# Deriving demand function

Deriving Demand Functions – Examples What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods XSL and xx , with respective prices Pl and up , and income m. Perfect Substitutes For perfect substitutes, we have to look at respective prices. After all, if goods are perfect substitutes, then the consumer is indifferent between them, and will have no problem adjusting consumption to get the good with the lowest price. . 1 The basic case (1:1) For 1:1 perfect substitutes, the situation is about as plain as can be. Say Pl > up . The consumer will spend all their income on good 2. How do we know without doing any of that fancy math stuff? If the consumer is Just as happy with a unit of good 1 as they are with a unit of good 2, and good 2 is less expensive, then they might as well use all their income on good 2 (they get more stuff that way). Similarly, if Pl < p2 , the consumer will choose only good 1. What if pl = p2 ?

Then any combination of good 1 and good 2 that uses all their budget is fine with them. So for each good, we have three possible demand functions depending on the prices. For example, demand for good 1 can be expressed as 0 O if Pl > up XSL (Pl . Up , m) = if Pl < p2 pl Any (xl ,x2) that satisfies pl xl + p2 m if pl = p2 and ssimilarly for good 2 (with the inequalities reversed, of course). 1. 2 A more complicated example (2:3) ProDlem : Let tne Inalvlaual nave a u ty Tunct Ion u x 2x1 + 3x2 ana an income of 120. They face prices pl = 2 and p2 = 6.

What is their demand for XSL ? For Disclaimer : This handout has not been reviewed by the professor. In the case of any discrepancy between this handout and lecture material, the lecture material should e considered the correct source. Despite all efforts, typos may find their way in – please read with a wary eye. Prepared by Nick Sanders, US Davis Graduate Department of Economics 2007 Solution : The easiest way to do this is to look at how much XSL they can buy with all their income and how much xx they can buy with all their income, then see which gives the higher m utility.

If they spend all their money on XSL , their utility is u(XSL ,0)=2 * Pl = 2 * 120вЂ” 120. 2 120 If they spend all their income on xx , their utility is u(O, xx)=3 * up = 3 = 60. Since hey get a higher utility from consuming only XSL , their demand functions will be XSL (Pl , up , m) = and XX(UP , UP, We could also have solved this by figuring out the slope of the budget line relative to the slope of the indifference curves (I. E. The MRS..). The slope of the budget line is -Pl /up = -1/3, and the MRS.. Is -2/3.

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Since the absolute value of the price ratio is lower than the absolute value of the MRS.. At all point, we know the individual is never going to trade XSL for xx , so they’ll spend all their income on XSL – the same result we found above. At what price would the individual would be willing to consume a non- zero amount offs ? Using our first method, that would happen whenever u(XSL , O) = 120 or UP UP Using our second method, we know they will be willing to trade some XSL for some xx when the absolute value of the price ratio is at least as great as the absolute value of the MRS.., or when 2/up 2/3, again giving up 3. Perfect Compliments n perfect complements, we solve Tort ten mean Tensions using two Deterrent equations. The first equation is the budget line, and the second is the optimal relationship between the amounts of XSL and xx consumed. . 1 When goods are perfect 1:1 compliments, the optimal relationship between XSL and xx is such that XSL =xx . Why? Say XSL >xx . Our utility is going to be the minimum of the two values (xx ) and any XSL above xx isn’t giving us any additional utility, even though it is eating up some of our budget.

We could increase our utility by getting rid of some of that extra XSL and buying more xx . We’d be in a similar situation if XSL

What is the demand for XSL ? What is the individual’s income? Solution : We know that at the optimal point, the individual will choose XSL and xx in a ratio such that ex. = ex. . Thus 5 4 = 20 = 25 We know that at the optimal point Pl XSL + up xx = m, and we know everything on the left hand side, so 10 *25+5 Problem : The individual has a utility function u(XSL ,xx)= min{XSL + ex.,xx + ex. }, faces prices Pl = 3 and up = 8, and has an income of 300. Find their demand for XSL and xx .

Solution : Using the tricks discussed above, we get our two equations XSL + 312 = + EX. 3 PIX +pike Plugging (3) into (4) and solving m + pix mm APP + APP xx (Pl , up , m) = mm where we got XSL by plugging the optimal xx into (3). Replacing Pl , up , and m with their known values, we get our quantities demanded. 2. 3 = 20 XX(UP , = 30 up , m) = A general result In general, if a utility function is of the form min{axle , Гџxx } then the demand functions will be Pl + app Cob-Douglas You should be plenty comfortable with Cob-Douglas preferences by now – we’ve dealt with them quite a bit.

Since it’s familiar territory, this first example will be speedy, and then we’ll solve the same problem using the Lagrange method. Problem : Let someone have a utility function defined by u(XSL , xx)= 3 xx xx . They have 8 1 an income of 150 and face prices Pl = 10 and up = 5. What is their demand for both goods? What is the ratio of total expenditure on good 1 to expenditure on good 2? Solution : First solve for the optimality condition by setting the negative of the price ratio equal to the marginal rate of substitution. We then use that to get XSL in terms of 2 (or the other way around, whichever suits you). XX 12 –15 XSL 12 p 2 Now plug this into the budget constraint and solve. M = app xx 212 Now we have our demand functions mm = 12. 5 APP 10 150 APP The ratio of expenditure on good 1 to expenditure on good 2 is Pl x 1 12. 5 p 212 which meaner that however much money we spend on good 1, we spend 5 times that amount on good 2. Remember back when we said that with a utility function representing Caboodles preferences of the form u(XSL , xx ) = ca XSL -a where O < a < 1, the a and (1 -a) 1 2 told you what fraction of your income you spent on each good? We Just verified that result.

If we transformed our utility function by raising it to the power of 1 (remember why? ) our 6 exponents would be 6 on good 1 and 6 on good 2, which would mean that we spend 5 times as much on good 1 as good 2 … TA-dad! 3. 1 Solving the problem using the Lagrange method Following the method shown in class (from the lecture on 1 1/1), we’re going to solve maximize , 12) = XX 81 subject to p lax + up Forming the Lagrange gives -NIP XSL + – m) We now take first derivatives and solve for first order conditions in XSL =xx-XP xx=XP 8 5 = – XP x = XP ,xx, and h.

ODL -p 1 XSL +pike- (7) m = O 1 XSL +pix Dividing (6) by (7) gives 12 Him . Looks familiar . Aha! It’s the good old tangency condition we’ve seen so much – compare (9) to (5). Now, we can use (9) to solve XSL in terms offs and plug that into (8), and from there on, it’s the same process we did before. U(XSL , xx) = xxГџ xx the demand functions will be Гџ+yup Qualifiers Qualifiers preferences are a bit harder to deal with, because we have to consider two possibilities – interior solutions and border solutions.