mathematical representation of substitution/income and price effect

[waqas izhar]

can anyone point out the mathematical proof/representation of substitution effect etc please

regards

[Toru]

Originally Posted by waqas izhar

can anyone point out the mathematical proof/representation of substitution effect etc please

regards

It's going to be a bit difficult (and daunting) in trying to manage putting down the correct mathematical notation as your requirement would typically make use of the lagrangian multiplier method, that would include partial derivatives and equations in terms of lambdas etc.

since you require the substitution (Hicksian) effect in mathematical notation, it will typically require using a cobb-douglas utility function ( since IC curves will be taken to be downward sloping ).

in substitution effect, what happens is you assume the utility is constant (by virtue of tangency with the budget constraint).

In a world of two goods, X and Y with prices 'Px' and 'Py', utility function can be

U(x,y) = X^a . Y^b and budget constraint can be taken to be I = Px.X + Py.Y and then forming a lagrangian multiplier

L(x,y,z) = X^a . Y^b - z(Px.X + Py.Y - I) where I have taken z: in place of Lambda.

Given the above, I would assume you would be able to work the math, taking partial differences with respect to x and y and setting to zeros. you will get the respective quantities as long as you keep total utility constant.

[Toru]

Originally Posted by waqas izhar

can anyone point out the mathematical proof/representation of substitution effect etc please

regards

I'm sorry for the poorly demonstrated effort since I wasn't able to make use of the keyboard any better with mathematical notation. But if you're a student of economics, I would assume you would at least have understood the gist of the math above.

and as far as the powers a and b go, 0.5 and 0.5 would be a good value to start.

Let it not worry you my friend, I don't suppose the examiner could give such a question. Depiction through proper graphs followed by proper intuition and explanation of your analysis will suffice. Do ensure mentioning the assumptions of your analytical framework before you embark upon this venture.

[waqas izhar]

Originally Posted by Toru

I'm sorry for the poorly demonstrated effort since I wasn't able to make use of the keyboard any better with mathematical notation. But if you're a student of economics, I would assume you would at least have understood the gist of the math above.

and as far as the powers a and b go, 0.5 and 0.5 would be a good value to start.

Let it not worry you my friend, I don't suppose the examiner could give such a question. Depiction through proper graphs followed by proper intuition and explanation of your analysis will suffice. Do ensure mentioning the assumptions of your analytical framework before you embark upon this venture.

Thank you sir but there is a glitch. With this method I will only get the optimum level of consumption. How do I incorporate substitution and income effect.

Regards

[Toru]

Originally Posted by waqas izhar

Thank you sir but there is a glitch. With this method I will only get the optimum level of consumption. How do I incorporate substitution and income effect.

Regards

There isn't a glitch brother. I wouldn't insist on it owing to the poor effort i have made in making the point by not being able to use proper notation. Therefore I wouldn't really blame you for it

I will assume you have solved for the utility maximizing values for both x and y. it should give you ( y/x = Px/Py ) ---> Px.X = Py.Y

substitute either one in the budget constraint. That will lead you to getting (in terms of x if substituted for y) --> X = I/2Px (and same for Y)

substituting these in the initial utility function, U(x,y) would get you
U(X,Y) = (I/2Px)^0.5 . (I/2Py)^0.5 where I have, for simplicity, taken the values of a and b as 0.5.

Utility is now a function of Income (I). Price of X (Px) and Price of Y (Py) which is also called the indirect utility function based on its constituents; can be written for as f(I,Px,Py)

Following from above, you now know that U(x,y) = X^0.5 . Y^0.5 = (I/2Px)^0.5 . (I/2Py)^0.5 which is shown to be equal to f(I,Px,Py)

Re-writing, we get I/2[(Px.Py)^5] = f where cross multiplication leads to I = 2.f.(Px.Py)^0.5

Since you had optimal values of both x and y in terms of I and px/py, you can now successfully substitute for I from the above and get optimal quantities in terms of prices for a CONSTANT utility that demonstrates the impact of a substitution effect. The functions we have deduced here are Compensating Demand Functions.

I hope the effort has been made in a better way relatively. I will, how ever, be open to any questions.

Regards.

[Toru]

Originally Posted by waqas izhar

Thank you sir but there is a glitch. With this method I will only get the optimum level of consumption. How do I incorporate substitution and income effect.

Regards

I just took note of you asking for the income effect.
It's going to follow the same process, only to keep INCOME (which you will get in terms of utility and respective prices) constant in place of Utility. In other words, we will be working for the ordinary demand function in a similar fashion to this which was a compensated demand function.

[waqas izhar]

Originally Posted by Toru

I just took note of you asking for the income effect.
It's going to follow the same process, only to keep INCOME (which you will get in terms of utility and respective prices) constant in place of Utility. In other words, we will be working for the ordinary demand function in a similar fashion to this which was a compensated demand function.

thank you sir! i will get my register and try to push it all in my pea brain and then come back to you.

Secondly sir, are VMPL and MRPL both the same thing i.e. MPL x Price ?

[Toru]

I take it you mean Value of Marginal Product of Labour (VMPL) ?
If that's so, then yes, MRP and VMPL are the same.

I will be more than pleased to offer any help that you may require.

[waqas izhar]

Originally Posted by Toru

I take it you mean Value of Marginal Product of Labour (VMPL) ?
If that's so, then yes, MRP and VMPL are the same.

I will be more than pleased to offer any help that you may require.

Sir, there is a question in PU past papers which asks for the difference between the two. i am appearing in previous next month. do i take it then that they meant MPPL?

regards

[Toru]

Originally Posted by waqas izhar

Sir, there is a question in PU past papers which asks for the difference between the two. i am appearing in previous next month. do i take it then that they meant MPPL?

regards

I just re-read your question.

Value of Marginal Product of Labour (VMPL): where, by value, i gather it implies 'worth'. and MPL is the additional output that labour produces. The ''value/worth'' should, intuitively, mean price of the product since a product is typically worth the same as its price.. Price of a MP is called MR (marginal revenue).

since MPL x MR = MRPL, I think the former and VMPL are the same as i previously mentioned.

according to an online page that I just googled,

Value of Marginal Product

The value of marginal product is the value to a firm of hiring one more unit of a factor of production. The value of marginal product equals the price of a unit of output multiplied by the marginal product of the factor of production. As long as a worker’s value of marginal product exceeds the wage, the worker is hired. But because the marginal product is diminishing, eventually so many workers will have been hired that the value of the marginal product of an additional worker would be less than the wage. At this point the hiring will stop. A firm hires labor up to the point at which the value of marginal product equals the wage rate. (My own note: equilibrium condition is MRPL = wage (or Marginal Factor Cost: MFC; This further confirms my point) If the value of marginal product of labor exceeds the wage rate, a firm can increase its profit by employing more workers. If the wage rate exceeds the value of marginal product of labor, a firm can avoid the reduction of its profit by employing fewer workers.

source can be referred here:
http://microecon201.wikispaces.com/V...rginal+Product

I hope it proves useful.

Regards