Department of Eonomis Universit of Marland Eonomis 35 Intermediate Maroeonomi Analsis Pratie Problem Set Suggested Solutions Professor Sanja Chugh Spring 0. Partial Derivatives. For eah of the following multi-variable funtions, ompute the partial derivatives with respet to both x and. Solution: In order to ompute the partial derivative with respet to x, momentaril pretend that is a onstant (for example, imagine momentaril that = 5) and proeed to differentiate using the usual rules of alulus. Likewise, in order to ompute the partial derivative with respet to, momentaril pretend that x is a onstant (for example, imagine momentaril that x = 5) and proeed to differentiate using the usual rules of alulus. Appling this algorithm to eah of the given funtions: a. f ( x, ) = x We have fx ( x, ) = and f ( x, ) = x. b. f ( x, ) = x+ 3 We have fx ( x, ) = and f ( x, ) = 3.. f ( x, ) = x 4 We have fx( x, ) x 4 = and f ( x, ) 4x 3 =. d. f ( x, ) = lnx+ ln We have f ( x, ) = / xand f ( x, ) = /. x e. f ( x, ) = x+ Reall from priniples of basi mathematis that we an write this funtion as / / / f ( x, ) = x +. Hene, the partial derivatives are (, ) f / x x= x = x / and f ( x, ) = = /.
f. f( x, ) = x Reall from priniples of basi mathematis that we an write this funtion as f ( x, ) = x. Hene, the partial derivatives are fx ( x, ) = = / and f ( x, ) = x = x/. g. f( x, ) = x Reall from priniples of basi mathematis that we an write this funtion as f ( x, ) = x. Hene, the partial derivatives are fx( x, ) = x = / x and f ( x, ) = x = / x. Properties of Indifferene Maps. For the general model of utilit funtions and indifferene maps developed in lass, explain wh no two indifferene urves an ever ross eah other. Your answer must explain the eonomi logi here, and ma also inlude appropriate equations and/or graphs. Solution: The proof proeeds b ontradition. Consider the following indifferene urves that ross eah other. B A C
The onsumption bundle A lies on both indifferene urves. Beause bundle A and bundle B lie on the same indifferene urve, the ield the same level of utilit. Likewise, beause bundle A and bundle C lie on the same indifferene urve, the must ield the same level of utilit. This then implies that bundle B and bundle C ield the same level of utilit (transitive propert of preferenes). But if this were true, then bundle B and bundle C should lie on the same indifferene urve, whih the do not b assumption. Thus, we have reahed a logial ontradition indifferene urves annot ross eah other. 3. A Canonial Utilit Funtion. Consider the utilit funtion σ u () =, σ where denotes onsumption of some arbitrar good and σ (the Greek letter sigma ) is known as the urvature parameter beause its value governs how urved the utilit funtion is. In the following, restrit our attention to the region > 0 (beause negative onsumption is an ill-defined onept). The parameter σ is treated as a onstant. a. Plot the utilit funtion for σ = 0. Does this utilit funtion displa diminishing marginal utilit? Is marginal utilit ever negative for this utilit funtion? b. Plot the utilit funtion for σ = /. Does this utilit funtion displa diminishing marginal utilit? Is marginal utilit ever negative for this utilit funtion?. Consider instead the natural-log utilit funtion u ( ) = ln( ). Does this utilit funtion displa diminishing marginal utilit? Is marginal utilit ever negative for this utilit funtion? d. Determine the value of σ (if an value exists at all) that makes the general utilit funtion presented above ollapse to the natural-log utilit funtion in part. (Hint: Examine the derivatives of the two funtions.) Solution: a. With σ = 0, the utilit funtion beomes the linear funtion u () =, whih has a simple graph: 3
Notie that utilit ma atuall be negative but reall that the units of utilit (utils) are ompletel arbitrar, so there is nothing wrong with onsidering negative values of utilit. This linear funtion learl does not displa diminishing marginal utilit beause its slope is onstant at one throughout so of ourse, marginal utilit never beomes negative either. b. With σ = /, the utilit funtion beomes u () =, whih looks like Notie again that here we have negative values of utilit, whih is fine sine utilit is measured in an arbitrar sale. The slope of this utilit funtion is u'( ) = / (reall the slope of a funtion is simpl the first derivative), whih is alwas positive as long as onsumption is positive, so marginal utilit is never negative. And learl as onsumption rises, the slope falls, so this funtion does displa diminishing marginal utilit. 4
. The natural log utilit funtion has graph The derivative of this funtion is u'( ) = /, whih is alwas positive when onsumption is positive, so this funtion also does not ever experiene negative marginal utilit. As onsumption rises, the slope of the utilit funtion falls, so this funtion does displa diminishing marginal utilit. d. The derivative of the log utilit funtion is u'( ) = /, while the derivative of the general utilit funtion presented above is u'( ) = σ. Clearl, for σ = the slopes of the two funtions (that is, the two marginal utilit funtions) are idential. This does not prove that the two funtions are idential, however, beause it ould be that the two funtions alwas have the same slope but are vertial translates of eah other. To see that the are indeed the same funtion, however, tr plotting the general utilit funtion for several values of the urvature parameter near one (that is, tr plotting the funtion for, sa, σ = 0.9, σ = 0.95, σ = 0.99, σ =.05, σ =.0, et) ou will see that as the urvature parameter approahes one from either diretion, the general utilit funtion approahes the log utilit funtion. 5
4. The Impliit Funtion Theorem and the Marginal Rate of Substitution. An important result from multivariable alulus is the impliit funtion theorem, whih states that given a funtion f ( x,, ) the derivative of with respet to x is given b d f / x =, dx f / where f / x denotes the partial derivative of f with respet to x and f / denotes the partial derivative of f with respet to. Simpl stated, a partial derivative of a multivariable funtion is the derivative of that funtion with respet to one partiular variable, treating all other variables as onstant. For example, suppose f ( x, ) = x. To ompute the partial derivative of f with respet to x, we treat as a onstant, in whih ase we obtain f / x=, and to ompute the partial derivative of f with respet to, we treat x as a onstant, in whih ase we obtain f / = x. We have desribed the slope of an indifferene urve as the marginal rate of substitution between the two goods. Imagining that is plotted on the vertial axis and plotted on the horizontal axis, ompute the marginal rate of substitution for the following utilit funtions. a. u (, ) = ln( ) + ln( ) b. u (, ) = + a a. u (, ) =, where a (0,) is some onstant. Solution: Simpl use the impliit funtion theorem in eah ase to ompute the slope of the indifferene urves (whih is the negative of the marginal rate of substitution): a. With the given utilit funtion, we have u/ = / and u/ = /, so that, b the impliit funtion theorem, the slope of eah indifferene urve is d / = = d / b. With the given utilit funtion, we have that u/ = 0.5/ and u/ = 0.5, so that, b the impliit funtion theorem, the slope of eah indifferene urve is d d 0.5/ = = 0.5/. With the given utilit funtion, we have that u/ = a and a a u/ = ( a) a a, so that, b the impliit funtion theorem, the slope of eah indifferene urve is 6
a a d a a = = a a d ( a) a Notie that in eah ase, the negative of the slope is a dereasing funtion of, whih graphiall is simpl our statement that indifferene urves generall beome flatter the further along the horizontal axis we travel (that is, indifferene urves are onvex to the origin). 5. Sales Tax. Consider the standard onsumer problem we have been studing, in whih a onsumer has to hoose onsumption of two goods and whih have pries (in terms of mone) P and P, respetivel. These pries are pries before an appliable taxes. Man states harge sales tax on some goods but not on others for example, man states harge sales tax on all goods exept food and lothes. Suppose that good arries a per-unit sales tax, while good has no sales tax. Use the variable t to denote this sales tax, where t is a number between zero and one (so, for example, if the sales tax on good were 5 perent, we would have t = 0.5 ). a. With sales tax t and onsumer inome Y, write down the budget onstraint of the onsumer. Explain eonomiall how/wh this budget onstraint differs from the standard one we have been onsidering thus far. b. Graphiall desribe how the imposition of the sales tax on good alters the optimal onsumption hoie (ie, how the optimal hoie of eah good is affeted b a poli shift from t = 0 to t > 0 ).. Suppose the onsumer s utilit funtion is given b u(, ) = log+ log. Using a Lagrangian, solve algebraiall for the onsumer s optimal hoie of and as funtions of P, P, t, and Y. Graphiall show how, for this partiular utilit funtion, the optimal hoie hanges due to the imposition of the sales tax on good. Solution: a. The sales tax on good is paid in addition to its prie P. Thus, the total expenditure on good (i.e., the total amount of mone the onsumer will pa out of his poket to purhase good ) is ( + t) P. For example, in Massahusetts the sales tax rate is 5 perent, so we would have t = 0.05. This means that ou would pa 05 perent ( + t =.05) of the pre-tax prie. Beause there is no tax on good, the budget onstraint is ( + t) P + P = Y. b. If we solve the budget onstraint for in terms of, we get 7
P( + t ) Y = +, P P so learl the slope of the budget line is affeted. This is an important general lesson: the reason wh taxes levied on onsumers have (or do not have) effets is beause in general the alter the budget onstraint. Intuitivel, if the onsumer now spent all of his inome on good, he would be able to bu less than without the sales tax, but if spent all of his inome on good, the quantit of good he ould bu would be unaffeted b the imposition of the sales tax on good. Graphiall, the budget line beomes steeper b pivoting around the vertial interept, as shown in the figure below. Beause the onsumer s optimal hoie of (, ) is desribed b the tangen of an indifferene urve with the budget line, the figure also shows how the optimal hoie hanges in this ase, the optimal hoie moves from point A to point B. As drawn in the figure, the new optimal hoie features less onsumption of good and more onsumption of good the onsumer has substituted some good for some good in the fae of a rise in prie (inlusive of tax) of good. But see part below for more on this substitution effet. B A Post-tax budget line Pre-tax budget line After the imposition of the sales tax on good, the optimal hoie moves from point A to point B, whih features less onsumption of good.. The onsumer s problem is to maximize ln + ln subjet to the budget onstraint Y P P = 0. The Lagrangian for this problem is L (,, λ) = ln+ ln + λ( Y ( + t) P P), where λ denotes the Lagrange multiplier on the budget onstraint. The firstorder onditions of L with respet to,, and λ are, respetivel, 8
λ( + t) P = 0 λp = 0 Y ( + t ) P P = 0 We must solve this sstem of equations for and. There are of ourse man was to solve this sstem, whih onl differ in the exat order of equations used. One useful wa of proeeding is to first eliminate the multiplier λ. To do this, from the seond equation, we get λ =. Substituting this into the first equation gives P P( + t) = 0. P From here, we an solve for P, whih is simpl total expenditure on good : P = ( + t) P This expression states that total expenditure on good equals total expenditure on good (inlusive of the tax on good ). Note that this result need not alwas hold it holds here beause of the given utilit funtion. Proeeding, substitute the above expression into the budget onstraint to get Y ( + t) P ( + t) P = 0, from whih it easil follows that the optimal hoie of onsumption of good is * Y =. ( + t) P If we had numerial values for the objets on the right-hand-side, learl we would know numeriall the optimal hoie of onsumption of good. This solution reveals that a rise in t (holding onstant Y and P ) results in a fall in. To obtain the optimal hoie of onsumption of good, return to the above expression P = ( + t) P, from whih we get that onsumption of good is related to ( + t) P onsumption of good in the following wa: =. Inserting the optimal P hoie of good found above, we have that the optimal hoie of good is * Y =. P Thus, even though the onsumer splits his inome evenl on expenditures on good and good, onsumption of good and good are not the same unless t = 0. But notie from the above expression that the optimal hoie of good is independent of the tax rate on good! Thus, if we plot this utilit funtion and the assoiated optimal * 9
hoies as in part b above, we would get a movement of the optimal hoie straight left from point A, the original optimal hoie, meaning the optimal hoie of good is idential while the optimal hoie of good dereases. 0