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Tutorial 5: Consumer choice (cont.)

 

Finding the Consumer's Optimal Choice

In Tutorial 5a we learned that a consumer seeks a market basket of goods which generates the maximum level of utility. In Tutorial 5b we learned that one's goal of maximizing utility is subject to the constraint imposed by one's money income and the prices of the goods.

Stated formally, the consumer's objective is to maximize utility subject to a budget constraint. That is, 

Consumer's objective is to maximize utility from F and C, subject to an Income constraint.

 

The function u(F, C) describes the level of satisfaction ("utility") received when the bundle of goods (F*, C*) is consumed. This form of the utility function allows us to manipulate the relative strength of the consumer's preference for F or C by changing the parameters a and b. [Review Tutorial 5 Question 3 in the Consumer Choice Workbook.]

Stating the consumer choice problem this way makes explicit the fact that, whereas one desires to maximize the satisfaction from a bundle of goods (Max u(F, C)), one is constrained in that quest by one's income and the prices of the goods (i.e., by one's budget).

The complete model of consumer choice is shown in Figure 7 below. Graphically, the consumer's utility-maximizing choice must meet two conditions:

  1. The market basket chosen must place the consumer on the highest indifference curve attainable;
  2. The market basket must be located on the budget line (i.e., no "saving").

By the Axiom of Greediness, any basket on U1 is preferred to any basket on U or U0. But attainment of U1 is blocked by the budget constraint.


At points C or D, for example, the budget constraint is satisfied, but U0 is not the highest level of utility attainable. There are a number of indifference curves higher than U0. In fact any point on B1 that lies between bundles C and D would place the consumer on a higher indifference curve.


Market basket A exhausts the consumer's income ($5*8 + 20*$2 = $80) and results in a higher level of utility than bundles C, D, or any other.

Figure 7

Figure 7 - Graph of three indifference curves and a budget constraint.
 

Therefore,  this graphical model shows that the maximum attainable level of satisfaction takes place when the budget line and indifference curve are tangent. At the point of tangency, A, income is exhausted, and:
The slope of the budget line
=
The slope of the highest IC attainable
|Pf/Pc|
=
|MRSfc|
The rate at which one can trade C for F
=
The rate at which one is willing to trade C for F
MC (of one more unit of F)
=
MB (of one more unit of F)

 

In contrast, at point C, MRSfc > Pf/Pc. So the rate at which the consumer is willing to trade C for F, exceeds the amount s/he is required (by the market) to trade. So if 15 units of C is traded for 6 units of F -- that's 2.5C for 1F, what the market requires -- the consumer will be better off as they move to point A on indifference curve U1.

Similarly, at point D, MRS < Pf/Pc. The rate at which the consumer is willing to trade F for C, exceeds the amount s/he is required (by the market) to trade. So if 6 units of F is traded for 15 units of C -- that's 1C for 2/5F, what the market requires -- the consumer will be better off as they move to point A on indifference curve U1.

 

Some "Mathemagics" Top of page.

Remember that the consumer's objective is to maximize the satisfaction received from the consumption of a market basket of goods, subject to the constraint that spending on the goods does not exceed income, i.e.,

Consumer's objective is to maximize utility from F and C, subject to an Income constraint.

Graphically, we were able to determine that this constrained maximum occurs at the point where the slope of the budget line, Pf/Pc, equals the slope of the indifference curve, referred to as the marginal rate of substitution (MRS). In this section we will learn how to calculate the MRS.

Once again, the function u(F, C) describes the level of satisfaction ("utility") received when the bundle of goods (F*, C*) is consumed. An indifference curve simply describes all the combinations of F and C that generate a constant level of utility, U1, to the consumer. So, the change in utility along an indifference curve is equal to zero. But as you give up some clothing, for example, your utility level changes (drops). The change in utility from a one unit change in clothing is called marginal utility (MUc). So the size of the change in total utility from a reduction in C depends on the size of the change in C and on the consumer's MUc, e.g., DC*MUc.

Similarly, when you get more food, your utility level changes again (rises). This time the change in total utility from an increase in F is equal to DF*MUf.

Mathematically, the two changes cancel each other out:

Change in U must be zero along an indifference curve..

Subtracting DC*MUc from both sides, the change in C times the marginal utility of C must equal the change in F times the marginal utility of F..
Divding both sides by DF and by MUc, and cancelling similar terms,
The negative of the ratio of delta C to delta F must equal the ratio of the marginal utilities of F over C. .
   
So the slope of an indifference curve is measured by the ratio of the marginal utilities of F to C..

 

Now all we have to do is figure out how to calculate marginal utilities. As mentioned above, marginal utility measures the additional satisfaction one receives from consumption of another unit of a good. Mathematically, marginal utility is the partial derivative of a total utility function. Starting with the utility function noted above:

Utility as a function of food and clothing.,

differentiating with respect to F, gives us MUf =
differentiating with respect to C, gives us MUc =

 

To calculate MRS = MUf/MUc

The marginal rate of substitution is equal to the alpha over F divided by beta over C.

 

Now you can see the importance of the terms a and b from your answers to Tutorial 5 Question 3 in the Consumer Choice Workbook.

Now it's time to "do the thing".

Click on the following link to download the Consumer Choice Workbook. Work through Tutorial 5 Questions 6 - 7 to improve your understanding of the consumer choice model.

Return here when you have finished.

Need help downloading the Excel file?

 

This is a good time to check your understanding of the consumer choice model by working through the Applications in Tutorial 5 Questions 8-10. Answer the questions without the use of the consumer choice template, and hand draw the graphical model you use to illustrate your answer. [If you give yourself some time pressure to answer the three questions, this will be excellent practice for the Literacy Test and the end of the semester...] 

Now it's time to "do the thing".

Click on the following link to download the Consumer Choice Workbook. Work through Tutorial 5 Questions 8 - 10 to improve your understanding of the consumer choice model.

Return here when you have finished.

Need help downloading the Excel file?

 

Now we are ready to use the consumer choice model to derive the individual consumer's demand curve.