No, the price elasticity of demand (PED) is not the same along a linear demand curve.不，需求价格弹性 (PED) 沿着线性需求曲线并不相同。

A linear demand curve has a constant slope, which means the change in quantity demanded per unit change in price remains constant. However, PED, which measures the percentage change in quantity demanded relative to the percentage change in price, varies along the curve.线性需求曲线具有恒定的斜率，这意味着每单位价格变化的需求量变化保持恒定。然而，衡量需求量相对于价格变化百分比的 PED 沿曲线变化。

Here's why it changes:这就是它改变的原因：

• At higher prices and lower quantities (upper part of the demand curve), the PED is relatively elastic. Small changes in price result in larger percentage changes in quantity demanded because the base quantity is smaller.在较高价格和较低数量（需求曲线的上部）的情况下，PED 相对有弹性。由于基本数量较小，价格的微小变化会导致需求量的百分比变化较大。
• At lower prices and higher quantities (lower part of the demand curve), the PED becomes inelastic. Here, changes in price lead to smaller percentage changes in quantity demanded because the base quantity is larger.当价格较低且数量较多时（需求曲线的下部），PED 变得缺乏弹性。在这里，价格变化导致需求量百分比变化较小，因为基本数量较大。

At the midpoint of a linear demand curve, PED is typically unit elastic, meaning a 1% change in price results in a 1% change in quantity demanded.在线性需求曲线的中点，PED 通常具有单位弹性，这意味着价格变化 1% 会导致需求量变化 1%。

Certainly! Here’s a table where we calculate the Price Elasticity of Demand (PED) using the endpoint method for each interval along a linear demand curve where price goes from 0 to 6, and quantity goes from 6 to 0.当然！在下表中，我们使用端点法计算线性需求曲线上每个区间的需求价格弹性 (PED)，其中价格从 0 到 6，数量从 6 到 0。

Step-by-Step Explanation分步说明

1. Endpoint Method Formula for PED:PED 终点法公式：

   $$\text{PED} = \frac{\left( \frac{Q_2 - Q_1}{Q_2 + Q_1} \right)}{\left( \frac{P_2 - P_1}{P_2 + P_1} \right)}$$

   where $Q_1$ and $Q_2$ are the initial and final quantities, and $P_1$ and $P_2$ are the initial and final prices in each interval.在哪里 $Q_1$ 和 $Q_2$ 是初始数量和最终数量，并且 $P_1$ 和 $P_2$ 是每个区间的初始价格和最终价格。

2. Intervals: We’ll calculate PED for each interval as the price decreases from 6 to 0 and the quantity increases from 0 to 6.间隔：当价格从 6 减少到 0、数量从 0 增加到 6 时，我们将计算每个间隔的 PED。

Table: PED Calculation Using Endpoint Method表：使用端点法的 PED 计算

Analysis of the Table表中分析

• Higher Prices (Intervals 1-2): PED is greater than 1, indicating elastic demand.价格较高（区间 1-2） ：PED 大于 1，表明需求富有弹性。
• Midpoint: Around intervals 2-3, the PED nears 1, suggesting unit elasticity.中点：在区间 2-3 附近，PED 接近 1，表明单位弹性。
• Lower Prices (Intervals 4-6): PED drops below 1, indicating inelastic demand.较低的价格（区间 4-6） ：PED 降至 1 以下，表明需求缺乏弹性。

This confirms that on a linear demand curve, PED is high (elastic) at the top, unit elastic in the middle, and low (inelastic) at the bottom.这证实了在线性需求曲线上，PED 在顶部较高（有弹性），在中间具有单位弹性，在底部较低（无弹性）。

You are absolutely correct! Let’s clarify this:你是绝对正确的！让我们澄清一下：

On a linear demand curve, the Price Elasticity of Demand (PED) behaves differently at the two endpoints:在线性需求曲线上，需求价格弹性 (PED) 在两个端点处表现不同：

1. Intersection with the Price Axis (Vertical Intercept):与价格轴的交点（垂直截距） ：

   • At this point, price is at its maximum, and quantity demanded is zero.此时，价格处于最高点，需求量为零。
   • PED here is negative infinity because a very small percentage change in price would lead to a large percentage change in quantity demanded (from zero to some positive amount).这里的 PED 是负无穷大，因为价格的一个非常小的百分比变化将导致需求量的一个大的百分比变化（从零到某个正数）。

2. Intersection with the Quantity Axis (Horizontal Intercept):与数量轴的交点（水平截距） ：

   • At this point, quantity is at its maximum, and price is zero.此时，数量达到最大，价格为零。
   • PED here is zero because any percentage change in price from zero has no impact on quantity demanded (which remains at its maximum level).这里的 PED为零，因为价格从零开始的任何百分比变化都不会影响需求量（仍保持在最高水平）。

In summary:总之：

• Top of the Demand Curve (Price Axis Intercept): PED is $-\infty$ (perfectly elastic).需求曲线顶部（价格轴截距） ：PED 是 $-\infty$ （完全有弹性）。
• Bottom of the Demand Curve (Quantity Axis Intercept): PED is $0$ (perfectly inelastic).需求曲线底部（数量轴截距） ：PED 是 $0$ （完全无弹性）。

This highlights that along a linear demand curve:这凸显了沿着线性需求曲线：

• PED decreases from negative infinity to zero as we move from the price axis (high price, low quantity) down to the quantity axis (low price, high quantity).当我们从价格轴（高价格，低数量）向下移动到数量轴（低价格，高数量）时，PED 从负无穷减小到零。

Certainly, let’s break down why the Price Elasticity of Demand (PED) varies along a linear demand curve, focusing on why PED is higher (more elastic) at the upper part of the curve and lower (more inelastic) at the lower part of the curve.当然，我们来分析一下为什么需求价格弹性（PED）沿着线性需求曲线变化，重点关注为什么PED在曲线的上部较高（更有弹性）而在曲线下部较低（更缺乏弹性）。曲线。

Key Points to Understand需要理解的要点

1. Definition of PED: PED measures how responsive quantity demanded is to a change in price. The formula is:PED 的定义：PED 衡量需求量对价格变化的反应程度。公式为：

   $$\text{PED} = \frac{\%\ \text{change in quantity demanded}}{\%\ \text{change in price}}$$
   • If PED > 1, demand is elastic: quantity demanded is highly responsive to price changes.如果 PED > 1，则需求具有弹性：需求量对价格变化高度敏感。
   • If PED < 1, demand is inelastic: quantity demanded is less responsive to price changes.如果 PED < 1，则需求缺乏弹性：需求量对价格变化的反应较差。

2. Percentage Change Calculation: PED depends on percentage changes in both price and quantity. A key factor here is the base value used in calculating percentage changes. The base value affects the size of the percentage change: smaller initial quantities lead to larger percentage changes for the same absolute change.百分比变化计算：PED 取决于价格和数量的百分比变化。这里的一个关键因素是计算百分比变化时使用的基值。基值影响百分比变化的大小：对于相同的绝对变化，较小的初始数量会导致较大的百分比变化。

Why the Upper Part of the Curve is More Elastic (PED > 1)为什么曲线的上部更有弹性 (PED > 1)

• Higher Prices, Lower Quantities: At the upper part of the demand curve, prices are high, and quantities are low.价格较高，数量较少：在需求曲线的上部，价格较高，数量较低。
• Impact of a Small Price Change: A small decrease in price from a high level causes a relatively modest percentage change in price.小幅价格变动的影响：价格从高位小幅下跌会导致价格相对适度的百分比变化。
• Large Percentage Change in Quantity: However, because the base quantity demanded is small, even a small increase in quantity represents a large percentage change.数量的大百分比变化：然而，由于需求的基本数量很小，即使数量的小幅增加也代表着大的百分比变化。
• Result: This combination (small percentage price change but large percentage quantity change) makes PED high at the upper part of the curve, so demand is elastic here.结果：这种组合（价格变化百分比较小，但数量变化百分比较大）使得 PED 在曲线的上部较高，因此此处的需求是有弹性的。

For example:例如：

• Suppose price decreases from $10 to $9, a 10% decrease.假设价格从 10 美元下降到 9 美元，下降 10%。
• Quantity demanded increases from 1 unit to 2 units, a 100% increase.需求数量从 1 单位增加到 2 单位，增加 100%。
• Here, PED = $\frac{100\%}{10\%} = 10$, which is highly elastic.这里，PED = $\frac{100\%}{10\%} = 10$ ，具有高弹性。

Why the Lower Part of the Curve is More Inelastic (PED < 1)为什么曲线的下部更缺乏弹性 (PED < 1)

• Lower Prices, Higher Quantities: At the lower part of the curve, prices are low, and quantities demanded are high.价格较低，数量较多：在曲线的下部，价格较低，需求量较高。
• Impact of a Price Change: A change in price (say, a $1 decrease) from a low price level represents a larger percentage change in price.价格变化的影响：价格从低价格水平的变化（例如，下降 1 美元）代表价格的较大百分比变化。
• Smaller Percentage Change in Quantity: Since the base quantity is large, even a change in quantity represents a relatively small percentage change.数量变化百分比较小：由于基本数量较大，因此即使数量变化也代表相对较小的百分比变化。
• Result: Here, PED is lower, making demand inelastic at the lower part of the curve.结果：此处，PED 较低，使得曲线下部的需求缺乏弹性。

For example:例如：

• Suppose price decreases from $2 to $1, a 50% decrease.假设价格从 2 美元下降到 1 美元，下降 50%。
• Quantity demanded increases from 5 units to 6 units, a 20% increase.需求数量从 5 单位增加到 6 单位，增加了 20%。
• Here, PED = $\frac{20\%}{50\%} = 0.4$, which is inelastic.这里，PED = $\frac{20\%}{50\%} = 0.4$ ，这是无弹性的。

Summary概括

• At high prices and low quantities (upper part of the curve), small price changes lead to large percentage changes in quantity, making demand elastic.在高价格和低数量（曲线的上部）的情况下，较小的价格变化会导致数量的较大百分比变化，从而使需求具有弹性。
• At low prices and high quantities (lower part of the curve), price changes lead to smaller percentage changes in quantity, making demand inelastic.在低价格和高数量（曲线的下部）的情况下，价格变化导致数量变化的百分比较小，从而使需求缺乏弹性。

This explains why PED declines as you move down a linear demand curve from high to low prices.这就解释了为什么当您沿着线性需求曲线从高价格向低价格移动时，PED 会下降。

Let’s use calculations to understand why PED is $-\infty$ at the price-axis intercept and $0$ at the quantity-axis intercept of a linear demand curve.让我们用计算来理解为什么 PED 是这样的 $-\infty$ 在价格轴截距处和 $0$ 在线性需求曲线的数量轴截距处。

1. Intersection with the Price Axis (Vertical Intercept)1. 与价格轴的交点（垂直截距）

• At this point: The price is at its maximum, and quantity demanded is zero.此时：价格处于最高点，需求量为零。
• Explanation: If we reduce the price slightly from this high level, the quantity demanded moves from zero to a small positive amount.解释：如果我们将价格从这个高位稍微降低，需求量就会从零变为一个小的正数。

Let’s break down the PED calculation here:让我们在这里分解一下 PED 计算：

Scenario Setup:场景设置：

Suppose the demand curve intersects the price axis at $P = P_{\text{max}}$ when quantity demanded $Q = 0$.假设需求曲线与价格轴相交于 $P = P_{\text{max}}$ 当数量需求时 $Q = 0$ 。

Small Decrease in Price价格小幅下降

Let’s reduce the price by a small amount, say $\Delta P$, causing the quantity demanded to increase by a small amount, $\Delta Q$.让我们稍微降低价格吧 $\Delta P$ ，导致需求量小幅增加， $\Delta Q$ 。

PED Calculation at the Vertical Intercept垂直截距处的 PED 计算

Using the formula for PED:使用 PED 公式：

1. Percentage Change in Quantity: Since the original quantity demanded is $Q = 0$, any increase from zero leads to an undefined or extremely large percentage increase in quantity (essentially approaching infinity).数量变化百分比：由于原始需求数量为 $Q = 0$ ，任何从零开始的增加都会导致数量的未定义或极大的百分比增加（基本上接近无穷大）。
2. Percentage Change in Price: A small change in price, $\Delta P$, from a high initial price, $P_{\text{max}}$, leads to a relatively small percentage change in price.价格变化百分比：价格的微小变化， $\Delta P$ ，从较高的初始价格， $P_{\text{max}}$ ，导致价格相对较小的百分比变化。

Thus:因此：

This is why PED at the price-axis intercept is $-\infty$, indicating perfectly elastic demand at that point.这就是为什么价格轴截距处的 PED 是 $-\infty$ ，表明此时需求完全弹性。

2. Intersection with the Quantity Axis (Horizontal Intercept)2. 与数量轴的交点（水平截距）

• At this point: The price is zero, and quantity demanded is at its maximum, say $Q_{\text{max}}$.此时：价格为零，需求量达到最大，例如 $Q_{\text{max}}$ 。
• Explanation: Any change in price from zero does not lead to any change in quantity demanded, as it is already at its maximum.解释：价格从零开始的任何变化都不会导致需求量发生任何变化，因为需求量已经达到最大值。

Scenario Setup:场景设置：

Suppose the demand curve intersects the quantity axis at $Q = Q_{\text{max}}$ when $P = 0$.假设需求曲线与数量轴相交于 $Q = Q_{\text{max}}$ 什么时候 $P = 0$ 。

Small Increase in Price价格小幅上涨

Let’s increase the price by a small amount, say $\Delta P$, from zero.让我们稍微提高价格，比如说 $\Delta P$ ，从零开始。

PED Calculation at the Horizontal Intercept水平截距处的 PED 计算

Using the formula for PED:使用 PED 公式：

1. Percentage Change in Price: Since the initial price is $P = 0$, any increase in price from zero leads to an undefined or extremely large percentage increase in price (similar to dividing by zero).价格变化百分比：由于初始价格为 $P = 0$ ，任何从零开始的价格上涨都会导致价格出现未定义或极大的百分比上涨（类似于除以零）。
2. Percentage Change in Quantity: However, quantity demanded stays at $Q_{\text{max}}$, meaning there is no percentage change in quantity demanded.数量百分比变化：但是，需求数量保持不变 $Q_{\text{max}}$ ，意味着需求量没有百分比变化。

Thus:因此：

This is why PED at the quantity-axis intercept is $0$, indicating perfectly inelastic demand at that point.这就是为什么数量轴截距处的 PED 为 $0$ ，表明此时需求完全缺乏弹性。