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IV_ Using the budget constraints and the first order conditions, derive the optimal con- sumption pattern and the amount of

Question:

IV_ Using the budget constraints and the first order conditions, derive the optimal con- sumption pattern and the amount of capital investment and deposit for given real rate of interest on deposit, T_ [Marks 61 V Suppose that there is only one bank: What would be the equilibrium rate of interest, T , offered by the bank to the depositors? Marks 21


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Suppose that the T-account for First National Bank is as follows: a. If the Bank of Canada requires banks to hold 5 percent of deposits as reserves, how much in excess reserves does First National now hold? b. Assume that all other banks hold only the required amount of reserves. If First National decides to reduce its reserves to only the required amount, by how much would the economy's money supply increase?

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Alright for number 55. We're talking about maximizing the revenue in a bank situation. You've got money that is being went out and you've also got money that is sitting in savings accounts. Um, interest is paid on both of them. And the question is, is if the raising of in the interest on the checking and savings accounts means more people are going to put more money in right, then house come, they maximize their revenue overall.

But we know that revenues to find this price times demand. Right. So the price in this case is the interest. That's the price of having the money lent to you Or, you know, the use of the money, um that they would have to pay. For now, the demand is proportionate to the interest rate, which means as interest rate rises, there's more.

Demand has interest rate falls. There's less demand. If you look over here, the chart, Um, if you were saying, I'll give you 0% to put your money in my bank, nobody would do it. And on the other side of things, if you said, um, sure, 18% interest. Come on, everybody.

Well, that would be great, But then you'd be, you know, loaning it all back out again. Then your net would be nothing. You would gain nothing from it. So you would have no revenue that way either. But there's a sweet spot that exists somewhere in the middle.

So if you think of our demand right as being the interest rate minus whatever the prices like that controls the amount, the size of that demand, then we can multiply the price, times the demand and come up with our function. So I function here for our of I is negative. I squared plus 18 i 18 being the interest rate that they're able to charge to lend money. So if we go ahead and apply the lessons from this chapter, then we take the derivative and we said it equal to zero to find out where that Vertex is of the curve. And if we do that when we come up with this equation and then down below, we can work out.

What is I and I equals nine. So we should set it at 9%. Okay, we should set our price at 9% which is actually smack in the middle, which makes a lot of sense.

Already. So this problem, we are given a function of f which basically describes the amount of invested capital. Yeah. They given time key. Um And so then the differential equation of that, there's going to be equal to the rate of net investment.

So are we investing more, putting more resources into it? Are we pulling them out? Are we selling stocks, etcetera? Someone determined that the optimum optimum level of investment, so that would be A. F. T. Is equal to see, I guess we want to write a differential equation for why? Because Fft for the rate of net investment. And so one thing to note is if the optimal level is, see, you could even think of that as some of these population models already looked at where there was a carrying capacity.

So when we go over the carrying capacity, the population falls back down towards it. When we're under the carrying capacity of the population grows. This is the same way if we're over the optimum investment, we want to invest less and pull out so we want to decrease the investment and if were under we want to increase our investment. So we're really mimicking those those models that we've already seen in some of these other problems. Yes, we're gonna have, you know, some factor.

Okay. And then we multiply that by c minus Y and we would need K to be positive. Okay. And so you can see here similarly to what I described, if our current invested capital is under C, this will be positive and it'll grow, but if it is greater than C than this expression here becomes negative and it will shrink back down towards C. Okay, so this uh differential equation of K.

Times and then in brackets here, C minus Y is going to be the solution. And just to give a quick try any bit more information, um You would expect it, let's see here. You expect it to either, um it's going to grow towards sea or it's going to fall down towards the right, depending on where our initial condition is. Okay, so I hope that helped..

So in this case we've been given the supply and the demand equations for 13 commodity. So the supply in question is modeled by the equation P is equal to under root of 0.1, Q plus nine minus two. And the demand equation is modeled by P is equal to under root of 25 minus 0.1. Cute! Right? So now what we are supposed to find over here simply we have to return mine like mental equilibrium be reached or we can say we have to determine the um equilibrium demand. Find the equilibrium demand for this case.

So equilibrium demand is reached when supply is equal to demand. Right? So for that I'll just take both. The equations equal to each other. Supply should be equal to demand. Right? So for that I just a body questions equal to each other that means and root of 0.1 Q plus nine minus two is equal to under root of 25 minus 0.1 Q.

Right now you can just change the decimals into factional forms so I can just write 0.1 as one upon 10-Q plus nine minus two is equal to and the root of 25 minus one upon 10-Q. Now just square buddha sides, square budo sites. So when you do that you'll get one upon 10-Q plus nine minus four times of under road one upon 10 cube plus nine plus four is equal to 25 minus one upon 10. Here right now I just need to combine all the like terms together and just simplify it. So I'm doing that I'll get one upon five q minus 12 minus four times off.

Under route one upon 10 Q Plus nine equal to zero. Okay all I can say one upon five q minus 12 is equal to four and rude, one upon 10-Q plus nine. Now you can just square both the sides. On doing that will get you squared upon 25 plus 140 for minus 24 upon five Q. Is equal to what will get on the other side.

You have to square the other side as well. Right? So on the other side we'll get That will be equal to eight upon 5 Q Plus 144. Now just uh like combine all the like terms together. And on doing that we'll get huge square upon 25 minus 20 for Q upon five minus eight upon five Q. Is a call to zero.

Now you just need to multiply both the sides by 25. So I'm doing that. What will get simply Q square upon 25 minus 32 upon five kiko to zero. Right now, I just combined these two terms together. Now, I just take the L C M s 25 we'll get you squared minus 32 times 25.

You equal to zero. You're right now. Uh further what you will get over here? Yeah, this will be five over here. Since there was a denominator. No for five.

So you just get five over here. Now, this will further give you Q square minus 160. Q is equal to zero. Or I can just characterize out cue from here. So we get q minus 160 equal to zero.

That means you will be equal to zero, or Q is equal 260. That means 162 of 160 units will help you restore equilibrium state right now, moving on to the next part, we have to find out the equilibrium price over here. So for that, I'm just going to substitute you as 160 in equation too. So just put q equal to 160 in equation too. If you want, you can just put it in the equation one as well.

So we'll get P is equal to and root of 25 minus 16. We shall give you and the root of night, and this will be equal to plus minus three. And obviously since we are finding out the price, that means we have to take the positive value. So P is equal to three. Hence the equilibrium price will be $3 Right? So this is how you find the questions based on the equilibrium conditions.

So were you?.

There are multiple steps involved in this problem. First, we need to figure out what the equilibrium constant is. Equilibrium constant is typically just a ratio of the rate constants in the forward direction, divided by the rate constant in the reverse direction. So if this is our generic reaction than K is going to be equal to the pressure of sea, or KP in particular is going to be equal to the pressure of sea over the pressure of a multiplied by the pressure of B de que value. Or, more specifically, the case see value is in the ratio of the rate constant in the forward direction, divided by the rate constant in the reverse direction.

This is Casey because rate constants are a function of concentrations, so K C then is 4.14 now to determine the final pressures at equilibrium. We need a K p value because we are given initial pressures and we're working with pressures, not a Casey value. So we need to convert the K C value into the KP value. Before we do that, let's figure out what the equilibrium pressures would be in terms of an unknown there in terms of the variable X. If we have start with a 1.6 atmosphere of A and I will 0.44 atmospheres of B and nothing to see that it has to shift to the right in order to establish equilibrium because we can't have any pressures at zero.

So the B minus X minus X and plus X because all of the coefficients are one, then it equilibrium will have 1.6 minus X for a 0.44 minus x for B and just X for C. So now let's go back to calculate in our KP value from RKC value, KP is going to be equal to Casey multiplied by our, which is 0.8 to 1 as a constant multiplied by t, which they gave to us in Kelvin at 3 23 All of that raised to the Delta End and Delta N is going to be one full of gas minus two moles of gas or negative one. So when we calculate our KP value, recognizing that this is now raised to the negative one, we get 0.156 Now that we know are KP value and we know our equilibrium concentrations while equilibrium pressures expressed in terms of the variable X we can then solve for X. Our expression will then be the KP value is going to be equal to the pressure of C, which is X divided by the pressures of A and B in the denominator multiplied by each other, which will be for a 1.6 minus X and for B 0.44 minus X, then to save some space. And some time I'll just show you the answer to X.

Once you perform the algebra, the algebra gives us an X value of 0.84 Now that we know the X value than solving for the the pressures of A B and C at equilibrium is pretty straightforward, See is just going to be equal to X, which will be 0.84 a will be 1.6 minus X or 1.52 and then see, I'm sorry. B will be point for four minus X or 40.356 atmospheres.

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