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1 Intermediate Microeconomic Theory: ECON 5: Alication of Consumer Theory Constrained Maimization In the last set of notes, and based on our earlier discussion, we said that we can characterize individual choice as the maimization of an individual s utility, ma, ma, (,, subject to the individual s budget constraint U u ) y +. That is the roblem is a constrained roblem, where the individual chooses good and to maimize her utility subject to the constraint her income imoses on her. We can solve this using the agrangian Method which we will discuss later. However, that some roblem can be re-eressed as a unconstrained roblem. y When we changed the utility function to U u,, what we had done is to change the roblem from a constrained roblem to an unconstrained roblem in terms of only one choice, choice of good. We can then solve for the otimal choice of good for the individual once we have chosen good based on the budget constraint. We will now take secific eamles by assuming secific utility functional forms and eamine how choices change. Cobb-Douglas et Norman be an analyst for Merrill ynch, who makes investments of his own on the side based on his ecellent knowledge of economics. He discilines himself by dedicating only $y from his total income to his investments. He has to choose between investing in government bonds (a safe bet!) and his favorite, the real money makers, N. American blue chi stocks like lackberry. et us denote the quantities of each tye of instruments as and for the bonds and blue chi shares resectively. His utility ( ) ( ) function is of the Cobb Douglas form, U. Following our earlier analysis, we can rewrite his roblem as a unconstrained roblem. First, let us write down his budget constraint y +. The Norman solves in his mind the following roblem, ma or equivalently subject to, ( ) ( ) y + y ma ( ) Note that in this form, Norman only solves for his choice of. To solve this maimization roblem, he needs first to differentiate the utility function with resect to. The first order condition or the oint where his utility is greatest is

2 Intermediate Microeconomic Theory: ECON 5: Alication of Consumer Theory ( ) ( ) ( ) y y y ( ) y * + Using the budget constraint, it is easy for you to show that ( ) y * + There are several things we can say based on these solutions.. As rice of the blue chis rise, the otimal choice of investment in blue chis falls. As you vary rice of blue chis holding rices of all other instruments, and income fied, you will trace out the Price Consumtion Curve (Diagram 5. of your tet on age 08).. The greater the coefficient and is, the greater the otimal levels of shares and bonds. 3. The greater the amount Norman dedicates to his investments, say because of increase in income, both his investment in shares and bonds will rise. That is as an individual s income rises, holding relative rices constant, consumtion of all goods rise (assuming all goods are normal. We will talk about this shortly). The line that traces the changes in consumtion when income changes gives you the Income Consumtion line on the same diagram of the indifference curve. The diagram of the relationshi between income and consumtion levels is known as the Engel Curve (It is on figure 5. on age 0 of your tet). Question: What determines the rate of change in investments in shares and bonds as his income rises? 4. More generally, the two levels of consumtion describe Norman s demand for the two roducts. Why? Well, what relationshi does this equation tell you? The relationshi between quantities demanded against the rice of the good. If you hold rice of the other good and income constant, isn t that your standard definition of demand? Can we write it in a form that vaguely resembles what we have seen in demand and suly analysis? ure, take natural logs on both sides say for bonds ln * ln y ln ln + ln y ( ). Diagram will + ( + ) be shown in class. It is as in your tet in chater After thought. We have assumed that Norman s choice is urely derived from having the shares and bonds. ut in and of itself, that stack of aer has absolutely no value. Can you think about how you can model the risk involved with the two tyes of instruments? y 0

3 Intermediate Microeconomic Theory: ECON 5: Alication of Consumer Theory Question: uose Hutan is similarly a young ustart trader with Goldman achs, who has made his first $0m by the time he turned 5 (Of course, we haven t accounted for his ta liabilities!). However, he is really gung ho, and chooses only between foreign currency futures, and otions on those futures, i.e. he is going to use his otions to limit his down side eosure on the futures. ut suose we are not concerned with the intricacies of his secret. et s say all we know is that it seems like he gets utility from just owning the instruments. ma F, O subject to y F F + O O What is his otimal holding of the two instruments, or his demand for them? What haens to the otimal levels consumed when relative rice changes? O F Quasilinear Utility Consider a simle maimization roblem where the individual chooses between two goods. This is one form of the quasilinear utility function subject to the usual budget constraint. ma, + ( ) subject to y + The marginal utility from consuming good is, and the marginal utility of consuming good is ( ). Then the otimal choice for consumtion using the equilibrium choice condition is ( ) Note that in the case of a quasilinear utility, the non-numariare good, is not deendent on the income realization of the individual. This is a feature of the quasilinear utility function. ut this is true only for some levels of demand for good, when the solution, or the choice made by the individual is ositive for both goods. There will come a oint when it has to be deendent on income. To see this, we first find the otimal level of the numeraire good, good. And by the budget constraint, the consumtion of good is y y + + y 3

4 Intermediate Microeconomic Theory: ECON 5: Alication of Consumer Theory Note that it is ossible that consumtion level for good can be negative. We know that cannot be true, since we cannot consume negative amounts. First let s find the condition when consumtion of good is set to 0, i.e. when the solution is a corner solution. That is when y. When this condition is fulfilled the demand function for good changes from to that derived from the budget constraint when consumtion of good is zero, which is y. Question:. et Karly s utility that she derives from consuming clean air and all other goods that we as humans tyically desire be U a C + b A, where the subscrit C and A denote clean air, and all other goods resectively. et it be a C that >, where a and b are ositive constants. What is the otimal level b A of clean air and all other goods consumed by Karly? Deict the equilibrium on a well labeled diagram. What if the inequality is reversed? (When solving, be aware that the sloe of the budget constraint is negative) [No Calculus needed]. What is her utility is of the Cobb-Douglas form? (Hint: Use the formulas I ve gave you in class) 3. What if her utility is of the eontieff form such as U min{ a C, b A }. For starters, let a be and b be. Find the solution to the otimal level of consumtion. Can you generalize now to any value of a and b? (Hint: Draw out the various indifference curves for given levels of consumtion, and see if you can see a attern.) [No Calculus needed] Other Variations There are several ways in which changes in the budget constraint can alter individual behavior. Consider the simle eamle of the quota we have discussed earlier.. Consider the simlest Cobb Douglas Utility, under what circumstance would a quota affect the individual s choice, and when it would not.. Consider the imact of subsidies on goods only when consumtion is below certain level deemed desirable. 4

5 Intermediate Microeconomic Theory: ECON 5: Alication of Consumer Theory abor eisure Choice o far we have talked about an individual s choice for goods, which generates a demand function for goods. When we talk about an individual s choice of labor suly, consumer theory can also be used, but the otimal choice generates a suly function. How can we structure such an eloration? imle, consider first what are some of the constraints that would bind on an individual. For simlicity, let s consider the choice as between leisure, and consumtion. What this does, is that it converts the goods into goods. et leisure be, and consumtion be C. The first constraint she faces is the time constraint. et total time be T, and labor time be H. The constraint is then, T H + The net constraint is the budget constraint, and is written as C wh + Y where C is the individual s consumtion, and Y is unearned income of the individual (Think of it as rior savings). We have imlicitly normalized the rice of consumtion as, since the coefficient to C is just. Notice that this two constraints can be combined into one. C w( T ) + Y. Now let the individual s utility be U u ( C, ). o the choice is between consumtion and leisure, both goods. Question is will the same equilibrium choice formula s derive. Of course! et s show this: We can rewrite the roblem as ma u( C w( T ) + Y, ). o the individual s utility is highest, or maimized dc uc + u 0 d u w + u 0 C ( ) u w uc This is just the equating of the marginal rate of substitution between consumtion and leisure and the relative rice. This is the demand for leisure function. However, we can substitute the solution of leisure here, *, and substitute this into the time constraint to get the labor suly function. Consider the simle case of a Cobb-Douglas utility, U C. The otimal choice for leisure is C C w( T ) + Y w C w T + Y w * H ( ) Y + wt w * ( + ) T * Y + wt T T Y ( + ) w ( + ) ( + )w 5

6 Intermediate Microeconomic Theory: ECON 5: Alication of Consumer Theory Here are some comarative statics from this solution (y comarative statics, we mean to eamine how the solution changes as the arameters changes, i.e. what is the individual s choice when wages, and unearned income changes.). When wages increase in this setu, labor suly increases.. When unearned income rises, labor suly falls. (The lottery effect) Question:. What is the labor suly when the utility function is U ac + b? Describe the comarative statics. Intertemoral Consumtion Choice We can also use the consumer choice theory we have learned here to think about how individuals chooses between consumtion in two eriods (This can easily be etended into multile eriods). How can we model such a situation? We can think of consumtion in each eriod as different goods. et C and C be the consumtion of goods in eriods and. We can let the utility be say of the Cobb Douglas form, then the otimal choice would be deendent on the coefficients. ut such a way of thinking about it ignores how each individual values the future alters how she chooses between when she consumes, and when she saves. et s consider the following utility u( C ) + u( C ). Where is just the discount factor. et each eriod have its own rice level, and let the individual s income in each eriod be w and w. et s allow the individual to borrow between the two eriod, that is she can consume more than her income would allow in the first eriod through borrowing, and consume less, thereby saving. What would the otimal choice be deendent on? 6