# Economics-indifference curves and Utility Function

I will try to understand the theory about indifference curves correctly. I have given a Utility function $$u(x_1,x_2)=x_1x_2^2$$. Then I have to illustrate the bundles that are indifference with the bundle $$(x_1,x_2)=(2,4)$$.

I think I the get the line $$x_2(x_1)=(\frac{u}{x_1})^{1/2}$$ for the indifference curve. If we find $$MRS(2,4)=-\frac{\frac{1}{2}x_1}{\frac{1}{2}x_2}=-\frac{1}{2}$$. I'm not quite sure how to plot that? I have not any values of u. Can someone help me?

• What are the prices of the goods? My guess that $p_1=p_2$ – callculus Nov 24 '20 at 16:55
• I have found now $MSR(2,4)=-1$. And then I plot $x_2(x_1)=(\frac{u}{x_1})^{1/2}$ where u=32 so the indifference curve pass through $(𝑥_1,𝑥_2)=(2,4)$ and a tangent in this point with coefficient -1? But how can I illustrate the bundles that are indifference with the bundle $(𝑥_1,𝑥_2)=(2,4)$. Is it all indifference curve with u>32? – Lifeni Nov 24 '20 at 17:09
• repetition:What are the prices of the goods? – callculus Nov 24 '20 at 17:11
• I have not any prices in my problem? $p_1$ and $p_2$ is just variables – Lifeni Nov 24 '20 at 17:13
• I think you have to draw the picture I´ve posted. If the budget line touches the utility line then the bundle is optimal. The bundles on the utility function are all indifferent with the bundle $(x_1,x_2)=(2,4)$ – callculus Nov 24 '20 at 18:45

You have $$u(x_1,x_2)=x_1x_2^2$$. The indifference curves are the level sets of that function, that is, all values of $$x_1,x_2$$ such that $$u(x_1,x_2)=K$$ for some constant $$K.$$ In your particular problem, you want all $$x_1,x_2$$ such that $$x_1x_2^2=2\times 4^2=32.$$ don't know the name of that curve, but it should be possible to plot by hand or program.
Calculating the MRS is a good idea, because with the data from the problem you have both a point on the indifference curve and its slope at that point. Of course in general the slope is $$-\frac{1}{2}\frac{x_2}{x_1}$$ so that can give you an idea of how the slope varies.