<p>(I'll be using formulae from Hal Varian.)</p>
<pre><code>Given, U(x,y)= 7.5 x^0.3 5y^0.2
Income (M) = 10000
Initial Price of good-x: 5
Price of good-y: 2.5 
New Price of good-x (Px'): 4
</code></pre>
<p>Solution:
This is clearly a Cobb Douglas Utility Function.
So, you could either derive demand curve for good-x using all the numbers given or directly put the value in the already derived demnd function of good-x.</p>
<p>The demand function for good-x is given by (it is independent of the coefficients of both the goods; you can check it by deriving the demand function);</p>
<pre><code>x* = (a.M)/(a+b)Px

At Px= 5, a= 0.3, b= 0.2 and M=10000, we get that
   
x* = (0.3* 10000)/ (0.5*5)
              x* = 1200
</code></pre>
<p>In order to calculate the <code>substitution effect</code>, one first needs to calculate the change in income required to keep the original bundle just affordable. As price of only good-x has changed, the change in income is given by</p>
<pre><code>Change in M= Change in price * (x*)
           = (-1)* (1200)
           = -1200

The new income, say M', is 10000 - 1200 = 8800
</code></pre>
<p>So, demand of good-x at new price and income level is <code>x'(Px',M')</code>:</p>
<pre><code>              x' = (0.3*8800)/(0.5*4)
              x' = 1320
</code></pre>
<p>The substitution effect is</p>
<pre><code>             x'(Px',M') - x*(Px,M)
           = 1320 - 1200 = 120
</code></pre>
<p>To calcuate <code>income effect</code>, the income is kept constant and new prices are taken into account such that demand for good-x is given by <code>x''(Px',M)</code></p>
<p>Hence, demand for good-x at
<code>Px' = 4 and M = 10000 is</code></p>
<pre><code>                x'' = 0.3*10000/0.5*4
                    = 1500
</code></pre>
<p>The change in quantity demanded of good-x due to income effect is thus given by:</p>
<pre><code>             x''(Px',M) - x'(Px',M')
             = 1500 - 1320
             = 180
</code></pre>
<p>Hence, total price effect is 120 + 180 = 300</p>

CC BY-SA 4.0
2024-04-04T13:04:16.570Z
2024-04-05T12:38:25.690Z
2024-04-05T12:38:25.690Z
Shreya
<p>We were given this information to compute the total, substitution, and income effects.</p>
<p><span class="math-container">$U=7.5x^{0.3} \cdot 5y^{0.2}$</span>,</p>
<p><span class="math-container">$\text{Income} = 10,000$</span>,</p>
<p><span class="math-container">$P_x = 5$</span>,</p>
<p><span class="math-container">$P_y = 2.5$</span>,</p>
<p><span class="math-container">$P_{x'}=4$</span>.</p>
<p>I am having a hard time trying to get the income and substitution effect.</p>

CC BY-SA 4.0
2024-04-03T14:57:38.230Z
Hi please show us your attempt first per our rules for homework questions
1muflon1
<microeconomics>
Total Effect, Substitution Effect, and Income Effect
Mikognito
# Total Effect, Substitution Effect, and Income Effect

(Asked by: username_0 on 2024-04-03)

We were given this information to compute the total, substitution, and income effects.

$U=7.5x^{0.3} \cdot 5y^{0.2}$,

$\text{Income} = 10,000$,

$P\_x = 5$,

$P\_y = 2.5$,

$P\_{x'}=4$.

I am having a hard time trying to get the income and substitution effect.

**Tags:** microeconomics

> **Comments:**
> **username_1**: Hi please show us your attempt first per our rules for homework questions  

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## 1 answers

### Answer by username_2 on 2024-04-04. Score: 1

(I'll be using formulae from Hal Varian.)

```
Given, U(x,y)= 7.5 x^0.3 5y^0.2
Income (M) = 10000
Initial Price of good-x: 5
Price of good-y: 2.5 
New Price of good-x (Px'): 4

```

Solution:
This is clearly a Cobb Douglas Utility Function.
So, you could either derive demand curve for good-x using all the numbers given or directly put the value in the already derived demnd function of good-x.

The demand function for good-x is given by (it is independent of the coefficients of both the goods; you can check it by deriving the demand function);

```
x* = (a.M)/(a+b)Px

At Px= 5, a= 0.3, b= 0.2 and M=10000, we get that
   
x* = (0.3* 10000)/ (0.5*5)
              x* = 1200

```

In order to calculate the `substitution effect`, one first needs to calculate the change in income required to keep the original bundle just affordable. As price of only good-x has changed, the change in income is given by

```
Change in M= Change in price * (x*)
           = (-1)* (1200)
           = -1200

The new income, say M', is 10000 - 1200 = 8800

```

So, demand of good-x at new price and income level is `x'(Px',M')`:

```
              x' = (0.3*8800)/(0.5*4)
              x' = 1320

```

The substitution effect is

```
             x'(Px',M') - x*(Px,M)
           = 1320 - 1200 = 120

```

To calcuate `income effect`, the income is kept constant and new prices are taken into account such that demand for good-x is given by `x''(Px',M)`

Hence, demand for good-x at
`Px' = 4 and M = 10000 is`

```
                x'' = 0.3*10000/0.5*4
                    = 1500

```

The change in quantity demanded of good-x due to income effect is thus given by:

```
             x''(Px',M) - x'(Px',M')
             = 1500 - 1320
             = 180

```

Hence, total price effect is 120 + 180 = 300


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