EconS Constrained Consumer Choice
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1 EconS Constrained Consumer Choice Eric Dunaway Washington State University eric.dunaway@wsu.edu September 21, 2015 Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
2 Introduction We have analyzed what consumers prefer, in the form of utility. We have also analyzed what consumers can actually buy, in the form of budget constraints. It s time to put them together and derive demand curves. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
3 Constrained Consumer Choice Given that consumers are constrained to choosing bundles that lie within their opportunity set, the want to choose the bundle that maximizes their utility. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
4 Constrained Consumer Choice z Budget Line Opportunity Set x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
5 Constrained Consumer Choice z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
6 Constrained Consumer Choice z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
7 Constrained Consumer Choice z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
8 Constrained Consumer Choice Intuitively, the consumer is going to choose the bundle on their budget line that is going to give them the highest utility. This corresponds to the indi erence curve that touches the budget line at only one point (a tangency point). We can nd this point mathematically. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
9 Constrained Consumer Choice A great feature of the tangency point is that the rate of change (slope) of the indi erence curve at that point is exactly the same as the slope of the budget line. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
10 Constrained Consumer Choice z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
11 Constrained Consumer Choice z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
12 Constrained Consumer Choice We actually calculated the rate of change of the indi erence curve last week. We called it the marginal rate of substitution. MRS = MU x MU z where MU x is the marginal utility with respect to x and MU z is the marginal utility with respect to z. (Note that I changed the y into a z from last week. No change in its meaning). Also, we calculated the slope of the budget line last week. We called it the marginal rate of transformation. MRT = p x p z where p x is the price of good x and p z is the price of good z. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
13 Constrained Consumer Choice Thus, the point where the indi erence curve is tangent to the budget line will be where the marginal rate of substitution equals the marginal rate of transformation MRS = MRT MU x p x = MU z p z MU x = p x MU z p z Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
14 Constrained Consumer Choice MU x MU z = p x p z This ratio is actually a representation of one of the most basic rules of economics. To see this, let s rearrange a few terms. MU x p x = MU z p z Remember that the marginal utility of x is the amount of utility the consumer gets for consuming one more unit of x. The price of x, p x is the price to consume one more unit of x. Same thing for good z. The consumer will want to consume until the relative gain to cost is the same across all goods. If it weren t the same, then they could get to a higher utility by switching the goods around. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
15 Constrained Consumer Choice Now that we have gured out where the tangency point is, we just need to use one more piece of information to solve for our demand. We know that the tangency point happens on the budget line. This tells us exactly what utility level we reach. Thus, our system of two equations and two unknowns to nd our solution are Let s look at an example. MU x = p x MU z p z p x x + p z z = Y Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
16 Example Consider the following utility function for two goods, x and z, Ū = x 0.75 z 0.25 The price for good x is p x = 3, the price for good z is p z = 4 and the consumer has an income of Y = 64. Derive the optimal quantity of x and z. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
17 Example Ū = x 0.75 z 0.25 Starting o, let s calculate our marginal utilities using the power rule: MU x = 0.75x 0.25 z 0.25 MU y = 0.25x 0.75 z 0.75 Next, the equation for our budget line is p x x + p z z = Y 3x + 4z = 64 Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
18 Example Let s gure out our marginal rate of substitution MRS = MU x = 0.75x 0.25 z 0.25 MU z 0.25x 0.75 z 0.75 = 3z x and our marginal rate of transformation MRT = p x p z = 3 4 Together, we can set up our tangency condition, MRS = MRT 3 z 3 = x 4 3 z x = 3 4 Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
19 Example 3 z x = 3 4 Let s rearrange some terms to make this a bit nicer. x = 4z Now, we just have this equation and the budget line to nd a solution 3x + 4z = 64 Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
20 Example I can substitute x = 4z into the budget line to get 3(4z) + 4z = 64 16z = 64 z = 4 Then, I put this value back into x = 4z to get my solution for x, x = 4z = 4(4) = 16 Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
21 Example z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
22 Example Ū = x 0.75 z x + 4z = 64 x = 16 z = 4 Does this answer make sense? Yes. From the utility function, we can see that the consumer gets a lot more utility from consuming good x than from good z. Also, good x is relatively cheaper than good z. We should expect in this case that x should be higher than z. Note: If x were more expensive than z, this might not be the case. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
23 Corner Solution A few notes: When utility functions are well behaved, our solution will always lie somewhere along the budget line. By this, I mean not at the end points. If the curvature of the indi erence curve is small (a relatively straight curve), we can get situations where the only intersection point is at an end point. When this happens, our tangency condition does not hold any more. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
24 Corner Solution z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
25 Corner Solution z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
26 Corner Solution If we let the budget line extend to the negative axis (either z is negative or x is negative), we would be able to nd a tangency point. But we don t consume negative quantities of goods. Therefore, if you re trying to solve a problem and you get that one of the goods is negative, set it to zero and solve for the other good. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
27 Corner Solution Lastly, let s talk about an extreme example: perfect substitutes. When we have perfect substitutes, the indi erence curves are straight lines. Remember that their marginal rate of substitution is constant no matter where on the line the bundle falls. Since it s constant, it s almost impossible for a tangency point to be found. This means that when we re dealing with perfect substitutes, we almost always have a corner solution The only exception is when the slope of the indi erence curve is the same as the slope of the budget line; then any bundle can be a solution! When dealing with perfect substitutes, people only buy the good that gives the highest utility. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
28 Deriving the Demand Curve And that s equilibrium. But what about deriving the demand curve? This is actually pretty easy from here. All we need to do is start changing prices, and the curve will develop. Let s look back at our gures. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
29 Deriving the Demand Curve z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
30 Deriving the Demand Curve z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
31 Deriving the Demand Curve z x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
32 Deriving the Demand Curve z p x x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49 x
33 Deriving the Demand Curve z p x x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49 x
34 Deriving the Demand Curve z p x x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49 x
35 Deriving the Demand Curve p x x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
36 Deriving the Demand Curve p x x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
37 Deriving the Demand Curve Mathematically, we can derive the demand curve by not picking values for prices or income. This should make sense, since usually these are factors that determine the demand for an item. Let s go back to our example. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
38 Example Recall the utility function from our earlier example Ū = x 0.75 z 0.25 Now, we don t specify any prices or income and we leave the budget line in a general form p x x + p z z = Y Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
39 Example Our marginal rate of substitution hasn t changed, it is still MRS = MU x = 0.75x 0.25 z 0.25 MU z 0.25x 0.75 z 0.75 = 3z x We can also just express our marginal rate of transformation in general terms MRT = p x p z Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
40 Example Now, we nd our tangency point Let s rearrange this for easier use MRS = MRT 3 z p x = x p z 3 z = p x x p z p x x = 3p z z Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
41 Example p x x = 3p z z Now, we use our tangency condition, along with the budget line and we can solve it for x and z p x x + p z z = Y Remember that the while we haven t given them numbers, we assume that we know what the prices and income are. We just need to solve for x and z as a function of those values. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
42 Example p x x = 3p z z p x x + p z z = Y Let s subtract both sides of the rst equation by p x x p x x + 3p z z = 0 p x x + p z z = Y and add the equations together (getting rid of p x x) which is the demand curve for z p x x + 3p z z + p x x + p z z = 0 + Y 4p z z = Y z = Y 4p z Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
43 Example z = Y 4p z Like before, we can plug this value back into our tangency point to get the demand for x p x x = 3p z z p x x = 3p z Y 4p z p x x = 3Y 4 x = 3Y 4p x which is the demand curve for x. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
44 Example x = 3Y 4p x z = Y 4p z These demand curves probably look strange. They re non-linear. It s actually really hard to pick well behaved utility functions that give linear demand curves. The important thing is that when their own price increases, the quantity demanded falls (Law of Demand). We can see this because p x is in the denominator. As it gets bigger, x gets smaller. The same thing happens with p z and z. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
45 Example p x x Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
46 Example x = 3Y 4p x z = Y 4p z When we plug in our original values, p x = 3, p z = 4 and Y = 64, we get the same solution from the rst example x = 3(64) 4(3) = 16 z = 64 4(4) = 4 Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
47 Summary Combining utility functions with budget constraints allows us to gure out solve the constrained consumer choice problem, which in turn allows us to derive demand curves. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
48 Preview for Wednesday We are going to analyze what happens to the constrained consumer choice problem when prices and incomes vary. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
49 Assignment 3-1 (1 of 1) 1. Consider the following market with two goods, x and z. The consumer s utility function is Ū = x 0.2 z 0.8 a. Derive the demand curves for x and z. (Remember to include p x, p z and Y in there as unknowns. Look at the last example if you need help.) b. Let p x = 2, p z = 4 and I = 50. Find the equilibrium quantities demanded of x and z. Eric Dunaway (WSU) EconS Lecture 12 September 21, / 49
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