Consumer Optimization Fundamentals

Transcription

1 Consumer Optimization Fundamentals The consumer optimization processes we use in this class find the point that brings the most utilit b simultaneousl testing all points along the budget constraint following these processes is the fastest wa to get to the answer. The goal of this worksheet, however, is to understand the concepts behind the processes. In each problem, ou will work with a particular tpe of utilit function, testing each point along the budget constraint individuall to find the one that brings the most utilit. You will then etract a general pattern from each answer, which will deepen our understanding of the step-b-step processes. Note: In general, consumers are allowed to consume fractions of goods, but to make things simple we will assume on this worksheet that the can onl consume in whole numbers. I) u(, ) = * m = $4 p = $1 p = $1 a) This is a utilit function b) Graph the budget constraint and label each whole-number coordinate (i.e. label (10, 10) not (8.5, 8.5); there will be five points) c) Test utilit at each of the five points (continued on the net page; the first line is alread filled out for ou) POINT X * Y = U ( 0, 4 ) 0 * 4 = 0

2 i) Utilit is maimized at (, ) ii) The MRS as a function of and =. The MRS is {diminishing, constant, or iii) At optimum (, ), MRS = and p / p = iv) Thus, when utilit is maimized for a tpe of utilit function, MRS { >, <, or =} p / p e) On our graph from part b, draw the IDC on which the utilit maimizing point lies (Hint: all points on the IDC will have the same utilit) II) u(, ) = + m = $4 p = $2 p = $1 a) This is a utilit function b) Graph the budget constraint and label each whole-number coordinate (there will be three points)

3 c) Test utilit at each of the three points POINT X + Y = U i) Utilit is maimized at (, ) ii) The MRS as a function of and =. The MRS is {diminishing, constant, or iii) At optimum (, ), MRS = and p / p = iv) Thus, for a tpe of utilit function, if MRS { >, <, or =} p / p, then the consumer chooses {all, all, or indifferent} e) On our graph from part b, draw the IDC on which the utilit maimizing point lies (Hint: all points on the IDC will have the same utilit) III) u(, ) = + m = $4 p = $1 p = $2 a) This is a utilit function b) Graph the budget constraint and label each whole-number coordinate (there will be three points)

4 c) Test utilit at each of the three points POINT X + Y = U i) Utilit is maimized at (, ) ii) The MRS as a function of and =. The MRS is {diminishing, constant, or iii) At optimum (, ), MRS = and p / p = iv) Thus, for a tpe of utilit function, if MRS { >, <, or =} p / p, then the consumer chooses {all, all, or indifferent} e) On our graph from part b, draw the IDC on which the utilit maimizing point lies (Hint: all points on the IDC will have the same utilit) IV) u(, ) = + m = $4 p = $1 p = $1 a) This is a utilit function b) Graph the budget constraint and label each whole-number coordinate (there will be five points)

5 c) Test utilit at each of the three points POINT X + Y = U i) Utilit is maimized at ii) The MRS as a function of and =. The MRS is {diminishing, constant, or iii) At, MRS = and p / p = iv) Thus, for a tpe of utilit function, if MRS { >, <, or =} p / p, then the consumer chooses {all, all, or indifferent} e) On our graph from part b, draw the IDC on which the utilit maimizing point lies (Hint: all points on the IDC will have the same utilit) V) u(, ) = min [, 3] m = $4 p = $1 p = $1 a) This is a utilit function b) Graph the budget constraint and label each whole-number coordinate (there will be five points)

6 c) Test utilit at each of the five points POINT min[ X, Y ] = U ( 0, 4 ) min[ 0, (3 * 4) ] = 0 i) Utilit is maimized at (, ) ii) For a tpe of utilit function, goods are alwas consumed in a fied ratio. In this case, utilit is maimized when * = *. (Now, compare the ratio statement to the original utilit function and see if ou notice an patterns.) The general formula for a tpe of utilit function is u(, ) = A * min [( / a), ( / b)]. Thus, in general, utilit for a tpe of function is maimized when * = *. e) On our graph from part b, draw the IDC on which the utilit maimizing point lies (Hint: all points on the IDC will have the same utilit)