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Chapter 11: Problem 7

This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Show that, for any linear demand curve, the vertical distance between the demand and marginal revenue curves is \(-1 / b \cdot q\) where \(b(<0)\) is the slope of the demand curve. c. Show that, for a constant elasticity demand curve of the form \(q=a P^{b}\), the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. d. Show that, for any downward-sloping demand curve, the vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results of parts (a)-(d) of this problem.

Short Answer

Expert verified
Answer: The marginal revenue curve bisects the distance between the vertical axis and the demand curve at any price for a linear demand curve because they have the same intercept on the price axis, but the marginal revenue curve has twice the slope of the demand curve. When the marginal revenue is zero, the quantity is halfway between the origin and the point on the demand curve, resulting in the vertical distance being equally divided between the demand curve and the marginal revenue curve.

Step by step solution

01

Assume a linear demand curve

Assume a linear demand curve in the form: $$ P = a - b \cdot q $$ where $$P$$ is the price, $$q$$ is the quantity demanded, and $$a$$ and $$b$$ are positive constants (with $$b>0$$).
02

Calculate Total Revenue

Total Revenue ($$TR$$) is the product of price ($$P$$) and quantity ($$q$$): $$ TR = P \cdot q = (a - b \cdot q) \cdot q $$
03

Derive Marginal Revenue

Marginal Revenue ($$MR$$) is the derivative of Total Revenue ($$TR$$) with respect to the quantity ($$q$$): $$ MR = \frac{d(TR)}{dq} = \frac{d((a - b \cdot q) \cdot q)}{dq} $$ Calculating the derivative, we get: $$ MR = a - 2b \cdot q $$ The marginal revenue curve is a straight line with the same intercept as the demand curve and twice the slope. To show that the marginal revenue curve bisects the distance between the vertical axis and the demand curve at any price, let's find the quantity at which the marginal revenue is zero: $$ 0 = a - 2b \cdot q \Rightarrow q = \frac{a}{2b} $$ Plugging this back into the demand curve, we find the price at this point: $$ P = a - b \cdot q = a - b \cdot \frac{a}{2b} = \frac{a}{2} $$ At this point, the marginal revenue curve goes halfway between the vertical axis and the demand curve. b. Calculating the vertical distance between demand and marginal revenue curves for a linear demand curve.
04

Calculate the difference in the slopes

For a linear demand curve, the difference in slopes between the demand curve and the marginal revenue curve is given by $$b$$.
05

Calculate the vertical distance

The vertical distance between the demand and marginal revenue curves is: $$ \Delta P = \frac{1}{b} \cdot q $$ c. Analyze the relationship between demand and marginal revenue curves for a constant elasticity demand curve.
06

Assume a constant elasticity demand curve

Assume a constant elasticity demand curve in the form: $$ q = a \cdot P^{b} $$ where $$a$$ is a positive constant and $$b$$ is the price elasticity of demand, which is negative for a downward-sloping demand curve.
07

Calculate Total Revenue

Total Revenue ($$TR$$) is the product of price ($$P$$) and quantity ($$q$$): $$ TR = P \cdot q = P \cdot (a \cdot P^{b}) $$
08

Derive Marginal Revenue

Marginal Revenue ($$MR$$) is the derivative of Total Revenue ($$TR$$) with respect to the price ($$P$$): $$ MR = \frac{d(TR)}{dP} = \frac{d(P \cdot (a \cdot P^{b}))}{dP} $$ Calculating the derivative, we get: $$ MR = a \cdot P^{b} + b \cdot a \cdot P^{b-1} \cdot P $$ This shows that the vertical distance between the demand and marginal revenue curves is a constant ratio depending on price elasticity $$b$$. d. Finding the vertical distance using a linear approximation.
09

Linear approximation

By using a linear approximation to the demand curve, we can calculate the vertical distance as we did in part (b). The vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results.
10

Linear demand curve and marginal revenue curve

Graph the linear demand curve ($$P = a - b \cdot q$$) and the marginal revenue curve ($$MR = a - 2b \cdot q $$) to show their relationship, and the vertical distance between them.
11

Constant elasticity demand curve and marginal revenue curve

Graph the constant elasticity demand curve ($$q = a \cdot P^{b}$$) and the corresponding marginal revenue curve ($$MR = a \cdot P^{b} + b \cdot a \cdot P^{b-1} \cdot P$$) to show their relationship, and the vertical distance between them.
12

Downward-sloping demand curve and marginal revenue curve

Graph any downward-sloping demand curve and its respective linear approximation, and the corresponding marginal revenue curve found using the linear approximation. Show the vertical distance between them.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Linear Demand Curve
When studying economics, the concept of a linear demand curve is fundamental. It's depicted as a straight line representing the relationship between the quantity demanded and the price of a product. This line slopes downward, reflecting the law of demand: as price increases, the quantity demanded typically decreases. The equation representing a linear demand curve is \( P = a - b \cdot q \), where \( P \) denotes price, \( q \) denotes quantity demanded, and \( a \) and \( b \) are constants that indicate the curve's position and steepness, respectively.

One of the primary characteristics of the linear demand curve is that the marginal revenue (MR) curve associated with it has the same \( y \)-intercept but is steeper. Specifically, the slope of the MR curve is double that of the demand curve. The consequence is that for any given price, the marginal revenue curve dissects the distance between the price on the demand curve and the vertical axis, illustrating that the rate at which revenue increases with each additional unit sold diminishes as more is sold.

For students visualizing the graph of a linear demand curve and its corresponding MR curve, it is apparent that the MR curve will eventually cross the horizontal axis, indicating a point where additional sales begin to reduce total revenue. This cross-section is crucial for businesses in determining optimal pricing and output levels.
Constant Elasticity Demand Curve Explained
The constant elasticity demand curve, unlike the linear model, isn't represented by a straight line but instead by a curve that shows a constant price elasticity of demand. Price elasticity of demand measures how much the quantity demanded responds to a change in price, a key concept in economics affecting pricing strategies and revenue analysis.

The formula for a constant elasticity demand curve is \( q = a \cdot P^{b} \), where \( q \) denotes the quantity demanded, \( P \) is the price, \( a \) is a positive constant, and \( b \) embodies the elasticity of demand. Here, \( b \) is negative to signify that the relationship between price and quantity demanded is typically inversely related. Important to note, this type of demand curve shows a proportional change in quantity demanded for a proportional change in price at any point on the curve, meaning elasticity is constant.

For businesses, understanding the implications of price changes on demand when the demand curve has constant elasticity can be critical for decision-making. It enables the prediction of revenue outcomes from various pricing strategies, which is cross-sectional to understanding how consumers will react to price variations within a market.
Price Elasticity of Demand Demystified
The price elasticity of demand is a nuanced concept that measures the responsiveness of the amount of a good or service demanded to a change in its price. It's an essential concept for both microeconomic theory and practical business application, gauging consumer behavior and the flexibility of the market.

Mathematically, it is expressed as the percentage change in quantity demanded divided by the percentage change in price. When the price elasticity of demand is high, consumers are sensitive to price changes, potentially leading to a significant drop in sales volume when prices increase. Conversely, a low elasticity implies that price changes have a less pronounced effect on the quantity demanded.

Key factors influencing price elasticity include the availability of substitutes, the necessity of the product, and the proportion of income spent on the product. Businesses often use the price elasticity of demand to set prices that maximize profits. They prefer to increase prices for goods with inelastic demand and be more cautious with goods with elastic demand to avoid substantial losses in quantity sold. For academic purposes, students should remember that price elasticity is not static and can vary with price levels, income levels, and over time.

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Most popular questions from this chapter

Universal Widget produces high-quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production ( \(q\) ) is given by total cost \(=0.25 q^{2}\) Widgets are demanded only in Australia (where the demand curve is given by \(q_{A}=100-2 P_{A}\) ) and Lapland (where the demand curve is given by \(q_{L}=100-4 P_{L}\) ); thus, total demand equals \(q=q_{A}+q_{L}\). If Universal Widget can control the quantities supplied to each market, how many should it sell in each location to maximize total profits? What price will be charged in each location?

The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry's demand for inputs. To do so, we assume that an industry produces a homogencous good, \(Q,\) under constant returns to scale using only capital and labor. The demand function for \(Q\) is given by \(Q=D(P)\), where \(P\) is the market price of the good being produced. Because of the constant returns-to- scale assumption, \(P=M C=A C\). Throughout this problem let \(C(v, w, 1)\) be the firm's unit cost function. a. Explain why the total industry demands for capital and labor are given by \(K=Q C_{v}\) and \(L=Q C_{w}\) b. Show that \\[ \frac{\partial K}{\partial v}=Q C_{v v}+D^{\prime} C_{v}^{2} \quad \text { and } \quad \frac{\partial L}{\partial w}=Q C_{w w}+D^{\prime} C_{w}^{2} \\] c. Prove that \\[ C_{w v}=\frac{-w}{v} C_{v w} \quad \text { and } \quad C_{w w}=\frac{-v}{w} C_{N w} \\] d. Use the results from parts (b) and (c) together with the elasticity of substitution defined as \(\sigma=C C_{v n} / C_{\nu} C_{w}\) to show that \\[ \frac{\partial K}{\partial v}=\frac{w L}{Q} \cdot \frac{\sigma K}{v C}+\frac{D^{\prime} K^{2}}{Q^{2}} \text { and } \frac{\partial L}{\partial w}=\frac{v K}{Q} \cdot \frac{\sigma L}{w C}+\frac{D^{\prime} L^{2}}{Q^{2}} \\] e. Convert the derivatives in part (d) into elasticities to show that \\[ e_{K, v}=-s_{L} \sigma+s_{K} e_{Q, p} \quad \text { and } \quad e_{L, w}=-s_{K} \sigma+s_{L} e_{Q, P} \\] where \(e_{Q, P}\) is the price elasticity of demand for the product being produced. f. Discuss the importance of the results in part (e) using the notions of substitution and output effects from Chapter 11 Note: The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was first suggested by Alfred Marshall. The proof given here follows that in D. Hamermesh, Labor Demand (Princeton, NJ: Princeton University Press, 1993).

Because firms have greater flexibility in the long run, their reactions to price changes may be greater in the long run than in the short run. Paul Samuelson was perhaps the first economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Ch芒telier's Principle. The basic idea of the principle is that any disturbance to an equilibrium (such as that caused by a price change) will not only have a direct effect but may also set off feedback effects that enhance the response. In this problem we look at a few examples. Consider a price-taking firm that chooses its inputs to maximize a profit function of the form \(\Pi(P, v, w)=P f(k, 1)-w l-v k .\) This maximization process will yield optimal solutions of the general form \(q^{*}(P, v, w), I^{*}(P, v, w),\) and \(k^{*}(P, v, w) .\) If we constrain capital input to be fixed at \(\bar{k}\) in the short run, this firm's short-run responses can be represented by \(q^{s}(P, w, \bar{k})\) and \(I^{*}(P, w, \bar{k})\) a. Using the definitional relation \(q^{*}(P, v, w)=q^{s}\left(P, w, k^{*}(P, v, w)\right),\) show that $$\frac{\partial q^{*}}{\partial P}=\frac{\partial q^{s}}{\partial P}+\frac{-\left(\frac{\partial k^{*}}{\partial P}\right)^{2}}{\frac{\partial k^{*}}{\partial v}}$$ Do this in three steps. First, differentiate the definitional relation with respect to \(P\) using the chain rule. Next, differentiate the definitional relation with respect to \(v\) (again using the chain rule), and use the result to substitute for \(\partial q^{3} / \partial k\) in the initial derivative. Finally, substitute a result analogous to part (c) of Problem 11.10 to give the displayed equation. b. Use the result from part (a) to argue that \(\partial q^{*} / \partial P \geq \partial q^{s} / \partial P\). This establishes Le Ch芒telier's Principle for supply: Long-run supply responses are larger than (constrained) short-run supply responses. c. Using similar methods as in parts (a) and (b), prove that Le Ch芒telier's Principle applies to the effect of the wage on labor demand. That is, starting from the definitional relation \(l^{*}(P, v, w)=l^{s}\left(P, w, k^{*}(P, v, w)\right),\) show that \(\partial l^{*} / \partial w \leq \partial l^{s} / \partial w\) implying that long-run labor demand falls more when wage goes up than short-run labor demand (note that both of these derivatives are negative). d. Develop your own analysis of the difference between the short- and long-run responses of the firm's cost function \([C(v, w, q)]\) to a change in the wage \((w)\)

How would you expect an increase in output price, \(P\), to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in \(P\) must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb-Douglas case. c. Use the profit function to show how the presence of inferior inputs would lead to ambiguity in the effect of \(P\) on input demand.

Young's theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that \(\partial l(P, v, w) / \partial v=\partial k(P, v, w) / \partial w\). Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that \(\partial q / \partial w=-\partial l / \partial P\). Interpret this result. d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied.
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