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Demand theory Ch. 3.doc

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McMaster University
Hannah Holmes

Chapter 3. Demand Theory Consumers face trade-offs in their decisions since incomes are limited and choices are numerous. When you buy more of one good you can only afford less of the other. In order to make choices, consumers combine 1. Preferences-what they would like to consume-We have analysed this in Chapter 2 2. Budget constraints-what they can afford. The Budget Constraint • We are concern with the consumer's choice of the optimal bundle of goods and services. Assumptions; -- Income (M) is exogenous --Prices are fixed and exogenous (p and1p fixed2. --There is only 1 period (No savings nor bequests) --Only consider two goods Given these assumptions Therefore total expenditure must be less than or equal to (≤) income or, p1 1+ p x2 2M. Thus given income (M) possible combinations are determined by the prices of the two goods. SEE FIG 3.1. • The equality divides the space into attainable and unattainable sets or bundles (given M and pis). The equality, known as the budget line, can be written as: x 2 (M - p x 1 1 2 • The slope is dx /d2 = -1(p /p )1 2 and the intercepts are: x 2 M/p 2 (when x =10) and x 1M/p 1 (when x =20) See problem 3.1 (We will talk about endowments later) • What if M, p , 1r p ch2nges? 1 If M increases, how will the budget line shift/change? If p1increases, how will the budget line change? If p2decreases, how will the budget line change? • Next, consider the interior of the consumption set. Would it ever make sense to consume in the interior of the budget line? Non-satiation (more is better) implies that there is always a preferred bundle on the boundary (budget line) – consider Fig 3.3 Therefore: The non-satiation assumption/axiom implies that the utility-maximising consumption bundle lies on the budget line. The Choice Problem: • We can now state the problem formally (for the 2 goods case) as: Maximize U = U(x ,x1) 2y choice of x and1x 2 subject to the constraint: p x +1 1x = 2 2 • One way of thinking of this is to find the highest indifference curve consistent with being in the attainable set (on the budget constraint). Figure 3.4 illustrates a solution (at point A) • Given the smoothness shown in the IC’s in Fig 3.4, it must be true at A, the highest IC attainable: 1) you are on the budget line x = (M - p x )/p 2 1 1 2 2) The slope of the IC = the slope of the Budget line. i.e. - MU /MU = - p /p 1 2 1 2 MRS=p /p 1 2 • The solution (such as at point A) if written for x as: 1 * x1= f(M, p ,p1) i2 called the x dema1d function For essential goods we will have an interior solution See figures 3.5a and 3.5b. Inessetila goods See Fig. 3.6a and 36b 2 • Fig. 3.6b-Here, given that at the vertical axis the budget line is steeper than the indifference curve going through that same point, the optimal solution is on the vertical axis. In this case at the optimumit is true that: 1) the individual is on the budget line 2) the budget line is steeper than the IC. Alternatively, we could say, MRS ≤ p /p . 1 2 3) the consumer spends all his/her income on x (x = 0)2 1 The solution is given by the demand functions: x1= 0 x2= M/p 2 • Note, this solution is common for a lot of goods you do not buy – eg helicopter rides,-In this case good 1 is an inessential good. (Note that it is due to the high price of x that t1e consumer doesn’t purchase it. • Note, alternatively, if it is everywhere true that the MRS is greater than the slope of the budget line, then, the solution must be on the horizontal axis with x = 0! 2 Conclusion: even with all our axioms and assumptions, one can have an interior solution- lies in the space between the two axes with MRS equal to the absolute value of the slope of the budget line. • Or one can have a corner solution with either x or x equ1l to 2ero. If a corner solution, then: slope of IC flatter than BL means x = 0,1x = M/p 2 2 slope of IC steeper than BL means x = 0, x = M/p 2 1 1 Example 1: Let p and 1 be uni2y so that the BC for an income of 100 becomes: 100 = x + 1 x2 Let U = x x1 2 Find the solution to the Utility maximizing problem. Problem could be stated as: Maximize U=x x 1 2 Subject to 3 x1+ x =200 Show that the solution is x =x1=502 Need to get MRS=p /p 1 2 This is an interior solution. Generally. Let U = x x1 2 Find the solution to the maximizingproblem in general M = p x1 1p x 2 2 and, MRS = x /x2= 1 /p 1 2 or, p 1 1 p x 2 2 Therefore x =2p x 1 1 2 on substituting in the budget constraint above M = 2p x1 1d x1= M/2p 1 similarly, x 2 M/2p 2 These are demand functions for x and x . 1 2 Example 2 ½ Let U = (x )1+ .25x and2M = 3, p = 1 and 1 = 1 2 solve for max. Maximize: U = (x ) +1.25x 2 Subject to:x + x =3 1 2 First, MRS = (1/2)/0.25√x ) = 1 (price ratio) thus, √x 1 2 or x = 41 Consequently, x = -1 < 0 2 Thus we cannot have a corner solution, x = 0 and2 x1= 3 We shall also talk about optimal bundles for the following in more details in class: 1. Perfect substitutes U= X + Y Px=1 and Py=2 M=200 4 Show optimal bundle is (X,Y)=( 200,0). Boundary solution. 2. Perfect complements Consider U(x,y)= min (2x,5y) Px=2 and Py=3 M=40 Show that optimal bundle is (12.5,5) 5 APPLICATIONS. Exercise Tax vs Lump-Sum tax • Suppose the government wants to raise a certain amount of tax revenue. Q. Are consumers better off if this tax takes the form of an Exercise tax (tax per unit or a percentage of price) or by lump sum tax? Exercise Tax-increases the relative price of the taxed good. Lump Sum-Alters the budget line See Figure 3.7 where good 1 is gasoline that attracts exercise tax. Good 2 is composite good measured in dollars. ACG- Pre-tax budget line Exercise tax-Budget line is AH Lump Sum-Budget line is BF Amount of tax DC • Not that segment DE lies above and to the right of indifference curve shown. Therefore consumers would always prefer lump-sum tax to exercise tax that yields the same revenue. • The same also applies to income tax vs a tax on a specific commodity. It can be shown that the government could levy an income tax with a higher yield than a specific tax without leaving the consumer worse in utility terms. 3.6.Comparative Statics • We look at responses of the optimums
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