Garrison Schlauch - CLAS. This handout covers every type of utility function you will see in Econ 10A.

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1 This handout covers every type of utility function you will see in Econ 0A. Budget Constraint Unfortunately, we don t have unliited oney, and things cost oney. To siplify our analysis of constrained utility axiization aking the best decision possible given a finite aount of oney let s iagine a siple world in which there are only two goods, and. These goods cost p and, respectively, and we have incoe. Then our budget constraint is p + =. Let be the x-axis variable and be the y-axis variable. Then solving for will give the budget constraint in slope-intercept for: = p. Slope = p p The x-intercept of p iplies that if we spend all of our incoe,, on good 2,, we can buy p units of. For exaple, suppose we have $0 to spend and haburgers cost $5. How any haburgers can we buy? Obviously, 2. But, where haburgers are good. Siilarly for. 2 = 0 5 = incoe price of haburger = p, The slope is the tradeoff rate between the two goods. Here s an exaple illustrating what that eans. Suppose we have $20 to spend, haburgers cost $5, and tacos cost $2. Let s say we always want to spend all our oney (frugality is overrated) and we are currently consuing 2 haburgers and 5 tacos. If we want to buy ore haburger, that akes 3 haburgers, which together cost $5. Then we only have $5 left to spend on tacos, so we buy 2.5 tacos. (Yes, we can buy half a taco #sipleeconoland). So by buying ore haburger, we had to decrease our consuption of tacos by 2.5 = 5 2 units. Now, let s use the given prices and incoe to write our budget constraint: = 20, where haburgers are good and tacos are good. Rewriting this in slope intercept for, we have =

2 Look at that! The tradeoff of buying ore unit of is slope. Note: interpreting the slope like this technically only works for linear functions, e.g. the budget constraint and the perfect substitutes utility function (see below). The distinction for non-linear functions is not crucial for this class, but it has been tested before. See Midter Spring 205 #3. Solving Utility Functions You will see five ain types of utility functions in this class. 0. Understanding Let s first explore the Cobb-Douglas utility function to further our understanding constrained utility axiization aking the best decision possible given a finite aount of oney. u(, ) = Ax α x β 2 Definition: Indifference Curve. A collection of bundles such that each bundle in the collection yields the sae utility, i.e. each point on a given indifference curve gives you the sae aount of utility, i.e. every point on a given indifference curve is just as good as another. For the Cobb-Douglas utility function, the indifference curves look like this: Adding in the budget constraint (reeber, our optial choice ust be affordable, i.e. on or below the budget constraint):

3 Assuing onotonicity, higher indifference curves are better, i.e. they give us ore utility. Thus we want to be on the highest possible, affordable indifference curve. The lowest indifference curve crosses the budget constraint twice, so we can afford it. But we can also afford the point arked by the black dot, which, since it s on a higher indifference curve, gives us ore utility. We can t afford any points on the highest indifference curve since every point on that curve is above the budget constraint. Consuing the cobination of goods given by the black dot axiizes our utility subject to our budget constraint. Looking at the graph, we see that the second indifference curve is tangent to our budget constraint at that point, i.e. they have the sae slope. We call this the tangency condition. Let s figure out how to find that black dot algebraically, using soe tools fro calculus. Definition: Marginal Utility. The aount of utility we get fro one ore unit of a good. (Again, technically only correct for linear functions, but don t worry about it). We can express this idea ore precisley with a derivative: MU x = u(x, y) x Definition: Marginal Rate of Substitution (MRS). The aount of good 2 the consuer needs to be just as happy if she gives up unit of good. (Only for linear functions, blah blah blah). Does that see failiar? We re looking at how uch the y-axis variable, good 2, changes if we give up unit of the x-axis variable, good. That s the slope, which is the sae thing as the derivative. Since u(, ) is a function of 2 variables, and, to find its slope we can use this forula: MRS = MU x = MU y u(, ) u(, ) Definition: Diinishing Marginal Rate of Substitution (DMRS). As we get ore and ore of one good, we re willing to trade ore of it to get unit of the other good, i.e. we prefer balanced consuption of both goods rather than extree consuption of one. A utility function with a DMRS will have indifference curves like the ones we ve seen ones that look like bananas. An indifference curve with an Increasing MRS will have indifference curves that look like...upside-down bananas:

4 To use our tangency condition, we need a DMRS. For an MRS to be diinishing, it ust be that MRS 0 and MRS 0 If you aren t cofortable with derivatives, check out the How to find a diinishing MRS video on the CLAS Econ0A website under the Skill Videos tab. Given that we have a DMRS, by using the tangency condition we can find and that axiize our utility subject to our budget constraint. At the point of tangency, the slope of the indifference curve the MRS is tangent to, i.e. equals, the slope of the budget constraint. Recall that the slope of the budget constraint is p. Thus, the black dot is the point such that u(, ) x MRS = = p u(, ) Since these are both negative, we ll usually ignore the inus sign and write the tangency condition as MRS = u(, ) = p () u(, ) We re alost done. To ake sure this point is affordable, we need to plug in our solution fro () into the budget constraint. Fro here we solve for and using both the budget constraint and equation (). 0.2 Cobb-Douglas Exaple: u(, ) =, (p,, ) = (, 2, 0). Find, that axiize utility subject to the budget constraint p + =.. Write down the budget constraint. p + = 2. Find MRS. u(, ) = u(, ) 3. Check for DMRS. = considering positive aounts of goods). 4. Tangency condition. MRS = = p 0 and = 0 for all > 0 and > 0 (we re only 5. Solve for one of the goods (it doesn t atter which). = p 6. Plug (5) into the budget constraint. p + ( p ) = x p = 2 2p Note: don t plug in the given values for p, and until the END of the proble. Later in the class, you will need to work with the general solution. Best get into good habits now. Also, we usually use x to denote our final solution. 7. Plug (6) into (5). = p ( 2p ) x 2 = 2 8. Plug in the given prices and incoe. x = 0 2 = 5, x 2 = = 5 2 Note: as a check, you should always ake sure your solution is affordable. In this case, = 0 = (5) + 2( 5 2 ) = p x + x 2.

5 0.3 Perfect Substitutes These utility functions take the for u(, ) = a + b + c. Method : Use the MRS Suppose u(, ) = +. Let s first draw soe indifference curves. Now for the budget constraint. There are two ain cases: Figure. MRS > p Figure 2. MRS < p In Figure, we can afford both the green and the black dot since they re both on our budget constraint. But the black dot is on a higher indifference curve, so we prefer the black dot to the green dot. At the black dot, we consue all and no ; hence our utility axiizing deands for goods and 2 are x = p, x 2 = 0. Siilarly, in Figure 2 our utility axiizing deands for goods and 2 are x = 0, x 2 =.

6 We call these corner solutions, since our solution is at a corner of the graph. The last case is where MRS = p. In this case, every point on the budget constraint gives us the sae utility, so every bundle of goods on the budget constraint axiizes our utility. Just pick one. Exaple: u(x, y) = 3x + 2y and (p x, p y, ) = (, 2, 0). Then MRS = 3 2 > 2 = p. Thus we consue all x and no y. Our utility axiizing deands are thus Method 2: Bang for Your Buck x = p x = 0 = 0, y = 0. For this ethod, we want to find which good gives us the ost bang for our buck, i.e. the ost utility per dollar. Exaple: u(x, y) = 3x + 2y and (p x, p y, ) = (, 2, 0). Then and MU x p x = MU y p y = u(x,y) x p x = 3 = 3 u(x,y) y p y = 2 2 = Hence, good x gives us ore utility per dollar. Therefore, we will spend all of our oney on good x and none on good y. Our utility axiizing deands are thus Method 3: Plug n Chug x = p x = 0 = 0, y = 0. For this ethod, we want to find which corner, i.e. intercept, gives us the highest utility. Recall that, in general, the corners are x = p x, y = p y. Exaple: u(x, y) = 3x + 2y and (p x, p y, ) = (, 2, 0). Then u( p x, 0) = 3 p x = 30 and u(0, p y ) = 2 p y = 0. Thus consuing all of good x gives us ore utility than consuing all of good y. Our utility axiizing deands are thus x = p x = 0 = 0, y = 0.

7 0.4 Leontief/Pefect Copleents These utility functions take the for u(x, y) = in{a, b }. The indifference curves look like this: Adding a budget constraint, Fro the graph, we see that the highest indifference curve we can hit is the one associated with the green dot. Notice that at this point, =. That s the key to solving these utility functions. Exaple: u(, ) = in{a, b }, (p,, ) = (2,, 0). subject to the budget constraint p + =. Find, that axiize utility. Write down the budget constraint. p + = 2. Set the interior of the in function equal to itself. a = b 3. Solve for one of the goods (it donsn t atter which). = b a

8 4. Plug (3) into the budget constraint. p ( b a x 2) + = = b a p + x a 2 = bp + a 5. Plug (4) into (3). = b a ( a bp + a ) x = b bp + a 6. Plug in the given prices and incoe. x = 0b 2b + a, x 2 = 0.5 Max These utility functions take the for The indifference curves look like this: u(x, y) = ax{a, b }. 0a 2b + a Adding a budget constraint, p

9 Fro the graph, we see that the highest indifference curve we can hit is the one associated with the black dots. Notice that at this point, either x = p and x 2 = 0 or x = 0 and x 2 =. Just like in the case of perfect substitutes, we have a corner solution. For these probles, you can use Method 3: Plug n Chug fro Perfect Substitutes. Exaple: u(, ) = ax{3, 2 }, (p,, ) = (2,, 0). subject to the budget constraint p + =. Find, that axiize utility. Plug the intercepts into the utility function. ax{3( p ), 2(0)} = 3 p, ax{3(0), 2( )} = 2 2. Plug in the given prices and incoe. Whichever value is greater, we will consue that corner since it gives us ore utility. Hence, 3 p = 30 2 = 5, 2 = 20 = 20. x = 0, x 2 = = Quasi-Linear These utility functions take the for u(, ) = + f( ), where f is soe non-linear function. We call this quasi-linear because half of the utility function is linear, and half is not. The indifference curves look like this: f is increasing f is decreasing In both cases, f > 0 since ore gives us ore utility. Here, I use increasing and decreasing to ean the rate at which f is increasing or decreasing. If f is increasing, it is increasing at an increasing rate; hence, f > 0. Siilarly, if f is decreasing, it is increasing at a decreasing rate; hence, f < 0. For exaple, f(x) = is increasing since f (x) = 2 > 0 for all x > 0, and f(x) = 2 is decreasing since f (x) = < 0 for all x > 0. 4x 3 2 Adding a budget constraint,

10 f is increasing f is decreasing Note that when f is increasing, we have a corner solution. When f is decreasing, our picture looks a lot like a Cobb-Douglas. Keep that in ind in the following exaples. Exaple (Increasing): u(, ) = + 2, (p,, ) = (, 2, 0). utility subject to the budget constraint p + =. Find, that axiize. Plug the intercepts into the utility function. u( p, 0) = p, u(0, ) = ( ) 2 2. Plug in the given prices and incoe. Whichever value is greater, we will consue that corner since it gives us ore utility. p = 0, ( ) 2 = 25. Hence, x = 0, x 2 =. Exaple (Decreasing): u(, ) = 2 +, (p,, ) = (, 2, 0). Find, that axiize utility subject to the budget constraint p + =.. Write down the budget constraint. p + = 2. Find MRS. u(, ) = u(, ) Check for DMRS. = 0 and 2 2 4x 3 2 only considering positive aounts of goods). 2 2 = 0 0 for all > 0 and > 0 (we re 4. Tangency condition. MRS = = p Solve for the good you can solve for. = p2 2 4 Note that since is already in ters of p,, and, this is our solution for. 6. Plug (5) into the budget constraint. p ( p2 2 4 ) + = x 2 = 4p Note that if < 4p, < 0, which is ipossible. In this case, x 2 = 0 and x = p

11 7. Plug in the given prices and incoe. First we ust check if < at the given prices and 4p incoe. = 0 2 = 5 > 2 =, 4p so we will use the deand functions Thus at the given prices and incoe, x = p2 2 4 and x 2 = 4p x = (2)2 4() 2 =, x 2 = (0) (2) (2) 4() = 9 2.

12 0.7 Practice Probles For each of the following, draw the indifference curves and budget constraint, and find the utility axiizing deands. Reeber, don t plug in the given prices and incoe until the end.. u(, ) =, (p,, ) = (, 2, 0) 2. u(, ) = 2 3 2, (p,, ) = (, 4, 20) 3. u(, ) = + 2, (p,, ) = (, 3, 20) (try all 3 ethods) 4. u(, ) = 2 + 5, (p,, ) = (3, 2, 30) (try all 3 ethods) 5. u(, ) = in{2, }, (p,, ) = (,, 0) 6. u(, ) = in{ 2, 4 }, (p,, ) = (,, 0) 7. u(, ) = ax{2, }, (p,, ) = (,, 0) 8. u(, ) = ax{ 2, 4 }, (p,, ) = (,, 0) 9. u(, ) = 2 + 2, (p,, ) = (, 2, 00) (first, check if increasing or decreasing) 0. u(, ) = + ln, (p,, ) = (5,, 6) (first, check if increasing or decreasing). u(, ) = + ln, (p,, ) = (5,, 4)