Microeconomics I. Antonio Zabalza. University of Valencia 1 Micro I. Lesson 5 : Consumer Equilibrium 5.1 Optimal Choice If preferences are well behaved (smooth, convex, continuous and negatively sloped), then at the optimal choice of the consumer, Slope of i.c. = Slope of b.c. MRS = Price ratio MUp xx= MUpyy y Optimal Choice x Why? Consider a situation in which the above equality does not hold. Also, remember that moving along the bc is equivalent to using the market. Moving along the ic shows the minimum amount of y I need to compensate for loss of x. If the market gives me an amount of y that is greater than the minimum I require, I will follow the market. Microeconomics I. Antonio Zabalza. University of Valencia 2 y C U(C) U(B) B B’ U(A) A x B’’ Microeconomics I. Antonio Zabalza. University of Valencia 3 At A, for instance, if I give up 1 unit of x (distance AB’’), the market gives me 1 unit of y (distance B’’B). But I would be satisfied with less; say, 0.25 units of y (distance B’B’’). Then, it is optimal for me to trade in the market, and go to point B where my utility U(B) is higher than that at point a, U(A). If I keep applying this reasoning I end up at point C. (Check that you understand this). Point C represents the best I can do, given my opportunities. Point C therefore represents the optimal choice, the equilibrium, of the consumer. Going beyond point C, would lower again my utility. Mathematics: Maximization of utility subject to a given budget constraint. Method of Lagrange. Max U=Uxy(,) xy, s.t. px+=pym xy éù L=Ux(,)y+l ëûm--pxyxpy Microeconomics I. Antonio Zabalza. University of Valencia 4 First order (necessary) conditions ddLU =-=l p 0 (1) ddxx x ddLU =-=l p 0 (2) ddyy y d L =m-px-=py 0 (3) d y xy This is a system of 3 equations in 3 unknowns: x, y and l. From (1) and (2) we have that ddUxMU l ==x pp xx ddUyMU l ==y ppyy MUMU MUp x=Þ=y xx pxpyMUpyx Then, to solve for x and y we consider equation (3) to get this more simplified form of the above system. MUp xx= (4) MUpyx m=+pxyxpy (5) Microeconomics I. Antonio Zabalza. University of Valencia 5 This is a system of 2 equations with two unknowns (x,y). (Notice that in general the marginal utilities will depend on x and y). Equation (4) is the equality of slopes of bc and ic discussed above. What the result above says is that this condition is not enough; we need also that the budget constraint is fulfilled (equation 5). Solving this system will, in general, give us the two demand functions for x and y that we are after. x= x(ppmxy,,) y= y(,ppmxy,) The purpose of this lesson is to find out how the three variables (,ppmxy,) influence the demand for x and y. Before, we give an example of this derivation for a particular utility function: Cobb- Douglas. Microeconomics I. Antonio Zabalza. University of Valencia 6 Example: Cobb-Douglas (CD) utility function: Ux(,)y= xyab Max U= xyab xy, s.t. px+=pym xy ab éù L=+xyl ëûm--pxyxpy Necessary conditions: d L =axab-1yp-=l 0 (1) d x x d L =bxabyp-1 -=l 0 (2) d y y d L =m--=pxpy 0 (3) d y xy Eliminate l from (1) and (2), and together with (3) you obtain axa--11ybbxyab = (4) ppxy m=+pxyxpy (5) System of two equations with two unknowns. Microeconomics I. Antonio Zabalza. University of Valencia 7 Equation (4) can be expressed in the form MUp xx= , MUpyy which for this particular case is ab-1 axy px ab-1 = bxypy or, ay p = x . bxpy So the system, in this simplified form is ay p = x (4) bxpy m=+pxyxpy (5) Solving for x and for y, we find the two demand equations: am x = a+ bp x bm y = a+ bpy With a CD utility function, the demand for each good depends on income (positively) and its own price (negatively). It does not depend on the price of the other good. Microeconomics I. Antonio Zabalza. University of Valencia 8 Another characteristic of this utility function is that the parameters of the function give information about the expenditure shares on each good. px a Share of expenditure on x : x = mab+ py b Share of expenditure on y: y = mab+ Sufficient condition Equations (1), (2) and (3) are the necessary conditions. They are not sufficient. For instance, consider the following situation: Point of tangency Point of maximum A utility At A, the first order conditions are met and yet utility is not maximized. You need also another set of conditions which are sufficient. These conditions boil down to the requirement that preferences have to be convex. See that in the figure they are concave. Microeconomics I. Antonio Zabalza. University of Valencia 9 Extreme examples Corner solutions Maximum utility, but at this point Slope of ic > slope bc MRS > price ratio A particular example of a corner solution is when the two goods are perfect substitutes: Suppose U=x+y and pxy=2; pm==4; and 12. ic slope: 1 bc slope: 1/2 3 Maximum U A 6 Microeconomics I. Antonio Zabalza. University of Valencia 10 Here pMU 1 xx<<; 1 pyyMU 2 Maximum is obtained at point A, where only x is consumed. So the demand function in this case is m x = px Kinky solutions Suppose the two goods are perfect complements with the following utility function: U= min,{xy} and pxy=2; pm==4; and 24. y 6 4 A x 4 12 The optimal choice must lie on the diagonal and on the budget constraint. Therefore, the optimal choice is found by solving the system Microeconomics I. Antonio Zabalza. University of Valencia 11 yx= m=+pxyxpy The solution is m xy== ppxy+ For the particular example used here 24 xy===4 6 5.2 Changes in the equilibrium position Now we want to investigate how the equilibrium just studied is altered (displaced) by changes in the exogenous variables of this problem. In particular, we want to know how the equilibrium changes when, m, ppxy and change. Or to put it in other words. The result of the previous analysis was the derivation of two demand curves x= x(ppmxy,,) y= y(,ppmxy,) We want to sign the partial effects of the three exogenous variables m, ppxy and on the demand of x and y. We will consider three types of changes: a) Simultaneous change in prices and income by the same proportion. b) Change in income only. Microeconomics I. Antonio Zabalza. University of Valencia 12 c) Change in one price only. Equiproportional change in prices and income If m, ppxy and all move by the same proportion, the bc does not change and therefore the point of equilibrium does not change either. Suppose initial bc is 000 m=+pxyxpy Multiply all prices and income by k (if, for instance, k=1.1, then all variables increase by 10%). The new bc is 000 km=+kpxyxkpy But k can be cancelled out by dividing both sides of the equation by k. So, the original bc remains unchanged. 000 m=+pxyxpy Change in m only We know that an increase in m moves the bc out. The position of the final equilibrium depends on whether the goods are normal or inferior. Suppose first that both goods are normal. Then, if there is an increase in income from mm0 to ¢, more of both goods will be bought. This is represented in the following figure. Microeconomics I. Antonio Zabalza. University of Valencia 13 y 0 B m Dy m¢ A x Dx The demand curve that treats prices as given parameters and income as a variable, is known as the Engle curve. It takes the form: x= xm(,,)ppxy. m (ppxy,) B m¢ 0 m A x 0 x x¢ Microeconomics I. Antonio Zabalza. University of Valencia 14 We say that a good is inferior if when income is raised, holding everything else constant, less of this good is bought. Say x is inferior. Then, y B 0 m Dy m¢ A x Ñx Exercise: Derive the shape of the Engle curve corresponding to good x, when x is an inferior good. Property: The sum of the income elasticities of each good, weighted by its corresponding expenditure share, must be equal to one. We start with the budget constraint: m=+pxyxpy Then, differentiating both sides of the equality by m, we find: Microeconomics I. Antonio Zabalza. University of Valencia 15 dmxy¶¶ =+pp dmmmxy¶¶ ¶¶xy 1 =+pp xy¶¶mm pxx æö¶¶xmpyy æöym 1 =+ç÷ç÷ mèø¶¶mxmèømy 1 =+ssxeexmyym As we were looking for, the weighted average of income elasticities must add up to 1. Implications of this result: a) Not all goods can be inferior. Not all Engle curves can have negative slope. b) Goods that take a large share of expenditures are unlikely to have either very large or very low income elasticities, since the average must equal one. Problem for home: Say we divide goods in two types: food and non-food. We know food takes 60% of expenditures, and the income elasticity of non- food is 2. What is the income elasticity of food? Exercise: Find the slope and graphical shape of the Engle curve for good x when the utility function is Cobb – Douglas.