Insurance/Future Consumption

[ZSchlaht]

Two quick questions:

1. I understand Gamma in the contingent consumption BC is the "price" of consumption in the bad state, does this just mean that each unit of consumption in the bad state costs (gamma/1-gamma), and is the price of consumption in the good state just assumed to be unity? 
2. On the Solving for C1 and C2 for C-D handout, since C2 is future consumption, does this mean that the demand for future consumption is: C2 = 1/2((1+r)m1 + m2) since this would be in terms of future values? 

Thank you! 

[J]

1. If you multiply the entire budget constraint by (1-Gamma), then the equation is

gamma*Cb + (1-gamma)*Cg = (1-gamma)*M + gamma*(m-L)

Let's compare this to the original budget constraint

Px*X +Py*Y = M

You can see from this that price of consumption in the bad state is Gamma and that the price of consumption in the good state is (1-gamma).

You can also derive this by looking at how we set MRS equal to the price ratio.

When solving for maximum utility, we generally set MRS = p1/p2. When solving for optimal consumption in insurance problems, we set MRS = gamma/(1-gamma). Looking at this it is easy to see that the price of consumption in the bad state (Cb, aka "good X") is gamma and the price of consumption in the good state (Cg, aka "good Y") is 1 - gamma.

2.

I don't believe there is a shortcut to future consumption for Cobb-Douglas. Because we set MRS equal to the price ratio, and the MRS is derived from the marginal utilities of the respective consumption in the function, it's going to be different every time. 

Also, just to relate this back to your first question,

recall that the budget constraint for future value, using both inflation and the nominal interest rate (r) looks like:

[(1+r)/(1+pi)]*C1 + C2 = [(1+r)/(1+pi)]*M1 + M2

Let's do the same thing we did last time and multiply everything by (1+pi).

(1+r)*C1 + (1+pi)*C2 = (1+r)*M1 + (1+pi)*M2

Again, comparing this to the original budget constraint, you can see that the price of each "good", that is, consumption in the present and future respectively, are (1+r) and (1+pi). And recall that when solving for optimal consumption, we set MRS = (1+r)/(1+pi) = "p1/p2"

Also take note that (1+r)/(1+pi) = 1+rho, rho (p) being the real interest rate. 

[J]

Here's a screenshot from the lecture slides. This might actually contradict the answer I gave you... But as far as I can tell about my second point, it sounds right in theory. Hopefully a TA or someone else more knowledgeable than I can give you a straighter answer. 

Take my words with a grain of salt. Or a whole pinch. :|

[Z]

That's what I was looking at when I initially made the post, but then I read your answer and it made sense too. Hopefully someone can confirm.

Also, for the Present/Future consumption problem, if the MRS is diminishing we set it equal to (1+p) to find our demand function,  but for C2 we are thinking in terms of future value, so do we replace the m in (m/p1) with the future formula [(1+P)m1 + m2]?

[J]

Your present value and future value budget constraints are technically equal, the future value constraint is just the present value constraint multiplied by 1+p.

 

So basically, whatever constraint you pick, you stick with. You don't have to change anything. Simply set your MRS equal to 1+p, or 1+r if there is no inflation (aka you are talking nominal, not real). Then isolate one of the variables (usually C2) and substitute it back into whatever budget constraint you chose to solve for the other one. Solve for it in terms of your interest rate and inflation if necessary, along with present and future income.

You can usually use this answer and plug it back into whatever equation you got from isolate C2

[Lindsay Appell]

Here's what my thoughts are on the contingent consumption budget constraint: technically, we will consider the price of cb to be y / 1-y and the price of cg to be 1. To me, this seems similar to what we do with c1 and c2 where we say that the price of c1 is 1 and the price of c2 is 1 + pi. However, writing it in the form that J did -- y*Cb + (1-y)*Cg = (1-y)*M + y*(m-L) -- and thinking of y as the price of cb and 1 - y as the price of cg is a great way to understand the intuition behind the budget constraint. 

Thinking about it graphically, the slope of the budget constraint is -(y / (1-y)), meaning that as we move from right to left on the budget constraint, we have a fall of “y” and a run of “1-y”. Since cg is on the y-axis (the “fall” axis) and cb is on the x-axis (the “run” axis), this means that as we move towards cb, we give up y and get 1-y. 

The budget constraint is originally derived from the formulas cg = m - yK and cb = m - L - yK + K —> cb = m - L + (1 - y)K. Moving towards cb graphically means that K, our insurance coverage, is increasing. Looking at the cg and cb formulas, we can see that each time K increases by a dollar, we lose $y of cg and gain $(1 - y) of cb. Therefore, you can think of $y as the price of cb because it's what we give up in the good state to get more money in the bad state; similarly, you can think of $(1 - y) as the price of cg because it's what we give up in the bad state to get more money in the good state.