Cobb Douglas Utility Function – With Example & Graph
What is Cobb Douglas Utility Function?
A utility function is an equation for calculating the preferences and choices by consumers at different options. There are many types of utility functions in economics (Cobb-Douglas utility function, Linear utility function, Von Neumann-Morgenstern and so on). Cobb Douglas Utility Function is an exponential formula that represents the utility (satisfaction) level of consumers from consuming different various goods and services. Usually in this utility equation, we represent the utility level of two goods.
Cobb Douglas Utility Function Formula
Cobb douglas utility function equation can be presented as, u(x, y) = x a y 1 – a
In the above function,
- U(x, y) represents the utility level.
- x and y are the two goods and “x” represents the quantity of x good consumed while “y” represents the quantity of y good consumed
- “a” is a positive constant between 0 and 1, indicating the relative importance of good x compared to good y. “1-a” represents the importance of the good y when compared to good x.
Cobb Douglas Utility Function MRS (Marginal Rate of Substitution)
Marginal rate of substitution (MRS) is a closely related concept to the utility of the consumers. So, firstly, let’s consider that what is the marginal rate of substitution (MRS) of the utility function.
MRS shows the ratio of a good (Let’s assume good “X”) is sacrificed by a consumer to increase the consumption of another good (Let’s assume good “Y”) while maintaining the same level of satisfaction. So, simply we can define MRS as the amount of good “x” is losing to increase the consumption of unit of good “y”.
How to find MRS from utility function? / How to calculate MRS from utility function?
MRS = – (MUx / MUy)
In the above function,
MUx = The partial derivative of the utility function with respect to good “x”
MUy = The partial derivative of the utility function with respect to good “y”
Divide MUx by MUy and multiply by -1 (When you can get a positive value).
How to find MRS from an indifference curve?
The slope of the indifference curve at a point represents the marginal rate of substitution of goods at that point. So, you can find the MRS by calculating the value of the slope at a particular point. Because MRS is the value of a good is sacrificed by a consumer to increase the consumption of another good
There is a diminishing marginal rate of substitution for a utility of a good. In other words, to increase the consumption of a unit of good x, the amount of consumption to be decreased from good y is continuously decreasing when additional units of x are rising.
Cobb Douglas Utility Function Example Problems – Cobb Douglas Utility Function Maximization
Let’s consider a Cobb-Douglas utility function example. In this question, you will learn how to maximize a Cobb-Douglas utility function.
Let’s assume that a consumer has two goods to consume. These goods will be good x and good y. The utility functions for these two goods are as follows.
u(x, y) = x 0.5 y 0.5
The consumer income level is $90. The price of a unit of x is $2 and the price of a unit of y is $4. So, budget constrain is equals to,
90 = 2x + 4y
Find the utility maximization quantity of good x and quantity of good y.
Answer
First thing that we should do is, determine the marginal utility of good x and good y.
MUx = Partial derivative of the utility function with respect to good “x”
u(x, y) = x 0.5 y 0.5
MUx = ∂u / ∂x = 0.5 x 0.5 – 1 y 0.5 = 0.5 x – 0.5 y 0.5
Muy = Partial derivative of the utility function with respect to good “y”
u(x, y) = x 0.5 y 0.5
MUy = ∂u / ∂y = 0.5 x 0.5 y 0.5 -1 = 0.5 x 0.5 y – 0.5
At the utility maximization point,
MUx / Px = Muy / Py
So,
MUx / Px = 0.5 x – 0.5 y 0.5 / 2
MUy / Py = 0.5 x 0.5 y – 0.5 / 4
0.5 x – 0.5 y 0.5 / 2 = 0.5 x 0.5 y – 0.5 / 4
0.5 x – 0.5 y 0.5 / 2 * 2 = 0.5 x 0.5 y – 0.5 / 4 * 2
0.5 x – 0.5 y 0.5 = 0.5 x 0.5 y – 0.5 / 2
0.5 x – 0.5 y 0.5 * 2 = 0.5 x 0.5 y – 0.5 / 2 * 2
x – 0.5 y 0.5 = 0.5 x 0.5 y – 0.5
y 0.5 /x 0.5 = 0.5 x 0.5 / y 0.5
y 0.5 * y 0.5 = 0.5 x 0.5 * x 0.5
y = 0.5 x
Let’s apply these values to budget constrain
90 = 2x + 4y
90 = 2x + 4 (0.5 x)
90 = 2x + 2 x
90 = 4 x
x = 22.5
90 = 2x + 4y
90 = (2 * 22.5) + 4y
90 = 45 + 4y
45 = 4y
y = 11.25
So, utility maximization good combination of goods x and y is 22.5 units of x and 11.25 units of y.
Cobb Douglas utility function graph
The following graph is showing an optimal consumption bundle from a consumer’s utility maximization problem.
In the above graph, “I” represents the budget constraint of the consumer. This consumer can choose any point on the budget constraint. Among different combinations of goods, the consumer chooses the combination of goods x and y that gives maximum benefit for them. Utility curve “u” gives the maximum utility for the consumers at this budget constraint. The goods combination that gives maximum utility is equal to point “E” and at this point consumer consumes y* and x*.
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