How to Calculate Marginal Utility and Its Impact on Consumption | U(x1,x2)

In summary, the marginal-utility function for each commodity is found by taking the partial derivative of the utility function with respect to that commodity. The value of the marginal utility of the first commodity when 3 units of each commodity are consumed is 90 utils.
  • #1
namy77
2
0
1. Utility function of an individual: [tex]U=U(x_1,x_2)=(x_1 +2 )^2 (x_2=3)^3[/tex]

Where [tex]x_1 and x_2[/tex] are the quantities of two commodities consumed.

2. Question: FInd the marginal-utility function of each of the commodities and the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.

My attempt:

[tex] dU = f_{x1} dx_1 + f_{x2} dx_2

= 2(x_2 + 3) ^2 (x_1 + 2) dx_1 + 3 (x_1 + 2)^2 (x_2 + 3)^3 dx_2[/tex]

For the latter question: I plugged in: [tex] x_1 = 3 = x_2 [/tex] and the result is 3060 utils... this sounds awfully lot - am I doing anything wrong? Thanks
 
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  • #2
I don't think you're supposed to take the total derivative, which is probably only used in the derivation of the marginal rate of substitution. You need to find MP1, the marginal product of good 1, and MP2, and this can be done by taking the partial derivative of U with respect to x1 and x2 respectively (so to find MP1, treat x2 as a constant and take the derivative with respect to x1). Then plug in x1 = 3 = x2 into MP1 to answer the second part.
 
  • #3
Would this be better?

[tex]\begin{flalign*}

\frac{\partial U}{\partial x_1} = 2(x_2 + 3)^2 (x_1 + 2) dx_1 \\*

x_1 = 3 : 2(0+3)^2 (3+2) = 2.9.5 = 90

\end{flalign*}
[/tex]

Thanks.
 
  • #4
Well the consumer is still consuming 3 units of each good, so that 0 in the place of x2 should be 3, but otherwise that looks fine.
 

1. What is marginal utility?

Marginal utility is a concept in economics that measures the additional satisfaction or benefit a consumer gains from consuming one additional unit of a good or service. It helps to explain the relationship between the quantity of a good consumed and the satisfaction or value that a consumer derives from it.

2. How is marginal utility calculated?

To calculate marginal utility, you need to divide the change in total utility by the change in the quantity of the good consumed. This can be represented by the formula: MU = ∆TU / ∆Q, where MU is marginal utility, ∆TU is the change in total utility, and ∆Q is the change in quantity consumed.

3. What is the law of diminishing marginal utility?

The law of diminishing marginal utility states that as a consumer consumes more and more units of a good, the additional satisfaction or benefit they derive from each additional unit will eventually decrease. This means that the more of a good a person consumes, the less valuable each additional unit becomes.

4. How does marginal utility affect consumer behavior?

Marginal utility plays a significant role in consumer decision-making. Consumers will continue to consume goods as long as the marginal utility is positive, meaning that the additional satisfaction or benefit they derive from each additional unit is greater than the cost. As marginal utility decreases, consumers will eventually reach a point where they are no longer willing to pay the cost for the additional units, leading to a decrease in demand for the good.

5. Can marginal utility be negative?

Yes, marginal utility can be negative. This occurs when the consumption of an additional unit of a good results in a decrease in total utility. This can happen when a consumer has already consumed a large quantity of a good and experiences diminishing marginal utility, or when a good is no longer desirable to the consumer. Negative marginal utility can also be observed in situations where a good has a negative impact on the consumer's overall well-being, such as with unhealthy or harmful products.

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