Cobb-Douglas Utility Functions

Cobb-Douglas Utility Functions:

The utility function used above is an example of a so called Cobb−Douglas utility function. This utility function is very popular since it represents well-behaved, i.e., monotonic and convex preferences. It is easy to verify that the MRS12 = Q2/Q1of this function is diminishing, as Q1 increases and Q2 decreases along an indifference curve. Remember now, our discussion about admissible transformations of utility functions, e.g., it was allowed to take the square root of the function U(Q) to represent the same preferences by a new utility function,

2338_cobb douglas.jpg

The exponent (1/2) actually shows the budget shares of each good, i.e., the total amount spent on each good divided by the total budget:

si = (piQi)/Y

If budget shares were: s1 = s2 = 1/2. A more general formulation of the Cobb-Douglas utility function is written as,

U(Q) = Qa1 . Qb2

If we make the following transformation of this function “raises the right-hand side by the exponent [1/(a+b)]J, we get a new utility function, again representing the same preferences,

2319_cobb douglas2.jpg

In this case, the budget shares are: s1 = a/(a+b), and s2 = b/(a+b). Note also that since s1 + s2 =1, we may as well define b = 1 − a, and V (Q) = Qa1Q1−a2.

The demand functions will in this case be,

Q1 (Y, p1) = (a . Y)/p1
Q2 (Y, p2) = [(1 − a) . Y]/p2

Example:

Let good 1 be equal to “food” and good 2 “all other goods” and assume that you make a field study of my purchase patterns, which shows that s1 = 0.10 and s2 = 0.90. Assume also that p1 = p2 = 1 (a so called “normalization”) and Y = 15000kr/month. Given these numbers my optimal bundle is Qkmm = {1500, 13500}, and my (maximized) utility is thus,

Ukmm = 15000.1 x 135000.9
= 10837.

It is mentioned several times, this number is of no particular consequence. However, it can be used to evaluate the consequences of some policy action by the government. For example, assume that the government (for political popularity reasons?) wants to subsidize food, so that its price to consumers decreases to p′1 = 0.9. The food subsidies must be paid by someone which of course is the tax payers, which of course are the same persons which benefits from lower food prices. Assume now that the food subsidies are financed by a higher tax on all other goods (a higher value added tax for example), which increases the price consumers must pay for all other goods to p′2 = 1.02. Am I better or worse o? by this policy initiative of the government? Well, my new bundle is, Q′kmm = {1666(2/3), 13366(1/3)}, and my new utility is: U′kmm = {1666(2/3)}0.1 x {13366(1/3)}0.9 = 10854. Hence, I’m better off and would vote for the government if it promised to implement this plan. Of course, this example is far from complete. For one thing how do I know that the prices will change in the way I assumed, and does the extra tax revenue really cover the cost of the subsidy? Well, we can’t answer these questions without knowing more about the supply side of the economy.

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