Finding marginal utility functions for U(x1,x2)=(x1+2)²(x2+3)³

[namy77]

1. Utility function of an individual: $$U=U(x_1,x_2)=(x_1 +2 )^2 (x_2=3)^3$$

Where $$x_1 and x_2$$ 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:

$$dU = f_{x1} dx_1 + f_{x2} dx_2<br /> <br /> = 2(x_2 + 3) ^2 (x_1 + 2) dx_1 + 3 (x_1 + 2)^2 (x_2 + 3)^3 dx_2$$

For the latter question: I plugged in: $$x_1 = 3 = x_2$$ and the result is 3060 utils... this sounds awfully lot - am I doing anything wrong? Thanks

 

[snipez90]

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.

 

[namy77]

Would this be better?

$$\begin{flalign*}<br /> <br /> \frac{\partial U}{\partial x_1} = 2(x_2 + 3)^2 (x_1 + 2) dx_1 \\*<br /> <br /> x_1 = 3 : 2(0+3)^2 (3+2) = 2.9.5 = 90<br /> <br /> \end{flalign*}$$

Thanks.

 

[snipez90]

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.