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4.4: More Practice with Deriving Demand

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    This section derives the demand curve from two different utility functions, quasilinear preferences and perfect complements, to provide practice deriving demand curves. Nothing new here, just practice applying the tools, techniques, and concepts of the economic way of thinking.

    Quasilinear Preferences

    We begin with the analytical approach. Rewrite the constraint and form the Lagrangean, leaving \(p_1\) as a letter so we can derive a demand curve.

    \[\max _{x_{1}, x_{2}, \lambda} L=x_{1}^{1 / 2}+x_{2}+\lambda\left(140-p_{1} x_{1}-10 x_{2}\right)\]

    STEP Follow the usual Lagrangean procedure to solve this problem. For help, refer back to section 4.2 where we solved this same problem except with \(m\) instead of \(p_1\).

    You should find reduced form expressions like this:

    x_{1}^{*} &=\frac{25}{p_{1}^{2}} \\
    x_{2}^{*} &=14-\frac{2.5}{p_{1}}

    The first expression, \(x_1 \mbox{*} = \frac{25}{p_1^2}\), is a demand curve for \(x_1 \mbox{*}\) because it gives the quantity demanded of \(x_1\) as a function of \(p_1\). If we rewrite the equation in terms of \(p_1\) like this, \(p_1^2 = \frac{25}{x_1 \mbox{*}} \rightarrow p_1 = \frac{5}{\sqrt{x_1 \mbox{*}}}\) then we have an inverse demand curve, with price on the y axis as a function of quantity on the x axis.

    The derivative of \(x_1 \mbox{*}\) with respect to \(p_1\) tells us the slope of the demand curve at any given price.

    x_{1}^{*} &=25 p_{1}^{-2} \\
    \frac{d x_{1}^{*}}{d p_{1}} &=-2 \cdot 25 p_{1}^{-3}=-\frac{50}{p_{1}^{3}}

    The own price elasticity of demand is:

    \[\frac{d x_{1}^{*}}{d p_{1}} \cdot \frac{p_{1}}{x_{1}^{*}}=-\frac{50}{p_{1}^{3}} \frac{p_{1}}{\frac{25}{p_{1}^{2}}}=-2\]

    The constant elasticity of demand for good 1 is a property of the quasilinear utility function. Notice that 2 is the reciprocal of the exponent on \(x_1\) in the utility function. In fact, with \(U = x_1^c + x_2\), the price elasticity of demand for \(x_1\) is \(-\frac{1}{1-c}\) for values of \(c\) that yield interior solutions.

    The expression for optimal \(x_2\) is a cross price relationship. It tells us how the quantity demanded for good 2 varies as the price of good 1 changes. The equation can be used to compute a cross price elasticity, like this:

    \frac{d x_{2}^{*}}{d p_{1}} \cdot \frac{p_{1}}{x_{2}^{*}}=\frac{2.5}{p_{1}^{2}} \frac{p_{1}}{14-\frac{2.5}{p_{1}}}=\frac{2.5}{p_{1}\left(14-\frac{2.5}{p_{1}}\right)}=\frac{2.5}{p_{1}\left(\frac{14 p_{1}-2.5}{p_{1}}\right)}=\frac{2.5}{14 p_{1}-2.5}\]

    Unlike the own price elasticity, the cross price elasticity is not constant. It depends on the value of \(p_1\). It is also positive (whereas the own price elasticity was negative). When \(p_1\) rises, optimal \(x_2\) also rises. This means that goods 1 and 2 are substitutes.

    Complements, on the other hand, are goods whose cross price elasticity is negative. This means that an increase in the price of good 1 leads to a decrease in consumption of good 2.

    Demand can also be derived via numerical methods.

    STEP Open the Excel workbook DemandCurvesPractice.xls, read the Intro sheet, then go to the QuasilinearChoice sheet.

    The consumer is maximizing satisfaction at the initial parameter values because the marginal condition, MRS = \(\frac{p_1}{p_2}\), is met at the point 6.25,12.75 (ignoring Solver’s false precision) and income is exhausted.

    We can explore how this initial optimal solution varies as the price of good 1 changes via numerical methods. We simply change \(p_1\) repeatedly, running Solver at each price, while keeping track of the optimal solution at each price. The Comparative Statics Wizard add-in handles the tedious, cumbersome calculations and outputs the results in a new sheet for us.

    STEP Run the Comparative Statics Wizard on the QuasilinearChoice sheet. Increase the price of good 1 by 0.1 (10 cent) increments.

    You can check your comparative statics analysis by comparing your results to the CS1 sheet, which is based on 1 (instead of 0.1) dollar size shocks. Of course, the numbers will not be exactly the same since the \(\Delta p_1\) shock size is different.

    The columns of price and optimal \(x_1\) are points on the demand schedule. The numerical approach via the CSWiz essentially picks individual points on the demand curve for the given prices. If you plot these points, you have a graph of the demand curve.

    The analytical approach, on the other hand, gives the demand function as an equation. You can evaluate the expression at particular prices and generate a plot of the demand curve.

    The two approaches, if done correctly, will always yield the same graphical depiction of the demand curve. They may not, however, yield the same slopes or elasticities.

    STEP Using your results, create demand and price consumption curves. Compute the own unit changes and elasticities for \(x_1 \mbox{*}\) and \(x_2 \mbox{*}\).

    The CS1 sheet shows how to do this if you get stuck. You can click on cells to see their formulas. Think about how the formulas work and how they compute the answer.

    It is critical that you notice that your own unit changes and elasticities are closer to the instantaneous rates of change in columns I and J of the CS1 sheet because you have smaller changes in \(p_1\) and, for this utility function, \(x_1 \mbox{*}\) is nonlinear in \(p_1\).

    Take a moment to reflect on what is going in the calculations presented in the CS1 sheet. The color-shaded cells invite you to compare those cells.

    Now, let’s walk through this slowly.

    STEP Click on cell F13 to see its formula.

    It is computed as the change in optimal \(x_1\) for a $1 increase in \(p_1\). There is a decrease of about 3.47 units when price increases by 1 unit.

    STEP Click on cell I12 to see its formula.

    It is computed by substituting the initial price, $2/unit, into the expression for the derivative (displayed as an equation above the cell). The result of the formula, \(-6.25\), is the instantaneous rate of change. In other words, there will be a 6.25-fold decrease in optimal \(x_1\) given an infinitesimally small increase in \(p_1\).

    STEP Go to your CSWiz results and, if you have not done so already, compute the change in optimal \(x_1\) for a $0.1 increase in \(p_1\).

    You should find that your slope is about \(-5.8\). The change in optimal \(x_1\) is about 0.58, but you have to divide by the change in price, 0.1, to get the slope. Notice that your answer is much closer to the derivative-based rate of change (\(-6.25\)). This is because you took a much smaller change in price, 0.1, than the one dollar change in price in the CS1 sheet and you are working with a curve.

    STEP Return to the CS1 sheet and compare cells G13 and J12.

    The same principle is at work here. Because the demand curve is nonlinear, the two cells do not agree. Cell G13 is computing the elasticity from one point to another, whereas cell J12 is using the instantaneous rate of change (slope of the tangent line) at a point.

    If you compute the price elasticity from 2 to 2.1 (using your CS results), you will find that it is much closer to \(-2\).

    Finally, you might notice that unlike the Cobb-Douglas utility function, which produced a horizontal price consumption curve (PCC), the quasilinear utility function in this case is generating a downward sloping price consumption curve. In fact, the slope of the price consumption curve tells you the price elasticity of demand: Upward sloping PCC means that demand is inelastic, horizontal PCC yields a unit elastic demand (as in the Cobb-Douglas case), and downward sloping PCC gives elastic demand (as in this case).

    Perfect Complements

    We begin with the analytical approach. \[U(x_1, x_2)=min\{ax_1,bx_2\}\] For \(a = b = 1\), we know that we can find the intersection of the optimal choice and budget lines to get the reduced form expressions for the endogenous variables, \(x_1 \mbox{*} = \frac{m}{p_1 + p_2}\) (which is the same for \(x_2 \mbox{*}\) since \(x_1 \mbox{*} = x_2 \mbox{*}\)).

    This solution says that when a and b are the same in a perfect complements utility function, the optimal amounts of each good are equal and found by simply dividing income by the sum of the prices.

    The reduced form expression contains Engel and demand curves. Holding prices constant, we can see how m affects consumption. Likewise, holding m and \(p_2\) constant, we can explore how optimal \(x_1\) varies as \(p_1\) changes. This, of course, is a demand curve for \(x_1\).

    As usual, we find the instantaneous rate of change by taking the derivative with respect to \(p_1\). The \(p_1\) elasticity of \(x_1\) is the derivative multiplied by \(\frac{p_1}{x_1 \mbox{*}}\).

    \frac{d x_{1}^{x^{*}}}{d p_{1}} &=-\frac{m}{\left(p_{1}+p_{2}\right)^{2}} \\
    \frac{d x_{1}^{x^{+}}}{d p_{1}} \cdot \frac{p_{1}}{x_{1}^{x^{*}}} &=-\frac{m}{\left(p_{1}+p_{2}\right)^{2}} \frac{p_{1}}{p_{1}+p_{2}}=-\frac{p_{1}}{p_{1}+p_{2}}

    We can also derive demand for a perfect complements utility function via numerical methods.

    STEP Proceed to the PerfCompChoice sheet and run the Comparative Statics Wizard with an increase in the price of good 1 of 0.1 (10 cents).

    Can you guess what we will do next? The procedure is the same every time: we solve the model then explore how the optimal solution responds to shocks.

    STEP Create demand and price consumption curves based on your comparative statics results. Compute the own units changes and elasticities for \(x_1 \mbox{*}\) and \(x_2 \mbox{*}\). The CS2 sheet shows how to do this if you get stuck.

    As before, you will want to concentrate on how your own units changes and elasticities are closer to the instantaneous rates of change than the \(\Delta p_1\) in columns F and G of the CS2 sheet because you have smaller changes in \(p_1\) and we are dealing with a nonlinear relationship.

    The lesson is clear: whenever the demand curve is not a line, that is, \(x_1 \mbox{*}\) is nonlinear in \(p_1\), then \(\Delta p_1\) will not exactly equal \(dp_1\). As the size of the discrete change in price gets smaller, the numerical method result will approach the result based on the derivative.

    Although the two methods might not exactly agree, they are usually pretty close. How close depends on the curvature of the relationship and the size of the discrete shock. This means you can always check your analytical work by doing a manual \(\Delta\) shock and computing the change from one point to another.

    Notice also that the price consumption curve is upward sloping and the price elasticity is less than one (in absolute value).

    Deriving Demand from the Consumer’s Utility Maximization Problem

    The primary purpose of this section was to provide additional practice in deriving demand with different utility functions. Clearly, the demand curve is strongly influenced by the utility function that is being maximized given a budget constraint.

    Two examples were used to demonstrate how the analytical and numerical methods are related. Calculus is based on the idea of infinitesimally small changes. You can see calculus in action by using the CSWiz to take smaller changes in pricewhich drives the numerical method ever closer to the derivative-based result.


    1. Return to the QuasilinearChoice sheet and click the Screen Shot 2021-07-09 at 09.18.59.png button. Now change the exponent on good 1 from 0.5 to 0.75. Use the Comparative Statics Wizard to derive a demand curve for this utility function.

    2. Working with the same utility function as in the first question, derive the demand for \(x_1 \mbox{*}\) via analytical methods. Use Word’s Equation Editor as needed. Show your work.

    3. Using your results from questions 1 and 2, compute the own price elasticity via numerical and analytical methods. Do they agree? Why or why not? Show your work and take screen shots as needed.


    The epigraph is from page 63 of Hal Varian’s best-selling, undergraduate textbook, Intermediate Microeconomics (7th edition, 2006). In the preface, Varian tackles head on the issue of calculus. "Many undergraduate majors in economics are students who should know calculus, but don’tat least not very well. For this reason, I have kept calculus out of the main body of the text."

    The book you are reading at this moment takes a different approach. Calculus is used extensively, but it is made accessible by consistent repetition along with the substantial support of numerical methods. If you are a student who struggles with analytical methods, you will never have a better opportunity to master calculus and algebra. Do the practice problems with care and match the analytical and numerical approaches in each application.

    This page titled 4.4: More Practice with Deriving Demand is shared under a CC BY-SA license and was authored, remixed, and/or curated by Humberto Barreto.

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