Adv. Micro Theor, ECON 6-9 Assignment Answers, Fall Due: Monda, September 7 th Directions: Answer each question as completel as possible. You ma work in a group consisting of up to 3 members for each group please turn in onl set of answers and make sure all group member names are on that set of answers. All group members will receive the same grade.. Consider the following utilit functions: u(x ; x ) = p x x and v(x ; x ) = ln(x ) + ln(x ): Verif that u and v have the same indi erence curves and the same marginal rate of substitution. Explain wh. The marginal rate of substitution for u (x ; x ) is: When we have v (x ; x ) the MRS is: MU x = x x MU x = x x MRS = x x x x MRS = x x MU x = x MU x = x MRS = x x MRS = x x Note that since the marginal rates of substitution are the same the indi erence curves will be the same. Also note that the utilit values will be di erent for the same bundles it is NOT required that the same bundles have the same utilit values. What is required is that all the bundles that have u (x ; x ) = z have v (x ; x ) = s where s tpicall will not be equal to z. Think of it this wa if I plot the indi erence curve that runs through the points (; 4), (; ), (4; ) basicall, the indi erence curve for u (x ; x ) =, I get the following picture (based on the fact that = x = x = implies x = 4 x ) drawn in green:
x 9 8 7 6 5 4 3 3 4 5 6 7 8 9 x If I now plot the indi erence curve that runs through the points (; 4), (; ), (4; ) basicall, the indi erence curve for v (x ; x ) = ln 4, I get the exact same picture (since ln 4 = ln x + ln x implies 4 ln x = ln x which further implies 4 x = x ) denoted b the dashed red line.. Graph an indi erence curve, and compute the marginal rate of substitution and the Marshallian demand functions for the following utilit functions: a Perfect substitutes: u(x ; x ) = x + x, where > and >. The marginal rate of substitution is simpl the slope of the indi erence curve (or the ratio of the marginal utilities). In this case: So we have: MU x = MU x = MRS = While the linear utilit function is di erentiable, if we attempt to use Lagrange s method (and assume an interior solution), we will get something that looks like: = p p Now, this ma or ma not be true. If it is true, then we have a Marshallian demand correspondence (and not a function), as an point along the budget constraint will be an optimal solution to the problem. Hence we would have the set: x R L + : px = If we were to have: p > p
then the marginal utilit per dollar spent on good is higher than that of good. would onl purchase good and we would be at a corner solution where: x = x = p So the consumer Finall, if p > p ; p. To sum- then the consumer would purchase onl good and the optimal bundle would be marize: if p x (p; ) = > p p if p < p x (p; ) = p if p > p if p < p x R L + : px = if p = p b Perfect complements: u(x ; x ) = minfx ; x g, where > and >. The indi erence curves for this utilit function look like: x 9 8 7 6 5 4 3 3 4 5 6 7 8 9 x Since this utilit function is nondi erentiable, we cannot use Lagrange s method. However, looking at the indi erence curves shows us that an optimal consumption bundle must be at the kinked point of the L-shape. This is because at an other point the consumer will be "wasting" mone b buing too much of one good or the other. This means that x = x at the optimum. Solving for x we 3
have x = x. Now we can plug this into the budget constraint and nd x : To nd x we have: = p x + p x x = p + p x = p x + p x p + p = x x = x x = x = p + p p +p For the marginal rate of subsitution we can look at the gure: MRS = when x > x (the vertical portion of the indi erence curve); MRS = when x < x (the horizontal portion of the indi erence curve), and MRS is not well de ned when x = x (the kink in the indi erence curve). 3. We have noted that u(x) is invariant to positive monotonic transformations. One common transformation is the logarithmic transform, ln (x). Take the logarithmic transform of the Cobb-Douglas utilit function; then using that as the utilit function, derive the Marshallian demand functions and verif that the are identical to those derived in class. The utilit function used in class was: u (x ; x ) = x x The demand functions we found were: x (p; ) = x (p; ) = ( + ) p ( + ) p Taking the natural log leads to: Setting up the Lagrangian we have: v (x ; x ) = ln (u (x ; x )) v (x ; x ) = ln x + ln x L (x ; x ; ) = ln x + ln x + [ p x p x ] We know there will be an interior solution (since ln () is unde ned), so we can take rst order conditions and set them equal to zero: @L @x = x p = @L @x = x p = @L @ = p x p x = 4
Using the rst two partial derivatives we have: Now using the budget constraint we have: For x : = x p x p x p = x p = p x + p x = p x p p = x p + p x = x p + p x = x p ( + ) p ( + ) = x x = x p p x = p x = ( + ) p p (+) + p x p So these demand functions are the same as the ones we derived in class. 4. A consumer of two goods faces positive prices and has a positive income. Her utilit function is Derive the Marshallian demand functions. u (x ; x ) = max fax ; ax g + min fx ; x g, for < a < For this problem the utilit of the consumer is determined b the relationship between x and x : if x > x : u (x ; x ) = ax + x if x = x : u (x ; x ) = ax + x if x < x : u (x ; x ) = ax + x Note that this creates a piecewise linear function, although the middle "equation" is not reall a line but a point since we need x = x. Graphing this for a = 3 and u =, this leads to the vertex being at x = x = 9. For the remaining pieces of the function we have: Plotting this we have: if x > x : 3 x = x if x < x : 36 3x = x 5
x 4 3 5 5 5 3 35 4 x Now, let s suppose that p = p = and that income is 8. the picture we have: Then plotting the budget constraint on x 4 3 5 5 5 3 35 4 x So in this instance we will have the optimal solution at the vertex where x = x (note that we. So we can nd the Marshallian demand b substituting x for x into a general budget constraint and solving for x : Since x = x we also have x (p; ) = = p x + p x = p x + p x p + p = x p +p. It looks like we are done, but when we had the simple linear function we had a corner solution with either ; p or p ; as the optimal solution. What if we had that p = 5 and p =? The budget constraint is now (with = 36): 6
x 4 3 5 5 5 3 35 4 x Because the budget constraint is steeper than either of the slopes of the pieces of the indi erence curve we will end up at a corner solution. The same will be true if the slope of the budget constraint is atter than either piece. If it is steeper we end up with all x, and if it is atter we end up with all x. So the actual Marshallian demand function is: if p ; p a : p if a < p < p a : x (p; ) = x (p; ) = p + p if p a : ; p p Technicall if the price ratio is equal to one of the slopes we have a demand correspondence (there are man bundles which will lie along both the indi erence curve and the budget constraint). 5. Bob consumes ice cream cones (x ) and hamburgers (x ). His utilit function is u(x ; x ) = (x ) (x ) Bob s income is $. The price of each hamburger is $. The price of ice cream depends on the quantit that Bob consumes. Speci call, he can bu the rst ten ice cream cones at the price of $ each. For each additional ice cream cone there is a discount, and Bob has to pa onl $ each. Derive Bob s budget constraint and compute his optimal consumption plan. Bob s budget constraint has a kink in it. The plot below provides his budget constraint: 7
hamburgers 5 4 3 3 4 5 6 7 8 9 ice cream When x, Bob s budget constraint is given b: = x + x x = 5 x When x >, Bob s budget constraint is given b: ( ) = (x ) + x x = 45 x The reason is that he has alread spent $ on his ice cream cones. If we set up the Lagrangian for the original budget constraint we nd that: L (x ; x ; ) = (x ) (x ) [ p x p x ] Now, we can work through the entire problem, but if we go to problem 3 we know that when u (x ; x ) = x x, we have the following Marshallian demand functions: x = x = ( + ) p ( + ) p Using our parameters with the rst budget constraint we have that Bob would consume: x = x = = 5 + = 5 + However, x >, so now Bob s "optimal" consumption bundle of (5; 5) is inside his feasible set and not on the budget constraint. To see this look at the gure below: 8
hamburgers 5 4 3 3 4 5 6 7 8 9 ice cream The diamond shaped green point is at (5; 5). constraint, with = 9 and p =, to nd: x = x = So we then use the parameters from the new budget 9 = 45 + 9 = :5 + Looking at the picture below, we see that at (45; :5) the indi erence curve is tangent to the budget constraint: hamburgers 5 4 3 3 4 5 6 7 8 9 ice cream 9