Change in Autonomous Expenditure
For instance, if the equilibrium expenditure changes from 4 trillion to 8 trillion real GDP due to a change in autonomous expenditure from 2 to 4 trillion, the change in equilibrium expenditure is 4 trillion, and the change in autonomous expenditure is 2 trillion. Thus, the multiplier is 4 / 2 = 2.
Step 4: Why the Multiplier is Greater than 1
The multiplier is always greater than 1 because any initial change in autonomous expenditure leads to a more than proportional change in equilibrium expenditure. This is due to the ripple effect in the economy, where an initial increase in spending leads to increased income, which in turn leads to further spending and income generation. Therefore, a multiplier less than 1 would not make sense as it would imply that the economy contracts with increased spending.
Step 5: Practical Example
Consider an initial equilibrium expenditure of 4 trillion real GDP at point A on the AE curve. If autonomous expenditure increases by 2 trillion, shifting the AE curve up, the new equilibrium expenditure might be 8 trillion real GDP at point B. The change in equilibrium expenditure is 4 trillion, and the change in autonomous expenditure is 2 trillion, resulting in a multiplier of 2 (4 / 2 = 2).
Step 6: Importance of the Multiplier
The multiplier is crucial for understanding the impact of fiscal policy and other economic interventions. It helps economists and policymakers predict the effects of changes in government spending, investments, and other autonomous expenditures on the overall economy. By knowing the multiplier, they can estimate the potential increase in real GDP and make informed decisions to stimulate economic growth.
FAQs
Here are some frequently asked questions about the multiplier effect and related concepts:
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What is the multiplier effect in economics?
The multiplier effect is an economic concept that describes how an initial change in spending can lead to a larger overall impact on the economy. It shows that the total increase in national income can be greater than the initial change in expenditure due to the ripple effect of spending throughout the economy.
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How is the multiplier calculated?
The multiplier is calculated using the formula: Multiplier = 1 / (1 - MPC), where MPC is the Marginal Propensity to Consume. For example, if the MPC is 0.8, the multiplier would be 1 / (1 - 0.8) = 5, meaning a $1 increase in spending could lead to a $5 increase in national income.
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How do imports and taxes affect the multiplier?
Imports and taxes reduce the multiplier effect. The Marginal Propensity to Import (MPI) and the Marginal Tax Rate (MTR) cause "leakages" from the circular flow of income, decreasing the overall impact of initial spending changes. The adjusted formula becomes: Multiplier = 1 / (1 - MPC + MPI + MTR).
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What is the relationship between the Aggregate Expenditure (AE) curve and the multiplier?
The slope of the AE curve is directly related to the multiplier. A steeper AE curve indicates a higher Marginal Propensity to Consume (MPC), which results in a larger multiplier. Conversely, a flatter AE curve suggests a lower MPC and a smaller multiplier effect.
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How do policymakers use the multiplier concept in economic decision-making?
Policymakers use the multiplier concept to estimate the potential impact of fiscal policies such as government spending or tax cuts. Understanding the multiplier helps them design more effective economic stimulus packages during recessions or plan for the broader economic effects of policy changes. However, they must also consider factors like time lags and potential crowding out effects when applying this concept.
Prerequisite Topics
Understanding the foundations of economics is crucial when delving into more advanced concepts like "The multiplier definitions." One of the key prerequisite topics that plays a significant role in grasping the multiplier effect is market equilibrium. This fundamental concept serves as a cornerstone for comprehending how changes in various economic factors can have amplified effects on the overall economy.
Market equilibrium, which refers to the state where supply and demand are balanced, is essential for understanding the multiplier effect. When we talk about the multiplier definitions, we're essentially exploring how initial changes in spending can lead to larger changes in economic output. This relationship is intrinsically linked to the concept of equilibrium in the goods market.
To fully appreciate the multiplier effect, one must first grasp how markets reach equilibrium. In a state of market equilibrium, the quantity of goods supplied matches the quantity demanded at a specific price point. This balance is crucial because it sets the stage for understanding how external factors can disrupt this equilibrium and trigger a chain reaction of economic events.
The multiplier definitions build upon this foundation by explaining how an initial change in spending can lead to a more significant change in national income. For instance, when government spending increases, it doesn't just affect the immediate recipients of that spending. Instead, it creates a ripple effect throughout the economy, as those recipients then increase their own spending, and so on. This cascading effect is what we refer to as the multiplier.
By understanding market equilibrium, students can better visualize how this initial disruption to the equilibrium state can propagate through the economy. It helps in comprehending why a small change in one sector can lead to much larger changes in overall economic output. This relationship between initial change and final impact is at the heart of the multiplier definitions.
Moreover, the concept of equilibrium in the goods market is crucial for grasping how the multiplier effect can vary in different economic conditions. In a perfectly competitive market at equilibrium, the multiplier effect might work differently compared to a market with imperfections or one that is not at equilibrium. This understanding allows for a more nuanced analysis of economic policies and their potential impacts.
In conclusion, mastering the prerequisite topic of market equilibrium is essential for students aiming to fully comprehend the multiplier definitions. It provides the necessary context for understanding how economic changes propagate and amplify, forming the basis for more complex economic analyses and policy decisions. By building on this foundational knowledge, students can develop a more comprehensive understanding of macroeconomic principles and their real-world applications.
The curve can be shifted up? How? By the increase of autonomous expenditure.
If the curve shifts up, then the equilibrium expenditure changes. How can we find the new equilibrium?
We need to introduce a new concept.
Multiplier: shows the magnitude in how much equilibrium expenditure has changed in proportion with the change in autonomous expenditure.
In other words,
The multiplier is important because it helps us see the magnitude of the increase in the curve.
Note: The multiplier is always greater than 1. This is because the change in equilibrium expenditure is always bigger than the change in autonomous expenditure.
Suppose we have the following graph
Notice that:
- Equilibrium expenditure is at point
- The change in autonomous expenditure shifts the curve up by 2 trillion
- The new equilibrium expenditure is at point
- The change in equilibrium expenditure is 4 trillion
- 2, which is greater than 1
Relationship Between the Multiplier & Slope of AECurve
If we know the slope of the curve, then we can also find the multiplier. How?
Recall that the change in real GDP is due to the changes in both induced expenditure and autonomous expenditure. In other words,
Where:
= change in real GDP
= change in induced expenditure
= autonomous expenditure
We also know the slope of the curve as
Substituting this to the other equation gives
Now dividing both sides by gives us
Example: If the slope of the curve is 0.5, what is the multiplier?
Note: There are other versions of the formula for the multiplier.
The Multiplier Applications
Imports and income taxes impacts the size of the multiplier. In fact, they make the multiplier smaller. How?
Recall that the multiplier is:
Notice that the smaller the slope of AE, the smaller the multiplier is. Using mathematical formulas, we can find that the slope of the curve is
Substituting this to our multiplier, we have
Where:
= marginal propensity to consume
= marginal tax rate
= marginal propensity to import
Therefore, the multiplier can be small if:
- is small
- (marginal tax rate) is large
- (marginal propensity to import) is large
Example: Find the multiplier if , , . Then find the multiplier when , , . What difference do you see?
The multiplier is smaller when there is marginal tax rate and marginal propensity to import.
Note: The smaller the multiplier is from import and income taxes, the less steep the slope of the AE curve is. This reduces the value of the multiplier, which makes the equilibrium expenditure lower.
The slope of the curve here is , and the multiplier is 2.
The slope of the curve here is , and the multiplier is 3.