2021S2 ECON8039/3004 HEALTH ECONOMICS TUTORIAL WEEK 10 Q1: TB CH9, Q17 17 In this exercise, we will formally show the two properties of indifference curves in IH- Is space we discussed in Section 9.2. To prove that indifference curves are downward- sloping, we calculate the slope of the indifference curves dls/alH directly. Recall that the individual with income IH in the healthy state and Is in the sick state, and with probability p of becoming sick, has an expected utility E[U]p of E[U]p=pul(Is)+(1 -p)U(IH) (9.3) a Take the total derivative of E[U]p. This will give a formula explaining how changes in In and Is contribute to changes in E[U]». the ( total ) differential b Because indifference curves, by definition, connect points with constant utility, set dE[U], equal to 0 and then solve for dIs dIH c Using what you know about the signs of p, U'(IH), and U'(Is), prove that the sign of dIs - is negative. dIH
Math Review Consider a function f : IR2 -> IR' The ( total ) derivative of f at point x =(x1, X2) . is denoted as D+(x ) or f'(x ) · is a 1×2 matrix. that is D + ( x ) = f ( x ) = | 2+(x) 2 ×1 2 ×2 2+(x) ) The ( total ) differential of fat x · is denoted as df . measures the change on function value of + inresponse to a small change on the independent variable X . · is a real number ( i.e ., a number in IR ). Consider h= (dx1, axa) a point in IR measures a small change on X. dx, of dx2 are small numbers. eg. X = ( 1 , 2 ) n = ( d x 1 , 0 x 2 ) = 1 0. 1 , 0 . 0 2 ) then X+ h = 1 1. 1, 2. 02 ) f ( x + h ) - f (x) = d+ = 2+(x) 2 X 1 dx2 2 X2 dx1 + 2+(x)
2021 S2 ECON8039/3004 HEALTH ECONOMICS TUTORIAL WEEK 10 Q1: TB CH9, Q17 17 In this exercise, we will formally show the two properties of indifference curves in IH- Is space we discussed in Section 9.2. To prove that indifference curves are downward- sloping, we calculate the slope of the indifference curves dis/aly directly. Recall that the individual with income IH in the healthy state and Is in the sick state, and with probability p of becoming sick, has an expected utility E[U]p of is a function of E[U]p=pl(Is)+(1 -p)U(IH) 5 Is and IH (9.3) a Take the total derivative of E[U]p This will give a formula explaining how changes in IH and Is contribute to changes in E[U]p. The b Because indifference curves, by definition, connect points with constant utility, set F differential de[U]) equal to 0 and then solve for dIs The total derivative - e 2 Is , 2 1H 2 E[U] ) - =