Question

Question 6: Logan has preferences over olives (x1) and ice creams (x2). He prefers to eat them separately but not together, which is represented by: U(X1, X2) = x + xỉ. Also suppose his income is $100 and the prices of olives and ice creames are px, = $1 and Px2 = $2, respectively. 1. Is this convex preferences or concave preferences? 2. Solve for the bundle that satisfis the tangency condition. 3. What's the level of utility using the bundle you just solved in Q6.2? 4. Consider spending all of your income on olives (xi). What's the utility from this bundle? 5. Which bundle does Logan prefer between the bundle in 26.2 and 26.4? Does the tangency condition lead to the optimal bundle? 6. What's the optimal bundle when price of olives is $3.

Answer 1

Questions 2 7 Logan has preference ova olive (21) and ice creams (XL) He Portues to eat them separately but not together, which is represuded by: U(K1, K2) - x² + 22² Hue Marginal utility of xii.c. MUA - JU(XI) KL) S aloni __ _ 221 and marginal utility of Xili:c. MULD OU (KI . . vedi 12.X20 2XL a Hunce slope of the indifference cover fo) nont ut - MUI MU2 X Marginal rate of 9 substitution (MRS) MRS.);xi tible ! well 222 262 OF MRSAH CUM Cuvatue of indiffuence Curver a d(MRS) - 10 dul 22' As d (MRS) 0 => Hice dx, wi Concave to the origin and indifference cure is MOLL this is concave) Preference. -00 L saj T (La La La Scanned with CamScanner

1 2 Now, P = $1 and Phra C and in come ME voor Trnce budgit constraint will be -Mex Px) + X2 Px2 2) 100 - X1 + 2X2 1 NT 2X2 - a Now, utility manimising tangency condition will bet Slope of indiffuene cure= Slope of budget line. We know, slope of indiffunice (MUC MUL). I MUL and slope of budget line PXL 10 - PX2 UM Hunce tangency condition will be - - - MUI MUZ PX2 2x - 1 2 22 T 2 .4. - " ! 22 - 2 2u1 = XL X - sro Now, substituting H2= 2x, in the budget constaint we gett hus spiro with SUNO 100 = X1 + 2x2 TOD = X/ +2_(2x), E) CamScanner

2) 5XL 2 180 =) X1 = 20 and as X2= 2XL - hence X2 = 2(20) = 40 Huce the bundle that satisfics the tangency condes will be (Xu X 2) - (20,90)- 32 Now, the level of utility when (X1, X2) = (20,90 will be found by substiputing (X1, X2) = (20,40) in the utility function, U (222) - x1 24222 - (20)2+ 1402 a 400 t1600 - = 2000 utils L 4 et all income is spurt on Olivy (X1) then, X2=0, and we substitute X2=0 in the budget constraint. 100 = X1 + 2(0) - 22 X = 100 Now, utlity from this bradle will be found by substituting X1 =100 and X220 in the Utility function. Himce u = X1²7X2² = (100)-+0 10,000 Scanned with CamScanner