CLAS. Utility Functions Handout

Transcription

1 Utility Functions Handout Intro: A big chunk of this class revolves around utility functions. Bottom line, utility functions tell us how we prefer to consume goods (and later how we want to produce) so that we maximize our utility (benefit). Some goods need to be consumed together in a fixed ratio, such as peanut butter and jelly (perfect complements). Some goods hurt us, or have negative utility, such as bad milk. Sometimes two goods serve the same purpose, and the consumption of one does not make the other better, such as Pepsi and Coke, since drinking Pepsi does not make Coke better (perfect substitutes). Most of the time our consumption will be restrained by a constraint, for example, a budget constrain. Using our utility functions (how we want to consume) and our constraint (the restriction on how much we can consume) we can find what our demand for each good is. We are going to spend a lot of time finding demand. Our functions include perfect substitutes, perfect complements, Cobb-Douglas, quasilinear increasing and decreasing, and max functions. Throughout this class, we will look at the SAME utility functions from DIFFERENT angles. At first, we look at the utility functions using the budget constraint. Later we will look at the utility functions from a different angle using the intertemporal budget constraint (a time constraint). When look at utility functions from different angles, their underlying characteristics STAY THE SAME, but solving the problems is slightly different. When we are looking at substitutes, regardless of whether we are using the budget constraint or the intertemporal budget constraint, we will want all of the good that gives the most bang for our buck. However, how we get there may be slightly different depending on what angle we are utilizing. When we go over a new angle at which to view preferences, the lecture notes and problems sets may only cover this new angle for some of the preferences (usually Cobb-Doug, maybe substitutes and complements). It would take the professor forever to cover every possibility. However, you are expected to know how to solve ALL of the applicable utility functions from the new angle. This sheet talks about the utility functions using the budget constraint, but the general rules learned here apply to preferences no matter what angle we are using. Learning how to identify, characterize, and solve all of the preferences now will make a significant portion of this class a lot easier. This guide is not exhaustive, nor does it replace lecture, section, hard work, ect. Take your time looking over this, and revisit it when you have done more problems. Best of luck, and I hope this helps!

2 Perfect Substitutes U(X,Y)=aX +by Examples: Pepsi and coke OR Ice-cream and frozen yogurt Since the marginal utility of x and the marginal utility of y (Mux and Muy) are constant, the marginal rate of substitution (MRS) is constant (EX MRS= -2 OR MRS = 5 OR MRS =1/2) Since the slope of the Indifference curves (IDCs) is the MRS, the IDCs are represented by a constant slope Examples: X + Y OR 4X + 5Y OR 10X +Y Any monotonic transformation of X + Y, Such as (X + Y) 3 OR X + Y +100 Think conceptually: Buy all of the good that gives the most BANG for your buck If both goods give the same bang for your buck, then you are indifferent between the two goods. You could consume all of good X, all of good Y, or some combination of the goods. o Methods (know all of the different methods!) 1. Sketch a graph: Compare the MRS (Mux/Muy) to the price ratio (Px/Py) 2. Compare Bang for Buck: Compare the utility per dollar for both goods (Mux/Px and Muy/Py) 3. Compare utility levels: find utility if all good 1 is purchased versus all good 2 purchased Check: am I getting the most utility possible?

3 Complements (Leontief) U(X,Y)=Min(aX,bY) Examples: Left and right shoes OR cups of coffee and sugar Examples: Min (X, Y/2) OR 5Min (X, Y) OR Min (5x, 5Y) Any Monotonic transformation of Min(aX, by) such as Min(aX, by) OR 3Min(aX,bY) (Note: monotonic transformations do NOT change the ratio at which you want to consume your goods. They DO change the total and marginal utility) Consume in a fixed ratio, where the two variables inside the Min function are equal. Think conceptually: I want to make the MOST bundles I can, while paying the least amount. This means I will not buy any extra of a good that will not contribute to a bundle. If we consume at U(x,y)= Min(10,50), our utility is 10, the lowest number. The extra 40 units of Y are not helping us at all, and it would receive the same amount of utility if we did not have them: Min(10,10)=10. We are buying 40 extra units of Y that are not increasing our utility at all! o Making peanut butter jelly sandwiches Example: Min(4X, 2Y). X= peanut butter and Y=jelly I want to make the most sandwiches I can. 1. Set the interior equal to find the fixed ratio at which to consume a. Consume at a fixed ratio 4X = 2Y X = Y/2. This means that for every unit of peanut butter, we want 2 units of jelly. This is the ratio needed to make a sandwich 2. Plug X=Y/2 into the budget constraint 3. Once one variable is solved, plug it into the budget constraint or the ratio (from setting the interior of the Min function equal) to get the other 4. Check: do I have an excess of either good?

4 Cobb-Douglas U(X,Y)=aX α by β Examples: 2X 1/2 Y 1/3 OR 2X 2 Y Any monotonic transformation of ax α by β MRS is always diminishing (meaning our graph is convex), allowing you to use the tangency condition. Tangency tells us where are budget line (what we can afford) intersects our IDC (what we want) The slope of the budget line is P1/P2 and the slope of the IDC is the MRS, so tangency is where MRS=Px/Py This tells us where we want to consume, AKA the best IDC we can afford Think Mathematically: what are the steps to find demand in a Cobb-Douglas function? 1. Find MRS 2. Test for diminishing MRS (also known as DMRS)* 3. Set MRS = Px/Py 4. Plug answer from 3 (solved for either X or Y) into the budget constraint 5. Plug in solved variable into the budget constraint or MRS=Px/Py to find the other Variable. Check: is my math right? *Testing for DMRS is not necessary with Cobb-Douglas, since we know it has DMRS. It is, however, a good step to show on tests, and a good habit to have. Test for DMRS X Y If MRS decreases as X increases and Y decreases, the function Has DMRS. If MRS is constant for an increase in X, but is MRS? decreasing for a decrease in Y, it is still has DMRS. When one is constant, the other dominates.

5 Quasi-Linear U(X,Y)=V(X) + by For decreasing quasi-linear Examples: X + ln(y) OR 2X + Y 1/2 OR X 2 + Y Any monotonic transformation of V(X) + by If diminishing MRS Ex ( X + 2Y 1/2 ) The function is convex Notice that Mux is constant, while the Muy is diminishing. This means that at some point the next unit of x will give more utility than the next unit of Y, and you will not demand any more Y Y in this case is a zero-income effect good, meaning the amount of income you have does not affect how much Y you want An example would be pencils and all other goods. Assume you need 10 pencils to make it through the school year. No matter what your income is, you demand 10 pencils. More will not help you and less won t be enough to finish off the year. In this case, pencils are a zero-income effect good meaning income does not change the demand for pencils. When you solve for the demand for the diminishing Mu good, you will notice that income (M) is not a part of the equation. This is how you know it is a zero income effect good, since the level of M does not change the amount demanded 1. Find MRS 2. Test for DMRS 3. Decreasing quasi-linear has DMRS, so use tangency 4. MRS=Px/Py (you will solve for the variable with diminishing Mu in this step) 5. Plug the solved variable into the budget constraint, or into MRS=Px/Py

6 If increasing MRS Ex (X 2 + 2Y) The function does not have DMRS and is therefore not convex. You cannot use tangency Function has increasing MRS, meaning the graph is concave You buy all of one or the other. Consume which ever gives the most utility. One good (in this case Y) has constant MU, while the other good (in this case X) has increasing MU. Eventually the good with increasing MU will be better than the good with constant MU. For example and chocolate and an addictive drug: the first couple units of chocolate may be as good or better than the addictive drug, but if you are able to afford lots of units, the addictive drug becomes much better (Don t do drugs, kids!) Ex U(X,Y)=(X 2 + 2Y) If you could only consume 1 unit total you would buy all Y (chocolate). If you could consume 10 units, you would buy all of X (addictive drug). 1. Find MRs 2. Test for increasing MRS 3. Increasing MRS, can t use tangency, consume all of one or the other 4. Consume all of the good that gives the most utility Max U(X,Y)=Max(X,Y) Examples Max (3X, Y) OR 6Max(X, 3Y) OR Max(2X, 2Y) +100 Any monotonic transformation of Max(X,Y) Notice how the cheapest point on each IDC is on the corner, this tells us we will consume all of one or the other U(X,Y)=Max(100,0) = 100 AND Max(100,99)=100 Our utility in both cases is 100. Notice that only the highest number maters. There is no point in purchasing 99 units of good Y. Purchase all of one or the other. Buy all of the good that gives the most utility Compare Bang for Buck: Compare the utility per dollar for both goods Compare utility levels: find utility if all good 1 is purchased versus all good 2 purchased

7 Demand Problems Substitutes 1. U(X, Y) = 5X + 2Y Px=6 Py=2 M=24 What is X* and Y*? 2. U(X, Y) = 2X + 3Y Px=2 Py=2 M=12 What is X* and Y*? 3. U(X, Y) = 3X + Y No prices/income given What is X* and Y*? Complements 1. U(X, Y) = Min(2X, Y) Px=10 Py=2 M=28 What is X* and Y*? 2. U(X, Y) = 5Min(X, Y/5) Px=5 Py=1 M=50 Y=20 What is X*? a. What would X* and Y* be if Y was not fixed at 20? Cobb-Douglas 1. U(X, Y) = X 2 Y No prices/income given What is X* and Y*? Quasi-Linear Decreasing 1. U(X,Y) = 2X 1/2 + Y No prices/income given What is X* and Y*? 2. U(X, Y) = X + ln(y) No prices/income given What is X* and Y*? Quasi-Linear Increasing Max 1. U(X, Y) = X 2 + 5Y Px=5 Py=3 M=30 What is X* and Y*? 1. U(X, Y) = 2Max(3X, Y) Px=4 Py=1 M=20 What is X* and Y*?