Please help!

[rosyposy]

Hi, I wondered if anyone might be able to offer some assistance with a gap in my lecture notes. The module is microeconomics and I thought this was probably the best forum to get help from considering the problem. I'm providing the background which may look a bit confusing if you haven't studied economics, but I think my actual issue is a mathematical one:

Here is the background to the problem:

Two (budget) constraints of the type i individual, i = h, l:

y₁ = y-P_i = y-

_iq_i
y₂ = y-L-P_i+q_i = y-L+(1-

_i)q_i

The decision maker's problem:

Choose qi to maximise expected utility subject to the budget constraints.

MaxU= (1-

_i)u(y₁)+

_iu(y₂)
=(1-

_i)u(y-

_iq_i)+

_iu(y_-L+(1-

_i)q_i)

Now the actual issue is how the professor has worked out the first-order condition:

U/

q= -(1-

_i)u' (y₁)

_i +

_iu'(y₂)(1-

_i)=0 for q_i>0

I should probably mention that u stands for utility, it just marks a function. and all of the rest of the symbols and letters including

are variables (I don't think the meaning of each is important to this question)

I really hope someone can help me! I'd appreciate if you could show me the mechanics of however he's worked this out, and if any particular derivative rules are involved. I've spent hours staring at it and I'm sure the answer is right in front of my face.

Thanks in advance Rosy

 

[JeffM]

Hi, I wondered if anyone might be able to offer some assistance with a gap in my lecture notes. The module is macroeconomics and I thought this was probably the best forum to get help from considering the problem. I'm providing the background which may look a bit confusing if you haven't studied economics, but I think my actual issue is a mathematical one:

Here is the background to the problem:

Two (budget) constraints of the type i individual, i = h, l:

y₁ = y-P_i = y-

_iq_i
y₂ = y-L-P_i+q_i = y-L+(1-

_i)q_i

The decision maker's problem:

Choose qi to maximise expected utility subject to the budget constraints.

MaxU= (1-

_i)u(y₁)+

_iu(y₂)
=(1-

_i)u(y-

_iq_i)+

_iu(y_-L+(1-

_i)q_i)

Now the actual issue is how the professor has worked out the first-order condition:

U/

q= -(1-

_i)u' (y₁)

_i +

_iu'(y₂)(1-

_i)=0 for q_i>0

I should probably mention that u stands for utility, it just marks a function. and all of the rest of the symbols and letters including

are variables (I don't think the meaning of each is important to this question)

I really hope someone can help me! I'd appreciate if you could show me the mechanics of however he's worked this out, and if any particular derivative rules are involved. I've spent hours staring at it and I'm sure the answer is right in front of my face.

Thanks in advance Rosy

Rosy I am sorry to be dense. I am fairly confident that this is a simple problem of using Lagrangian multipliers, but I am having trouble with understanding the problem, which may be in part the notation. (It has always driven me nuts that a lot of economics does not use standard math notation.)

Furthermore, this looks like microeconomics or maybe some initial steps in general equilibrium analysis. I believe I can help you if we walk through what is going on before we get to the solution.

What is y_i? What is i indexing? How is u defined? What is P_i. What is \(\displaystyle \pi.\) The latter is probably the Lagrangian multiplier. In short, I need to understand more about the problem before I can be sure that it is a simple problem of Lagrangian multipliers. Frequently a budget constraint is its own function. Here there seem to be two, which I do not quite understand.

You may not want to bother with educating me on this problem if you already know about Lagrangian multipliers. They crop up repeatedly in economics.

 

[JeffM]

Maybe there is a shortcut. Bolded small letter are vectors; unbolded small letters are scalars.

Suppose you want to optimize \(\displaystyle F(x)\) subject to m constraints \(\displaystyle C_i(x) \le a_i\ for\ 1 \le i \le m.\)

You set up \(\displaystyle H(x, \lambda) = F(x) + \displaystyle \sum_{i=1}^m(\lambda_i\{a_1 - C_i(x)\}.\)

Now optimize H(x), lambda).

\(\displaystyle \dfrac{\delta H(x, \lambda)}{\delta \lambda_i} = a_i - C_1(x).\)

So among your first order conditions is

\(\displaystyle a_i = C_i(x).\) You have now optimized subject to m constraints.

Pay no attention to this. LaTeX is not working in this post.

 

[rosyposy]

Rosy I am sorry to be dense. I am fairly confident that this is a simple problem of using Lagrangian multipliers, but I am having trouble with understanding the problem, which may be in part the notation. (It has always driven me nuts that a lot of economics does not use standard math notation.)

Furthermore, this looks like microeconomics or maybe some initial steps in general equilibrium analysis. I believe I can help you if we walk through what is going on before we get to the solution.

What is y_i? What is i indexing? How is u defined? What is P_i. What is \(\displaystyle \pi.\) The latter is probably the Lagrangian multiplier. In short, I need to understand more about the problem before I can be sure that it is a simple problem of Lagrangian multipliers. Frequently a budget constraint is its own function. Here there seem to be two, which I do not quite understand.

You may not want to bother with educating me on this problem if you already know about Lagrangian multipliers. They crop up repeatedly in economics.

Hi Jeff, thanks for offering your help!

Firstly, you're completely right. I wrote macroeconomics instead of microeconomics by mistake. Sorry to add more confusion! And yes, I believe the problem involves Lagrangian multipliers, which I understand somewhat, but this one has lost me!

Sorry about not being clear with my explanation.I'll try to answer your questions as best as I can:
I think you're misreading y₁ as y_i. If this is the case, y₁ just represents the first budget constraint.
u stands for utility, for example u(y₁) would mean that utility is a function of y₁.
P_i is the price offered to the individual i.

We have worked with multiple budget constraints earlier in the course and the way of dealing with them seems to be to multiply each one by a new lagrangian multiplier and add it to the lagrangian function. Hope that explains it.

I think the main problem I can't get my head around is how the first order condition is calculated. I know that it is a partial derivative of the utility function (MaxU) but I don't know the mechanics of how the lecturer worked it out. Are you able to help me with this? Am I missing something obvious?

Thanks so much for your help!

Rosy

 

[JeffM]

Hi Jeff, thanks for offering your help!

Firstly, you're completely right. I wrote macroeconomics instead of microeconomics by mistake. Sorry to add more confusion! And yes, I believe the problem involves Lagrangian multipliers, which I understand somewhat, but this one has lost me!

Sorry about not being clear with my explanation.I'll try to answer your questions as best as I can:
I think you're misreading y₁ as y_i. If this is the case, y₁ just represents the first budget constraint.
u stands for utility, for example u(y₁) would mean that utility is a function of y₁.
P_i is the price offered to the individual i.

We have worked with multiple budget constraints earlier in the course and the way of dealing with them seems to be to multiply each one by a new lagrangian multiplier and add it to the lagrangian function. Hope that explains it.

I think the main problem I can't get my head around is how the first order condition is calculated. I know that it is a partial derivative of the utility function (MaxU) but I don't know the mechanics of how the lecturer worked it out. Are you able to help me with this? Am I missing something obvious?

Thanks so much for your help!

Rosy

Let's take this in steps.

First show me how the utility function would be expressed. That is U(x) = what or U(y) = what. Of course if y is a function not a variable, you need to put in variables. Maybe those are the q's. What function would you optimize if there were no constraints?

 

[rosyposy]

Let's take this in steps.

First show me how the utility function would be expressed. That is U(x) = what or U(y) = what. Of course if y is a function not a variable, you need to put in variables. Maybe those are the q's. What function would you optimize if there were no constraints?

Ok, well it doesn't actually say anything about the problem other than what I included in my first post, but I think we're maximising the utility of the type i individuals, i=h,l (h for high risk individuals and l for low risk individuals- the lecture is on insurance). The decision makers are the consumers because they have to decide which insurance contract maximises their utility.

Sorry if this doesn't help

 

[JeffM]

Usually a utility function will be expressed in terms of quantities like \(\displaystyle U(\vec{q}),\)

Budget constraints may be expressed in terms of quantities and prices like \(\displaystyle C_i(\vec{q}, \vec{p}).\)

 

[JeffM]

Ok, well it doesn't actually say anything about the problem other than what I included in my first post, but I think we're maximising the utility of the type i individuals, i=h,l (h for high risk individuals and l for low risk individuals- the lecture is on insurance). The decision makers are the consumers because they have to decide which insurance contract maximises their utility.

Sorry if this doesn't help

OK I think now I get it.

We have \(\displaystyle U_L\) is the utility function a for risk avoider, and \(\displaystyle U_H\) is the utility function for a risk seeker. Am I ok that far?

And I am guessing that \(\displaystyle \pi\) is the premium

 

[rosyposy]

Usually a utility function will be expressed in terms of quantities like \(\displaystyle U(\vec{q}),\)

Budget constraints may be expressed in terms of quantities and prices like \(\displaystyle C_i(\vec{q}, \vec{p}).\)

Oh gosh, I've never seem it expressed that way before! Perhaps just different techniques used by different tutors?

The only thing I can think of that looks similar is the insurance contract: {P;q} offered to the customer.
This may also be helpful if given with the expected profit in equilibrium:

(1-

_i)pq +

[pq-q] = 0; i = h,l
where

_i= p
(Premium cover = pq)

Sorry, looks like we might be at a dead end if that doesn't help!

 

[rosyposy]

OK I think now I get it.

We have \(\displaystyle U_L\) is the utility function a for risk avoider, and \(\displaystyle U_H\) is the utility function for a risk seeker. Am I ok that far?

And I am guessing that \(\displaystyle \pi\) is the premium

I think you're right, yes!

but pi seems to be the accident probability:

"Two types of individuals:

High-risks with accident probability

_h
Low-risks with accident probability

_l;

_h>

_​l"

 

[rosyposy]

OK I think now I get it.

We have \(\displaystyle U_L\) is the utility function a for risk avoider, and \(\displaystyle U_H\) is the utility function for a risk seeker. Am I ok that far?

And I am guessing that \(\displaystyle \pi\) is the premium

My reply to this doesn't seem to have come through...

Basically, yes you're right, except for that

i is the accident probability:

"Two types of individuals:
High risks with accident probability

_h
Low risks with accident probability

_l;

_h>

_​l"

 

[JeffM]

Oh gosh, I've never seem it expressed that way before! Perhaps just different techniques used by different tutors?

The only thing I can think of that looks similar is the insurance contract: {P;q} offered to the customer.
This may also be helpful if given with the expected profit in equilibrium:

(1-

_i)pq +

[pq-q] = 0; i = h,l
where

_i= p
(Premium cover = pq)

Sorry, looks like we might be at a dead end if that doesn't help!

No, not different techniques. It is just that many economics texts do not use the clearest mathematical notation. It would clarify a lot of economics to use vectors and matrices. I mean it is simpler and clearer to take advantage of the fact that:

\(\displaystyle F(x_1, ...\ x_n)\ and\ F(\vec{x})\ MEAN\ the\ same\ thing.\)

I am going to try to work on this with you. If someone else comes along who is more intelligent than I am, I hope that he or she will take over. In the meantime, let's push ahead. This may take some time because I need to understand the problem to set up the relevant functions.

So I now think that I guessed wrong about the two types of purchaser. One faces high risks, and one faces low risks. Risk appetite seems to be captured in the utility function.

The decision maker is the purchaser of insurance correct? So profit does not seem relevant to me, but utility does.

Moreover, we seem to be considering two types of purchaser: one with a high risk of loss and one with a low risk of loss. The indices seem to be that 1 means high risk purchaser and 2 means low risk purchaser. Does that seem sensible?

I think I am getting a glimpse of the meaning of the two constraints. One seems to be the amount of loss that can be absorbed. The other is the amount of premium that is affordable. And there is some unspecified utility function that has a trade-off between pain of loss and pain of premium. Does this sound anything like what was being talked about during your lecture?

And what does premium cover mean? Does q mean the amount of coverage purchased in dollars (or euros or whatever). Is P the premium per unit of q?

 

[JeffM]

Please look at your notes. Does it say anything about expected value? Now that I know pi is a probability, this looks like optimizing the expected value of utility, rather than utility. In that case, the y's look like random variables rather than budget constraints. If that is what this is about, the notation is awful, but then it is economics. I cannot say that the values of the random variables make much sense to me, but then I really do not know what the variables represent.