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May 9th, 2017, 03:51 AM  #1 
Member Joined: Mar 2017 From: Israel Posts: 70 Thanks: 2  Microeconomics
Hello Can you please help me about the following exercise: The straight graph y = Y0 > 0 cuts all the equivalent curves. Prove that the marginal rate of substitution decreases when moving from one cutting point to a second cutting point (from left to right). My answer: We have y = Y0 > 0 which is parallel to the axis of x, so that from one cutting point to a second cutting point we can see that the amount of x units increases during that the amount of y units doesn't change, it means that the producer doesn't give up on y units during that the amount of x units increases, that's why the marginal rate of substitution decreases when moving from one cutting point to a second cutting point (from left to right). Am i right? Thanks a lot! 
May 10th, 2017, 11:40 AM  #2 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
what do you mean by all equivalent curves? I guess this doesnot apply if the graph is concave up

May 10th, 2017, 12:48 PM  #3  
Member Joined: Mar 2017 From: Israel Posts: 70 Thanks: 2  Quote:
 
May 11th, 2017, 10:08 AM  #4  
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Quote: Quote:
 
May 11th, 2017, 10:39 AM  #5  
Member Joined: Mar 2017 From: Israel Posts: 70 Thanks: 2  Quote:
I tried to prove it with formula of https://en.wikipedia.org/wiki/Margin...f_substitution, but the value of marginal rate of substitution will be always 0 because y is constant (if I'm not mistaken). I don't know... do you have an idea?  
May 11th, 2017, 11:20 AM  #6 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
A point to the right on the next indifference curve is always better because the consumer gets more of product x and the same amount of product y. Because of the law of diminishing marginal utility, the consumer would be willing to give up less of product y to obtain an additional unit of product x at the second cutting point. Therefore, this implies that the MRS has decreases at the second cutting point. Cheers. This is enough of a proof without math 
May 22nd, 2017, 07:09 AM  #7 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,131 Thanks: 716 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Informal proof... Since for all indifference curves $\displaystyle \frac{dy}{dx} < 0$, then $\displaystyle \frac{\Delta y}{\Delta x} < 0$. Consider two coordinates $\displaystyle (x_1, y_1)$ and $\displaystyle (x_2, y_2)$ which represent points on a single indifference curve where the line $\displaystyle y = y_1$ and $\displaystyle y = y_2$ cut the curve. Consequently, $\displaystyle \frac{y_2  y_1}{x_2  x_1} < 0$ Since the second curve has $\displaystyle x_2 > x_1$, the condition above is only satisfied if $\displaystyle y_2 < y_1$. 

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