1) Consider a consumer who is choosing among combinations of two goods: cookies
ID: 1160033 • Letter: 1
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1) Consider a consumer who is choosing among combinations of two goods: cookies (c) and donuts (d). A bundle of c cookies and d donuts is given by (c, d). Assume that both c and d can take values 0,1,2, or 3. The consumer's preferences over bundles of cookies and donuts are as follows: For any two bundles (c, d) and (d,d'): (c, d) (c,d') whenever c and d 2d (3.2) and (d, d')-(0.1). Since 3-0 1, we can say that (3,2) (0,1). We can also check the "other direction", i.e., let (c, d) = (0,1) and (3, 2), and find that we cannot say that (0,1) (3,2). Thus, the consumer strictly prefers (3,2) to (0, 1); To understand how to use this definition, consider two bundles (c, d) and 2 c,d) a) Are these preferences complete? Hint: If they are complete, you should be able to rank any two bundles. If not, there is at least one pair of bundles that cannot be ranked b) Are these preferences transitive? Hint: One way to do this is to check every combination of three rankable bundles using the definition of the preference and transitivity. You can use this method, but t wll take a long time. Another way is to use three arbitrary bundles, say (ci,d), (c2, d2 and (C3,dy), and show that if (ci, di) (q,4) and (c2.4) (c3, ds), then (ci, d) (C3, ds) using the definition of the preferences given above. c) Suppose the consumer has $6 to spend and that the price of a cookie is Pe $2 and the price of a donut is F : $4. Can you make a good prediction about what bundle the consumer will purchase? Îf so, what is it? Îf not, why not? (Assume that the consumer has no use for unspent money. For example the $6 is on a gift card that is about the expire.)Explanation / Answer
a) No the prefences are not complete. This can be understood by taking an example: Let one bundle (c,d) = (3,0) and another bundle (c', d') = (0,3). In this case c > c' as 3 > 0. But d' >d because d' = 3 > d =0. Therefore, we cannot rank the two bundles according to the preference relation given. Since there exists atleast one pair of bundles which cannot be ranked, this prefernce relation os not complete.
b) To show the proof, we can show using contradiction. Consider three bundles (c1, d1), (c2, d2), (c3, d3) such that (c1, d1) > (c2, d2) and (c2, d2) > (c3, d3). Lets assume (c3 , d3) > (c1, d1). Given a choice between (c1, d1) and (c2, d2), the consumer will choose (c1, d1) according to the preference relation. This means c1 >= c2 and d1 >= d2. Also given (c2, d2) and (c3, d3) the consumer will choose (c2, d2) which means c2 >= c3 and d2 >= d3. But given a choice between (c1, d1) and (c3 , d3), consumer is choosing (c3, d3) which implies c3 >= c1, and d3 >= d1 which is not possible as c3 <= c2 <= c1 and d3 <= d2 <= d1. Hence given a choice between (c3 , d3) and (c1, d1) the consumer wil choose (c1, d1)Therefore this is a contadiction of our assumption. Therefore (c3 , d3) < (c1, d1) and the prefence relation is transitive.
c) There are two choices for the consumer if they spend all $6 on cookies and donuts: either buy all cookies or divide money amond donuts and cookies. Hence the two bundles will be (3, 0) and (1, 2). Here 3>1 but 0 >/ 2 hence we cannot predict which bundle will be purchased.
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