Utility Maximization Outline

model of tonsumer theory utility maximization model of consumer theory premises a consumer fares a budget constraint prices of the goods services he buys In S sb In s L L'm L preterenies based on his income and the in the from IT s of pleasure amount consuming budget constraint consumption decision constrained consumer choile constraint what you can have so_ of preferences many pretend no 30 ft 20 10 preserved I t bug books intome 500 a Pbooks is it possible that point under the budget line is most a Pmeals 100 10 preferred A cant be best constrained optimization o Ph The mail.FIy9TuniiiiIaenmy optimalpoint will be so in fi i is no 30 line I 520 iinTehaiuiuePYffanta nfYIn'et's optimal it we are consuming both goodsand preferences are rational monotonin to At a I 1 s no and convex 4 Mrs MRT if the consumer buys one more book he must give up fewer meals than he would have been to in order to remain as well off Thus he'swilling better off At B MRS MRT if the consumer gives up one book she is able to buy more meals on the so_ market mansneneeaedtoinorae.to remain as well off Thusshe's better off Sob is not hest If preferences are rational monotonic and convex then my optimal point will be on the budget line and will be at a point such that the slope of the indifference curve is to identiial the slope of the budget line almost always to o Ém the rate at which I am willing to trade the goods must equal the rate at whith the market dictates I trade them otherwise there is opportunity for arbitrage whith would make you better off mountibly Eiffel's ex MYI 48 4 4 b constrained consumer choice Max choices can be constrained graphical analysis of consumers stated mathmatically our ucx.gl subject to 1 p pay the optimum is still expressed as in the graphical MRS MRT P analysis these conditions hold if the utility function is quasi concave which implies inditarenie cauves are convex to the origin solution Langragian method d t Pax pyy L U x y first order conditions MUx Apx O May apy 0 1 Pax py 9 a YI MRS MRI y 0 example 8th In L 2 Foc 29112 d 12 y A 2 lay 2 of 20 10x l y a 112 54 0 Pxx 20 10x con s pyy Sy o of 2 y 10 112 1129 112 5 set equal a y 2X 2 19 112 9 5 10x 20 1 0 2x I 2 y there 2 1 2 exceptions where you cant use this method are convexity of preferences everything optimal bundle in which one of the goods is not corner solution consumes the budget line need not drives indifference curve apply condition tangency example U lox Max Muy MRS 9 10 0 ICs 0 g Y 0 did Cy downward co a d ytioxth 180 30 109 FOC 4 10 306 100 0 0 4 10 On 30s 0 4 80 30 104 80 4 YI's 0 30 180 10 60 345 10 10 3x 60 100 1 at A MRS MRT A is out of budget 8 at B MRS Bpo MRT et on highest at B y o Mrs MR T example u a y 14 as MRS a d 2 S 0 0 ward If 38 3 on.am A Yp B Y 4 0 0 y 4 42 0 16 py 8 U 02 8 8 the MRT the consumers perfectsubstitutes if the Mrs does not equal optimal bundle occurs at a corner solution a two approaches 0 Bangper 4572 444 buck FYI it If b an 2 Y 5 929ferburn is EYE alwayshigher than O O graphical 5 MRS is 44 MRT T.in anmaitiIn In a i n ieonthebuaaer line and at example i 2 i pie 3 Max U min 5 t 20 12 n 2x.y 44 5 2 T.is vertex a nM 2x s I's knife y 2 unknowns a min 2x y exiy 1 9,95 44 I 4 2x budget ionstraint 2x umincax.by 18 Lahgency 1C 141 4 2117 into budget 12 4 2x 2 u axtby Ion p notusual math IIines