1Consumer ChoiceECON 370: Microeconomic TheorySummer 2004 – Rice UniversityStanley GilbertEcon 370 - Consumer Choice 2Consumer Equilibrium• Now we have all the tools we need to analyze consumer choice• Suppose a consumer faces the constraints– p1x1+ p2x2≤ m and x1≥ 0, x2≥ 0.• The consumer’s problem is to choose and to maximize utility (i.e. to attain highest possible indifference curve) s.t. the above constraints,• That is, the consumer is choosing the most preferred point in the budget set. Econ 370 - Consumer Choice 3Optimal Choicex1x2PreferencesBudgetOptimal ChoiceEcon 370 - Consumer Choice 4Rational Constrained Choice• Provided that this is an “interior” equilibrium,– That is, that it involves strictly positive amounts of all goods• And assuming all our assumptions (A1 – A4) hold:• Then we have equality of– Slope of budget constraint (-p1/ p2)– Slope of indifference curve (MRS)– Ratio of marginal utilities (MU1/MU2)• And our budget is exhausted: p1x1*+ p2x2*= m• Interpretation– Willingness to trade (-slope of IC) = p1/ p2– Marginal utility per dollar equalized across goods2Econ 370 - Consumer Choice 5Characteristics of Optimal Choicex1x2• Interior solution• Slope of indifference curve = Slope of budget constraint •(that is) MRS = -p1/ p2• budget is exhausted: p1x1*+ p2x2*= mEcon 370 - Consumer Choice 6What about savings?Spending TodaySaving for Retirement• We said that generally all money is spent• That is p1x1*+ p2x2*= m• How do we deal with savings?– By assuming our consumption choices include spending today v. spending in the future.Econ 370 - Consumer Choice 7Finding the Optimum• We will demonstrate two methods for finding the optimal choice–MRS = p1/ p2– Maximizing the Utility function• (At least) one other that I will not cover:2211pMUpMU=– Mathematically, it is indistinguishable from the firstEcon 370 - Consumer Choice 8Example• Assume for this example that– Preferences are Cobb-Douglass– U = xαy1 – α– Budget constraint is pxx + pyy ≤ m3Econ 370 - Consumer Choice 9MRS Solutionααα−−=∂∂=11yxxUMUx()ααα−−=∂∂= yxyUMUy1yxMUMUyuxudxdyMRS =∂∂∂∂==()αααααα−−−−=yxyx111yxppxy=−αα1In order to calculate MRS, we first calculate Marginal UtilitiesThen we calculate MRS as:xyαα−=1Since MRS equals the slope of the budget line, we get:Econ 370 - Consumer Choice 10MRS Solution (cont)We will use x*and y*to represent our solutionSolving for y*:**1xppyyxαα−=Since all money is spent, we also have: pxx*+ pyy*= mSo:**1xpppxpyxyxαα−+ mxpx==α***1xpxpxxαα−+=yxppxy=−**1αα(from previous page)Econ 370 - Consumer Choice 11MRS Solution (cont 2)mxpx=α*If Thenxpmxα=*Gives()ypmyα−=1*Finally:So, our solution is:xpmxα=*()ypmyα−=1*Substituting this into**1xppyyxαα−=(from before)Econ 370 - Consumer Choice 12Maximization Solution• Our problem is:–max{u(x, y)} = max{xαy1 – α}– Subject to the requirement thatpxx + pyy ≤ m.• Remembering that all money will be spent, we have:– pxx + pyy= m• From here, the simplest way to proceed is to substitute the modified constraint into problem to be solved:– y = (m – pxx) / py• Substituting into the original problem gives:{}(){}111maxmax−−−−=αααααyxpxpmxyx4Econ 370 - Consumer Choice 13Maximization (cont)Remembering our calculus, maximizing a function requires:(){}11max−−−αααyxpxpmx()0=dxxdfand()022≤dxxfdSo()011=−⇒−−αααyxpxpmxdxd()()()[]01111=−−−−⇒−−−−αααααααxpmxpxpmxpxxxyEcon 370 - Consumer Choice 14()()()[]01111=−−−−−−−−αααααααxpmxpxpmxpxxxyMaximization (cont 2)Since all this mess equals zero, we can eliminate common terms:()01 =−−⎟⎠⎞⎜⎝⎛−⇒xxpxxpmαα()()xpxpmxxαα−−−⇒ 1xpxpxpmxxxααα+−−=0=−=xpmxαxpmxα=⇒Exactly as beforeEcon 370 - Consumer Choice 15()()()[]ααααααα−−−−−−−− xpmxpxpmxpxxxy1111Maximization (cont 3)Verifying the Second Order Condition:()⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−−−⎟⎠⎞⎜⎝⎛−=−−−αααααxxxypxmdxdppxmdxdp 111() ()⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−−−⎟⎠⎞⎜⎝⎛−−−=−−−−122111αααααααxxxypxmxmppxmxmp< 0dxd(reorganizing it a bit)Econ 370 - Consumer Choice 16Cobb-Douglas Observation• Note that for Cobb-Douglas utility function– Demands are linear in income– Expenditure shares are constant– Expenditure shares sum to onexpmxα=*()ypmyα−=1*5Econ 370 - Consumer Choice 17“Corner” Solutions• An “interior” solution was earlier described as:– involving strictly positive amounts of all goods– And all our assumptions (A1 – A4) hold• If a solution does not meet these requirements, it is called a “Corner” Solution• Other methods have to be used to solve itEcon 370 - Consumer Choice 18Corner Examplex1x2PreferencesBudgetOptimal