LECTURE NOTES ON MICROECONOMICS

LECTURE NOTES ON MCROECONOMCS ANALYZNG MARKETS WTH BASC CALCULUS William M. Boal Part : Consumers and demand Chater 5: Demand Section 5.: ndividual demand functions Determinants of choice. As noted in the revious chater, we assume each consumer chooses the most referred bundle that she or he can afford. Which bundle is chosen therefore deends on () references, as shown by indifference curves or a utility function, and () rices and income, as shown by the budget line. Note that any change in references, rice or income will, in general, cause the consumer to make a new choice. n this chater, we assume that references do not change. We focus on changes in consumer choice that are driven by changes in the budget line that is, changes in rices and income. Definition of demand function. f references do not change, then the uantities of each good chosen by a consumer deend on rices and income. For examle, suose there are only two goods. Denote the uantities of these goods and. Denote their rices and. Denote the consumer s income. Then the uantities chosen by the consumer can be written as demand functions.,, and,, (5.) Here, the asterisks indicate that these are not arbitrary uantities these are the uantities of the consumer s most referred affordable bundle. Put differently, they are the consumer s utility-maximizing choices, given the budget constraint. Note that these uantities are, in general, functions of all rices and income. (For some secial references, some of the arguments may dro out of the demand function, as we will see later.) Prices and income are taken as given by the consumer. By contrast, the uantities of other goods are not arguments of the demand functions, because other uantities are choices made by the consumer Section 5.: Overall roerties of legitimate demand functions Note that for each good there must be a corresonding demand function. A collection of functions that define the uantities of all goods chosen in the consumer s bundle is called a demand system. However, not all urorted demand functions or demand systems can be derived from utility-maximizing behavior by consumers. Certain roerties must hold for these urorted demand functions to be legitimate. 06 William M. Boal - - Part Chater 5

06 William M. Boal - - Part Chater 5 Budget constraint. Recall from the revious chater that the consumer s choice will always lie on the budget line. t follows that total sending imlied by a demand system must add u to the consumer s income. This is sometimes called the "adding-u roerty." For the two-good case, a demand system that satisfies the budget line must satisfy the following euation. (5.),,,, Examle: Assume that there are only two goods. Do the following functions satisfy the budget constraint? (5.3) 5 3 and 5 3 To answer this uestion, we must substitute the above formulas for and into euation (5.) and check to see if the euation balances: (5.4)? 5 3 5 3 5 3 5 3 The euation does indeed balance, so sending necessarily euals income and the budget constraint is indeed satisfied. Homogeneity. Another roerty of demand functions is homogeneity of degree zero in all arguments. A function that is homogeneous of degree zero is defined as one whose value does not change if all its arguments change roortionally. Why must demand functions be homogeneous of degree zero? Recall that if income and all rices rise as the same rate then the budget line does not change. f the budget line does not change, the consumer s most referred bundle will certainly not change. Therefore demand functions must be homogeneous of degree zero. Formally, for any arbitrary ositive number, the following euations must hold: (5.5),,,,,,,,. Homogeneity means that ure inflation will have no effect on a consumer's choices. Pure inflation is defined as a roortionate increase in income and all rices. Of course, ure Additional roerties follow directly from the budget constraint. f euation (5.) is differentiated with the resect to income, the so-called Engel aggregation condition is derived. f the same euation is differentiated with resect to one rice, the so-called Cournot aggregation condition is derived. More generally, a function f(x,x,,x n ) is said to be homogeneous of degree n if, for any number, f(x,x,,x n ) = n f(x,x,,x n ).

inflation is unusual in the real world. Actual exeriences of inflation may cause some comonents of income to fall relative to others (such as retirement benefits) and some rices to fall relative to others. Examle: s the function = /(3 ) + ( / ) homogeneous of degree zero? Relacing by, by, and by, and canceling yields the original demand function. So this demand function is indeed homogeneous of degree zero. Examle: Return to the functions used in a revious examle: 3 5 and Are these functions homogeneous of degree zero? To check this, we must multily all arguments by an arbitrary number call it and check to see if the uantity chosen is unchanged, as reuired by euation (5.5) above. Relacing by, by, and by yields the following. (5.6)?? 3 5 3 5 3 3 5 3 5 Though many of the s in this examle cancel, not all do so. Therefore, these functions do not satisfy homogeneity of degree zero and therefore cannot be legitimate individual demand functions. Often Cobb-Douglas or constant-elasticity functions are used to model demand. These functions take the form = b c d e, where b, c, d, and e are constants. Relacing by (), by ( ), and by ( ) yields the following: 5 (5.7)? b b b b c d c c d d e e c c d d e e c d cd e e e. Thus the s cancel if and only if (c+d+e) = 0, for only then will c+d+e eual one. So constant-elasticity demand functions are homogeneous of degree zero if and only if the exonents on income and all rices sum to zero. This rovides a shortcut check for constant-elasticity demand functions. n alied work, we can guarantee that a demand function satisfies homogeneity if we simly exress income and rices in real (or "constant dollar") terms. To do this, simly divide rices and income by some index of the overall rice level, such at the Consumer 06 William M. Boal - 3 - Part Chater 5

Price ndex ublished by the U.S. Bureau of Labor Statistics. 3 For examle, the following demand function is homogeneous of degree zero, because if,, and the CP all increase by the same ercentage, the uantity demanded of good does not change: (5.8) 5 3. CP CP Purorted demand functions that do not satisfy the budget constraint and the homogeneity roerty cannot be rooted in utility-maximizing behavior. To check whether a function can be a legitimate individual demand function, we can check whether these two roerties hold. Section 5.3: Finding demand functions Particular choices versus general functions. n the revious chater, we learned to solve for the consumer s most referred affordable bundle. Our aroach was to solve the budget line and the tangency condition together. Because we assumed articular values for the rices and income, we were able to calculate articular uantities chosen. We now use the same aroach but leave rices and income as variables, and so find the consumer s demand functions. Demand functions, like articular choices, deend on the consumer s articular references or utility (through the tangency condition). For some utility functions, demand functions are difficult or imossible to find exlicitly. However, for a number of realistic utility functions, demand functions can be found without great difficulty. Examle (). Suose a consumer has the following Cobb-Douglas utility function U(, ) =. What are the demand functions for good # and good #? The tangency condition here is given by MU /MU = /( ) = /, which after crossmultilying becomes =. Substituting this into the euation for the budget line = + and solving yields = /(3 ). Substituting into the budget line (that is, exloiting the adding-u roerty) and solving yields = /(3 ). (A detailed ste-by-ste solution is given in an aendix to this chater.) Examle (). Suose a consumer has the following Stone-Geary utility function U(, ) = ( -3). What are the demand functions for good # and good #? The tangency condition here is given by MU /MU = /( -6) = /, which after crossmultilying becomes = 6. Substituting ( ) into the euation for the budget line = + and solving yields = /(3 ) +. Substituting into the budget line (that is, exloiting the adding-u roerty) and solving yields = /(3 ) - /. (A detailed ste-by-ste solution is given in an aendix to this chater.) 3 For descrition and data, see htt://www.bls.gov. Similar indexes are available for most countries. 06 William M. Boal - 4 - Part Chater 5

Examle (3). Suose a consumer has the following CES utility function U(, ) = (/ ) (/ ). What are the demand functions for good # and good #? The tangency condition here is given by MU /MU = / = /, which can be rewritten as = ( / ) /. Substituting this into the euation for the budget line = + / and and solving yields, after some tedious algebra, / 0. 5. (A detailed ste-by-ste solution is given in an aendix to this chater.) Section 5.4: Effect of income on uantity demanded A change in income. Recall from chater 4 that if income rises but rices remain constant, then the budget line shifts u and out, away from the origin, in arallel fashion. ts sloe, the rice ratio, does not change, but both intercets rise in roortion to the increase in income. After such a shift, the consumer will choose a bundle on the new budget line (see figure 5.). Put differently, the increase in income results in an increase in sending. t follows that the consumer will urchase more of one or both goods. Moreover, the consumer will now reach a higher indifference curve, reresenting a higher level of utility or well-being. ncome exansion ath. magine the conseuences of a continuous increase in income. The consumer will choose a continuous seuence of bundles, each one at a oint of tangency between the current budget line and the highest attainable indifference curve. The resulting curve is called the consumer s income exansion ath, 4 shown as a dotted arrow in figure 5.. The euation for the income exansion ath is simly the roerty shared by all bundles on that ath namely the tangency condition introduced in the revious chater. MU (5.9) MU Examle: Suose a consumer has the utility function U(, ) = 3 and faces rices =$5 and = $3. What is the euation for the consumer s income-exansion ath? Here, MU = U/ = 3 and MU = U/ = 3. Alying euation (5.9), yields (3 )/( ) = 3/5 or = (/5). n this examle, the income-exansion ath turned out to be a straight line through the origin. 5 Not all income-exansion aths have this simle shae. 4 Also called the income consumtion ath or the income offer curve. 5 Utility functions are called homothetic if they yield exansion aths that are always straight lines through the origin. t can be roven that all utility functions which are homogeneous are also homothetic. For examle, the utility functions in examles 4, 6, and 7 are homogeneous of degrees 3, -, and 5, resectively, so they are all homothetic. However, the utility function in examle 5 is not homogeneous, so it is not homothetic in fact, its exansion aths intersect the axis at =3 instead of assing through the origin. 06 William M. Boal - 5 - Part Chater 5

Figure 5.. Effect of increase in income Figure 5.. Consumer s income exansion ath Examle: Suose a consumer has the utility function U(, ) = ( -3) and faces rices =$3 and = $4. What is the euation for the consumer s income-exansion ath? Here, MU = U/ = and MU = U/ = -3. Alying euation (5.9), yields /( -3) = 4/3 or = (4/3) - 4. This income-exansion ath is not a straight line through the origin. 06 William M. Boal - 6 - Part Chater 5

Examle: Suose a consumer has the utility function U(, ) = (/ ) (/ ) and faces rices =$ and = $3. What is the euation for the consumer s incomeexansion ath? Here, MU = U/ = / and MU = U/ = /. Alying euation (5.9), yields / = 3/ or 3/. This income-exansion ath is a straight line through the origin. Normal versus inferior goods. The income-exansion ath shown in figure 5. sloes uward, indicating that the consumer urchases more of both goods as her or his income rises. However there is no mathematical reason why this need always be the case. Figure 5.3 shows a different consumer who urchases more units of good # but fewer units of good # as her or his income rises. f a consumer urchases more units of a good as her or his income rises, that good is called a normal good. f a consumer urchases fewer units, that good is called an inferior good. nferior goods are rare. Broad categories of goods like clothing, transortation, or food are not inferior goods. nferior goods tend to be narrow, low-uality categories of goods that consumers abandon in favor of high-uality alternatives as their income rises. Examles of inferior goods include used clothing, mass transit, and macaroni-and-cheese dinners. Whether a good is normal or inferior, of course, may vary with individual references. Some consumers might have no inferior goods. However, every consumer must have at least one normal good otherwise sending could not kee ace with rising income. Engel curves and artial derivatives. A grah of the uantity demanded of a single good against income, holding all rices constant, is called an Engel curve (see figure 5.4). A good is a normal good if its Engel curve sloes u and an inferior good if its Engel curve sloes down. n figure 5.4, good A is a normal good at all levels of income, while good B becomes an inferior good for sufficiently high levels of income. The sloe of an Engel curve is the increase in uantity demanded when income is increased by one dollar, holding all rices constant. This sloe is given by the artial derivative of the demand function with resect to income. For a normal good, the artial derivative is ositive: / > 0. For an inferior good, the artial derivative is negative: / < 0. Examle: Suose the demand for good # is given by = /( ). s good # a normal good or an inferior good? First, note that / = /( ), which is necessarily ositive, assuming is ositive. So an increase in income always brings an increase in the uantity demanded. t follows that good # is a normal good. Examle: Suose the demand for good # is given by = /(3 ) + /. s good # a normal good or an inferior good? Note that / = /(3 ), which is necessarily ositive, assuming is ositive. So an increase in income always brings an increase in the uantity demanded. t follows that good # is a normal good. 06 William M. Boal - 7 - Part Chater 5

Figure 5.3. ncome exansion ath with one inferior good Figure 5.4. Engel curves for normal (A) and inferior (B) goods Quantity demanded Good A Good B ncome Examle: Suose the demand for good # is given by = = / ( + ( ) / ). s good # a normal good or an inferior good? Note that / = /( + ( ) / ), which is necessarily ositive, assuming is ositive. So an increase in income always brings an increase in the uantity demanded. t follows that good # is a normal good. 06 William M. Boal - 8 - Part Chater 5

Section 5.5: Effect of own rice on uantity demanded A change in the good s own rice. f the rice of good # falls, while income and the other rice remain constant, then the budget line rotates out, away from the origin, but anchored on the axis of the good whose rice does not change. Assuming good # is on the vertical axis, the budget line becomes steeer. After such a shift, the consumer will choose a bundle on the new budget line (see figure 5.5). Since the new budget line is farther from the origin, the consumer will urchase more of one or both goods. Moreover, the consumer will now reach a higher indifference curve, reresenting a higher level of utility or well-being. Ordinary versus Giffen goods. Consider the consumer s urchases of good #, the good whose rice has decreased. Figure 5.5 seems to indicate that this consumer has resonded to the rice decrease by urchasing more of good #, but there is no mathematical reason why this need always be the case. Figure 5.6 shows a different consumer who urchases fewer units of good # as its rice decreases. f a consumer urchases more units of good as its rice decreases, that good is called an ordinary good. f a consumer urchases fewer units, that good is called a Giffen good. Ordinary goods are so common that the negative relationshi between the uantity demanded of a good and its own rice is often called the Law of Demand. By contrast, Giffen goods are extremely rare in the world, if any exist at all. We will show later that Giffen goods are in theory a secial, extreme case of inferior goods. Demand curves and artial derivatives. A grah of the uantity demanded of a single good against its own rice, holding income and other rices constant, is called a demand curve (see figure 5.7). t is traditional in economics to lace rice on the vertical axis and the uantity of the same good on the horizontal axis. Regardless of the choice of axes, a good is an ordinary good if its demand curve sloes down and a Giffen good if its demand curve sloes u. With rice on the vertical axis, the sloe of the demand curve is the change in rice reuired to increase the uantity demanded by one unit. The recirocal of the sloe is given by the artial derivative of the demand function with resect to the good s own rice. For an ordinary good, the artial derivative is negative: / < 0. For a Giffen good, the artial derivative would be ositive: / > 0. Examle: Suose the demand for good # is given by = /( ). s good # an ordinary good or a Giffen good? First, note that / = /( ), which is necessarily negative, assuming and are ositive. So an increase in rice always brings a decrease in the uantity demanded. t follows that good # is an ordinary good, following the "Law of Demand." 06 William M. Boal - 9 - Part Chater 5

Figure 5.5. Effect of decrease in rice of good # Figure 5.6. Effect of decrease in rice of good # when good # is a Giffen good 06 William M. Boal - 0 - Part Chater 5

Figure 5.7. Demand curves for ordinary and Giffen goods Price Ordinary good Giffen good Quantity demanded Examle: Suose the demand for good # is given by = /(3 ) + /. s good # a normal good or an inferior good? Note that / = /(3 ) /, which is necessarily negative, assuming,, and are ositive. So an increase in always brings a decrease in the uantity demanded. t follows that good # is an ordinary good, following the "Law of Demand." Examle: Suose the demand for good # is given by = = / ( + ( ) / ). s good # a normal good or an inferior good? Note that, using the chain rule, / = [/( + ( ) / ) ] [ + (/)( ) -/ ] [ ]. This exression is comlicated, but note that each exression in brackets is necessarily ositive, assuming,, and are ositive. Since the whole exression begins with a negative sign, the whole exression is necessarily negative. So an increase in always brings an increase in the uantity demanded. t follows that good # is an ordinary good, following the "Law of Demand." Section 5.6: Effect of another rice on uantity demanded A change in another rice. f the rice of good # falls, while income and the other rice remain constant, then the budget line rotates out, away from the origin, but anchored on the axis of good #. Assuming good # is on the horizontal axis, the budget line becomes flatter. After such a shift, the consumer will choose a bundle on the new budget line (see figure 5.8). Again, the consumer will urchase more of one or both goods and will now reach a higher indifference curve. 06 William M. Boal - - Part Chater 5

Figure 5.8. Effect of decrease in rice of good #: the case of substitutes Substitutes versus comlements. Consider the consumer s urchases of good #, the good whose rice has not changed. Figure 5.8 seems to indicate that this consumer has resonded to the decrease in the rice of good # by urchasing less of good #, but there is no mathematical reason why this need always be the case. Figure 5.9 shows a different consumer who urchases more units of good # as the rice of good # decreases. f a consumer urchases fewer units of good as the rice of another good decreases, the goods are called substitutes. f a consumer urchases more units, the goods are called comlements. 6 Substitutes, as the name suggests, are tyically goods consumed in lace of each other. Similar foods like oranges and graefruits are substitutes. Different forms of energy like electricity and natural gas are substitutes. Different brands of the same item like Chevrolet cars and Ford cars are close substitutes. By contrast, comlements are tyically goods consumed together. Comuter hardware and software are comlements. Large automobiles and gasoline are comlements. 6 More recisely, these are the definitions of gross substitutes and gross comlements. Alternative concets, called net substitutes and net comlements, are defined with resect to movements along the same indifference curve. 06 William M. Boal - - Part Chater 5

Figure 5.9. Effect of decrease in rice of good #: the case of comlements Partial derivatives. The sign of the derivative of the demand function for one good with resect to the rice of another good reveals whether the goods are substitutes or comlements. For substitutes, the artial derivative is ositive: / > 0. For comlements, the artial derivative is negative: / < 0. Of course, many goods are unrelated in demand. For unrelated goods, the artial derivative is zero: / = 0. 7 Examle: Suose the demand for good # is given by = /( ). Are goods # and # substitutes, comlements, or unrelated? First, note that / = 0, since does not aear in the demand function. So has no effect on. Goods # and # are unrelated. Examle: Suose the demand for good # is given by = /(3 ) + /. Are goods # and # substitutes, comlements, or unrelated? Note that / = /, which is necessarily ositive, assuming is ositive. So an increase in always brings an increase in the uantity demanded of good #. t follows that goods # and # are substitutes. Examle: Suose the demand for good # is given by = = / ( + ( ) / ). Are goods # and # substitutes, comlements, or unrelated? Note that, using the chain rule, / = [/( + ( ) / ) ] [(/)( ) -/ ] [ ]. This exression is comlicated, but note that each exression in brackets is necessarily ositive, assuming,, and are ositive. Since the whole exression begins with a negative sign, the whole exression is necessarily negative. So an increase in always brings an 7 The definitions of gross substitutes and comlements are sometimes ambiguous. For examle, sometimes / < 0, but / = 0, as in examle 5 above. t can be roven that this ambiguity does not occur in the definitions of net substitutes and comlements. 06 William M. Boal - 3 - Part Chater 5

decrease in the uantity demanded of good #. t follows that goods # and # are comlements. Section 5.7: Summary Holding references constant, a consumer s choice of uantities to urchase is determined by her or his income and the rices she or he faces. Functions relating these uantities to income and rices are called demand functions. Demand functions must satisfy an adding-u roerty, which insures that sending euals income, and homogeneity of degree zero, which ensures that ure inflation has no effect on uantities demanded. For a number of realistic utility functions, demand functions can be found using the budget line and the tangency condition. The artial derivative of a demand function with resect to income is ositive for normal goods (the most common case) and negative for inferior goods. The artial derivative with resect the good s own rice is negative for ordinary goods and ositive for Giffen goods. However, real-world Giffen goods are rare or nonexistent. The artial derivative with resect to another good s rice is ositive for substitutes, negative for comlements, and zero for unrelated goods. 06 William M. Boal - 4 - Part Chater 5

Aendix to Chater 5: Demand Detailed ste-by-ste solutions for Examles (), (), and (3) given in Section 5.3: "Finding demand functions" Examle (): The consumer is assumed to have utility function U(, ) =. We seek formulas for the demand functions = (,,) and = (,,). Our method is to solve the budget line and the tangency condition jointly, treating,, and as fixed constants. Thus we have two euations (the budget line and the tangency condition) in two unknowns ( and ). The budget line is the same as always: + =. The tangency condition says that the sloe of the budget line must eual the sloe of the indifference curve. The sloe of the budget line, with on the vertical axis and on the horizontal axis, is as usual negative /. The sloe of the indifference curve is the negative of the consumer's marginal rate of substitution in consumtion (MRSC), whose formula deends on the utility function. The formula for the MRSC is found the usual way: MRSC = MU /MU, where the marginal utilities MU and MU are the artial derivatives of the utility function. For this articular utility function, MU = U/ = and MU =, so MRSC = / ( ) = /( ). So the tangency condition for this utility function is /( ) = /. Now we must solve these two euations jointly. There are many ways to do this. One way to begin is to multily both sides of the tangency condition by ( ) to get / =. Then we can substitute the left-hand side for ( ) in the euation for the budget line: + = + ( /) = (3/) = Dividing both sides by (3/) gives the demand function for good #: = ()/(3 ) Substituting this into the budget line gives + = ()/(3 ) + = (/3) + = Subtracting (/3) from both sides gives = /3 Dividing both sides by gives the demand function for good #: = ()/(3 ) Examle (): The consumer is assumed to have utility function U(, ) = ( -3). Again, we seek formulas for the demand functions = (,,) and = (,,). Our method is to solve the budget line and the tangency condition jointly, 06 William M. Boal - 5 - Part Chater 5

treating,, and as fixed constants. Thus we have two euations (the budget line and the tangency condition) in two unknowns ( and ). The budget line is the same as always: + =. The tangency condition says that the sloe of the budget line must eual the sloe of the indifference curve. The sloe of the budget line, with on the vertical axis and on the horizontal axis, is as usual negative /. The sloe of the indifference curve is the negative of the consumer's marginal rate of substitution in consumtion (MRSC), whose formula deends on the utility function. The formula for the MRSC is found the usual way: MRSC = MU /MU, where the marginal utilities MU and MU are the artial derivatives of the utility function. For this articular utility function, MU = U/ = and MU = ( -3), so MRSC = / ( ( -3)) = /( -6). So the tangency condition for this utility function is /( -6) = /. Now we must solve these two euations jointly. There are many ways to do this. One way to begin is to multily both sides of the tangency condition by (( -6) ) to get = ( -6). Then we can substitute the right-hand side for ( ) in the euation for the budget line: + = ( -6) + = (3 ) - 6 = Adding 6 to both sides gives 3 = + 6 Dividing both sides by (3 ) gives the demand function for good #: = ( + 6 ) / (3 ), which can also be written as = /(3 ) +. Substituting this into the budget line gives + = + (/(3 ) + ) = + (/3) + = Subtracting ((/3) + ) from both sides gives = - (/3) +, = (/3) +, Dividing both sides by gives the demand function for good #: = ((/3) + ) /, which may also be written as = ()/(3 ) + /. Examle (3): The consumer is assumed to have utility function U(, ) = (/ ) (/ ). This utility function looks eculiar at first, because utility is necessarily negative for all ositive values of and. But the key assumtions do not reuire that utility be ositive. They only reuire that marginal utilities be ositive ("monotonicity") and that the marginal rate of substitution be diminishing, which turn out to hold for this utility function. 06 William M. Boal - 6 - Part Chater 5

Again, we seek formulas for the demand functions = (,,) and = (,,). Our method is to solve the budget line and the tangency condition jointly, treating,, and as fixed constants. Thus we have two euations (the budget line and the tangency condition) in two unknowns ( and ). The budget line is the same as always: + =. The tangency condition says that the sloe of the budget line must eual the sloe of the indifference curve. The sloe of the budget line, with on the vertical axis and on the horizontal axis, is as usual negative /. The sloe of the indifference curve is the negative of the consumer's marginal rate of substitution in consumtion (MRSC), whose formula deends on the utility function. The formula for the MRSC is found the usual way: MRSC = MU /MU, where the marginal utilities MU and MU are the artial derivatives of the utility function. For this articular utility function, MU = U/ = / and MU = / (which are both ositive as reuired by the assumtion of monotonicity). So MRSC = (/ ) / (/ ) = / (which diminishes as decreases and increases, as reuired). So the tangency condition for this utility function is / = /. Now we must solve these two euations jointly. There are many ways to do this. One way to begin is to divide both sides of the tangency condition by, and take suare roots: / = /( ), / = ( /( )) /. Multily both sides by to get = ( /( )) /. Then we can substitute the right-hand side for ( ) in the euation for the budget line: + = ( ( /( )) / ) + = Now simlify the first term on the left side as follows: ( ( /( )) / ) = / -/ / / = / -/ / / = / / / / = ( / ) / = (0.5 ) /. So the budget line becomes (0.5 ) / + = Now, is a common factor on the left hand side, so we can write [ + (0.5 ) / ] + =. Dividing both sides of the euation by the exression in brackets gives gives the demand function for good #: = / [ + (0.5 ) / ], which may also be written as. 0.5 Now earlier we found that = ( /( )) /. 06 William M. Boal - 7 - Part Chater 5

06 William M. Boal - 8 - Part Chater 5 Substitute the demand function for good # ( ) into this euation to get / 0.5 0.5 0.5 0.5 This may be further simlified as follows: 0.5 which may also be written as. [end of aendix]

Problems (5.) [Budget constraint] Assume there are only two goods and that the demand for good # is given by = /. Substitute this into the euation for the budget line to find the formula for the demand for good #. (5.) [Budget constraint] Let denote the consumer s income, the uantity of housing chosen, denote the uantity of other goods chosen, the rice of housing, and the rice of other goods. Assume a consumer always sends one-fourth of her or his income on housing regardless of income, the rice of housing, or the rice of other goods. a. Write an euation reresenting this assumtion. b. Solve the euation to find the demand function for housing. (5.3) [Budget constraint and homogeneity] Consider whether the following functions might be legitimate demand functions for an individual consumer. 3 and 4 a. s the budget constraint satisfied by this demand system? (Assume there are only two goods.) Show your work, ste by ste. b. Are these functions homogeneous of degree zero in income and rices? Show your work, ste by ste. (5.4) [Budget constraint and homogeneity] Consider whether the following functions might be legitimate demand functions for an individual consumer. and 3 a. s the budget constraint satisfied by this demand system? (Assume there are only two goods.) Show your work, ste by ste. b. Are these functions homogeneous of degree zero in income and rices? Show your work, ste by ste. (5.5) [Budget constraint and homogeneity] Consider whether the following functions might be legitimate demand functions for an individual consumer. and 3 3 a. s the budget constraint satisfied by this demand system? (Assume there are only two goods.) Show your work, ste by ste. b. Are these functions homogeneous of degree zero in income and rices? Show your work, ste by ste. (5.6) [Homogeneity] Are the following functions homogeneous of degree zero in income and rices? Show your work, ste by ste. a. = 5 + 0.05 + 4. b. = 5 0.7-0.9 0.. 06 William M. Boal - 9 - Part Chater 5

(5.7) [Homogeneity] Are the following functions homogeneous of degree zero in income and rices? Show your work, ste by ste. a. = (/ ) ( / ). b. = 3.5-0.9 0.. (5.8) [Homogeneity] Are the following functions homogeneous of degree zero in income and rices? Show your work, ste by ste. a. = / (5 ). b. = 7-0. -0.8 0.4. (5.9) [Homogeneity] Are the following functions homogeneous of degree zero in income and rices? Show your work, ste by ste. a. = 35 + 0. 3 0.. b. = 3-0.3 0. 0.. (5.0) [Finding demand functions] Suose a consumer has utility function U(, ) = /3 /3 and has income. The rice of good # is and the rice of good # is. a. Give the euation for the consumer s budget line. b. Give a formula for the consumer s marginal rate of substitution in consumtion (MRSC) of good # for good #. [Hint: This is the sloe of the consumer s indifference curve when good # is on the vertical axis and good # is on the horizontal axis.] c. Find an exression for the consumer s demand for good # ( ) as a function of,, and. [Hint: Begin by setting the MRSC eual to /. Solve this euation for. Substitute the resulting exression in the budget line and solve for.] d. Find an exression for the consumer s demand for good # ( ) as a function of,, and. (5.) [Finding demand functions] Suose a consumer has utility function U(, ) = /4 3/4 and has income. The rice of good # is and the rice of good # is. a. Give the euation for the consumer s budget line. b. Give a formula for the consumer s marginal rate of substitution in consumtion (MRSC) of good # for good #. [Hint: This is the sloe of the consumer s indifference curve when good # is on the vertical axis and good # is on the horizontal axis.] c. Find an exression for the consumer s demand for good # ( ) as a function of,, and. [Hint: Begin by setting the MRSC eual to /. Solve this euation for. Substitute the resulting exression in the budget line and solve for.] d. Find an exression for the consumer s demand for good # ( ) as a function of,, and. 06 William M. Boal - 0 - Part Chater 5

(5.) [Finding demand functions] Suose a consumer has utility function U(, ) = ( -5) and has income. The rice of good # is and the rice of good # is. a. Give the euation for the consumer s budget line. b. Give a formula for the consumer s marginal rate of substitution in consumtion (MRSC) of good # for good #. [Hint: This is the sloe of the consumer s indifference curve when good # is on the vertical axis and good # is on the horizontal axis.] c. Find an exression for the consumer s demand for good # ( ) as a function of,, and. [Hint: Begin by setting the MRSC eual to /. Solve this euation for. Substitute the resulting exression in the budget line and solve for.] d. Find an exression for the consumer s demand for good # ( ) as a function of,, and. (5.3) [Finding demand functions] Suose a consumer has utility function U(, ) = ( -5)( -4) and has income. The rice of good # is and the rice of good # is. a. Give the euation for the consumer s budget line. b. Give a formula for the consumer s marginal rate of substitution in consumtion (MRSC) of good # for good #. [Hint: This is the sloe of the consumer s indifference curve when good # is on the vertical axis and good # is on the horizontal axis.] c. Find an exression for the consumer s demand for good # ( ) as a function of,, and. [Hint: Begin by setting the MRSC eual to /. Solve this euation for. Substitute the resulting exression in the budget line and solve for.] d. Find an exression for the consumer s demand for good # ( ) as a function of,, and. (5.4) [Finding demand functions] Suose a consumer has utility function U(, ) = / + / and has income. The rice of good # is and the rice of good # is. a. Give the euation for the consumer s budget line. b. Give a formula for the consumer s marginal rate of substitution in consumtion (MRSC) of good # for good #. [Hint: This is the sloe of the consumer s indifference curve when good # is on the vertical axis and good # is on the horizontal axis.] c. Find an exression for the consumer s demand for good # ( ) as a function of,, and. [Hint: Begin by setting the MRSC eual to /. Solve this euation for. Substitute the resulting exression in the budget line and solve for.] d. Find an exression for the consumer s demand for good # ( ) as a function of,, and. 06 William M. Boal - - Part Chater 5

(5.5) [Finding demand functions] The following three utility functions must yield exactly the same demand functions: U(, ) = -(/ ) (/ ), V(, ) = -(/ ) (/ ), W(, ) = -(0/ ) (0/ ) + 5. Exlain why, without solving exlicitly for the demand functions. (5.6) [Linear exenditure system] Suose a consumer has the utility function U(, ) = ( -a) b ( -c) d, where a, b, c, and d are arbitrary constants. a. Find an exression for the consumer s demand for good # ( ) as a function of,, and. b. Find an exression for the consumer s sending on good # ( ) as a function of,, and. c. Exlain why this utility function and its associated demand functions are sometimes called a linear exenditure system. (5.7) [Cobb-Douglas utility] Suose a consumer has the utility function U(, ) = a b, where a and b are arbitrary constants. a. Find an exression for the consumer s demand for good # ( ) as a function of,, and. b. Find an exression for the consumer s demand for good # ( ) as a function of,, and. c. Find an exression for ln( / ) as a function of,, and. d. The so-called elasticity of substitution is defined as. Find an exression for the elasticity of substitution. (5.8) [CES utility] Suose a consumer has the utility function U(, ) = a b + b, where a and b are arbitrary constants. a. Find an exression for the consumer s demand for good # ( ) as a function of,, and. b. Find an exression for the consumer s demand for good # ( ) as a function of,, and. c. Find an exression for ln( / ) as a function of,, and. d. The so-called elasticity of substitution is defined as. Find an exression for the elasticity of substitution in terms of a and/or b. e. Exlain why U(, ) = a b + b is called a CES (constant elasticity of substitution ) utility function. (5.9) [Exansion ath] Suose a consumer has the utility function U(, ) = -(/ ) (/ ) and faces rices =$ and = $3. What is the euation for the consumer s income-exansion ath? 06 William M. Boal - - Part Chater 5

(5.0) [Proerties of demand functions] Suose the urorted demand function for good # is suosed to be given by = (/) -/3 -/3. a. s this function homogeneous of degree zero in income and rices? Why or why not? b. Find an exression for /. s good # a normal good or an inferior good? Why? c. Find an exression for /. s good # an ordinary good or a Giffen good? Why? d. Find an exression for /. Are goods # and # comlements or substitutes? Why? (5.) [Proerties of demand functions] Suose the urorted demand function for good # is suosed to be given by = 5-4/5 /5. a. s this function homogeneous of degree zero in income and rices? Why or why not? b. Find an exression for /. s good # a normal good or an inferior good? Why? c. Find an exression for /. s good # an ordinary good or a Giffen good? Why? d. Find an exression for /. Are goods # and # comlements or substitutes? Why? (5.) [Proerties of demand functions] Suose the urorted demand function for good # is suosed to be given by = 3 ( ) -/ (), where = ( /CP), = (/CP), and CP = an index of consumer rices. Note that does not aear in this function, excet through the CP. a. s this function homogeneous of degree zero in income and rices? Why or why not? b. Find an exression for /. [Hint: Use the chain rule.] s good # a normal good or an inferior good? Why? c. Find an exression for /. [Hint: Use the chain rule.] s good # an ordinary good or a Giffen good? Why? [end of roblem set] 06 William M. Boal - 3 - Part Chater 5