<p>If the consumer's utility really only depends on <em>total</em> mgs consumed, then the pills are perfect substitutes and the indifference curves are parallel lines. Utility maximization then always results in a corner solution whenever the price ratio deviates from 2:1. This invalidates results based on calculus.</p>
<p>The Hicksian demand for a good is the derivative of the expenditure function w.r.t. the good's price. Therefore the symmetry of the cross-price effects is a consequence of the <a href="https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives" rel="nofollow noreferrer">symmetry of second derivatives</a> of a function, i.e. of Young's Theorem. Unfortunately, Young's Theorem has to the best of my knowledge no &quot;explanation in plain language&quot;. There's an attempt to provide intuition in <a href="https://youtube.com/watch?v=DmB8I0TFlc8" rel="nofollow noreferrer">this video</a>, but it's still somewhat mathematical.</p>

CC BY-SA 4.0
2023-05-08T21:54:53.920Z
2023-05-08T21:54:53.920Z
VARulle
<p>It is a well known fact in consumer theory that for a Hicksian demand curve the cross-price effect of good i with respect to the price of good j equals the cross-price effect of good j with respect to the price of good i.</p>
<p>Is there way to express this in simple language--there are mathematical proofs many places.</p>
<p>Suppose there are two very close substitutes, say 10 milligram and 20 milligram pills for some drug.  The 10 mg pills cost 5 dollars and the 20 mg pills 10 dollars.  If the price of the 10 mg pills goes up by a penny, suppose a consumer reduces consumption by 2 pills and increases purchases of 20 mg pills by <strong>1</strong> pill.  Utility should remain the same because he is buying the same total milligrams.  Alternatively, suppose the price of the 20 mg pills goes up by a penny.  The consumer should reduce consumption by 1 20 mg pill or 20 milligrams. To stay at the same level of utility the consumer then needs to increase purchases of 10 mg pills by <strong>2</strong> pills, not <strong>1</strong> pill, to maintain the same level of total milligrams purchased and presumably same level of utility.</p>
<p>The cross effects are not the same.  Where is the disconnect in my reasoning?</p>

CC BY-SA 4.0
2023-05-04T12:30:06.937Z
If the consumer's utility really only depends on *total* mgs consumed, then the pills are perfect substitutes and the indifference curves are parallel lines. Utility maximization then always results in a corner solution whenever the price ratio deviates from 2:1. This invalidates results based on calculus.
VARulle
CC BY-SA 4.0
2023-05-05T14:58:55.020Z
OK, @VARulle, suppose the 10 mg pill is marginally preferred because it is a bit easier to swallow.  This will add some curvature to the indifference curves.  And suppose the change in consumption of  the 10 mg pills is very small in response to the price change.  How can you explain the symmetry in plain language, not mathematics.  I've been unsuccessful in finding such an explanation.  Might be useful if you had to explain this to a senior policy maker, or an undergraduate in a principles class.
Istvan
CC BY-SA 4.0
2023-05-08T12:45:44.787Z
The Hicksian demand for a good is the derivative of the expenditure function w.r.t. the good's price. Therefore the symmetry of the cross-price effects is a consequence of the symmetry of second derivatives of a function, i.e. of Young's Theorem. Unfortunately, Young's Theorem has to the best of my knowledge no "explanation in plain language". There's an attempt to provide intuition at https://www.youtube.com/watch?v=DmB8I0TFlc8, but it's still somewhat mathematical.
VARulle
CC BY-SA 4.0
2023-05-08T21:50:16.287Z
I'll turn this into an answer. Maybe it helps somebody.
VARulle
<demand>
Is there a simple language explanation for Hick's Reciprocity/Symmetry Theorem?
Istvan
# Is there a simple language explanation for Hick's Reciprocity/Symmetry Theorem?

(Asked by: username_0 on 2023-04-28)

It is a well known fact in consumer theory that for a Hicksian demand curve the cross-price effect of good i with respect to the price of good j equals the cross-price effect of good j with respect to the price of good i.

Is there way to express this in simple language--there are mathematical proofs many places.

Suppose there are two very close substitutes, say 10 milligram and 20 milligram pills for some drug. The 10 mg pills cost 5 dollars and the 20 mg pills 10 dollars. If the price of the 10 mg pills goes up by a penny, suppose a consumer reduces consumption by 2 pills and increases purchases of 20 mg pills by **1** pill. Utility should remain the same because he is buying the same total milligrams. Alternatively, suppose the price of the 20 mg pills goes up by a penny. The consumer should reduce consumption by 1 20 mg pill or 20 milligrams. To stay at the same level of utility the consumer then needs to increase purchases of 10 mg pills by **2** pills, not **1** pill, to maintain the same level of total milligrams purchased and presumably same level of utility.

The cross effects are not the same. Where is the disconnect in my reasoning?

**Tags:** demand
> **Comments:**
> **username_1**: If the consumer's utility really only depends on *total* mgs consumed, then the pills are perfect substitutes and the indifference curves are parallel lines. Utility maximization then always results in a corner solution whenever the price ratio deviates from 2:1. This invalidates results based on calculus.  
> **username_0**: OK, @VARulle, suppose the 10 mg pill is marginally preferred because it is a bit easier to swallow.  This will add some curvature to the indifference curves.  And suppose the change in consumption of  the 10 mg pills is very small in response to the price change.  How can you explain the symmetry in plain language, not mathematics.  I've been unsuccessful in finding such an explanation.  Might be useful if you had to explain this to a senior policy maker, or an undergraduate in a principles class.  
> **username_1**: The Hicksian demand for a good is the derivative of the expenditure function w.r.t. the good's price. Therefore the symmetry of the cross-price effects is a consequence of the symmetry of second derivatives of a function, i.e. of Young's Theorem. Unfortunately, Young's Theorem has to the best of my knowledge no "explanation in plain language". There's an attempt to provide intuition at https://www.youtube.com/watch?v=DmB8I0TFlc8, but it's still somewhat mathematical.  
> **username_1**: I'll turn this into an answer. Maybe it helps somebody.  

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## 1 answers

### Answer by username_1 on 2023-05-08. Score: 0

If the consumer's utility really only depends on *total* mgs consumed, then the pills are perfect substitutes and the indifference curves are parallel lines. Utility maximization then always results in a corner solution whenever the price ratio deviates from 2:1. This invalidates results based on calculus.

The Hicksian demand for a good is the derivative of the expenditure function w.r.t. the good's price. Therefore the symmetry of the cross-price effects is a consequence of the [symmetry of second derivatives](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives) of a function, i.e. of Young's Theorem. Unfortunately, Young's Theorem has to the best of my knowledge no "explanation in plain language". There's an attempt to provide intuition in [this video](https://youtube.com/watch?v=DmB8I0TFlc8), but it's still somewhat mathematical.


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