Dr m lives for two periods his utility is given by uc1c2c1c2...

Dr. M lives for two periods. His utility is given by U(c1,c2) = c1c2, where c1 is consumption in period 1 and c2 is consumption in period 2. In the first period, he earns $200 and in the second period he earns $920. The interest rate is 15%. The price of c1 and c2 is $2. His utility maximizing consumption bundle is: a) c1 = 247.5, c2 = 284.6 b) c1 = 284.6, c2 = 247.5 c) c1 = 271.12, c2 = 311.79 d) c1 = 500, c2 = 575 e) c1 = 250, c2 = 287.5

Transcript

00:01 So here we're going to maximize consumption over two periods.

00:03 We're told that the utility function is c1 times c2.

00:07 We're told that the first period income is equal to y1 is 200, y2 is equal to 920, and the interest rate is equal to 15%.

00:20 So what we want to do is maximize c1 times c2 subject to the present value of budget constraint, right? so we have to think about in the first period when we're buying c1, we have to set to c1, that is the expenditure, right? because we're told that the price of one equals the price of two is equal to two.

00:45 We have a set, so two times the number of c1 is the expenditure on c1.

00:53 Then we're going to have savings, right? and this has to have to be equal to y1, right? so my spending on consumption plus my saving or borrowing if savings is negative has to equal my income.

01:06 In the second period, right, what happens then? well, i have to spend money on consumption and period two.

01:17 I have to repay my debt or i get my savings with interest, right? again, depending on whether savings is negative or positive.

01:28 Oh, sorry, this is wrong, right? we're not saving again.

01:32 We're now receiving our savings, right? so this is equal to y2 plus 1 plus rs.

01:40 Right.

01:40 So if we saved money in the first period, the savings is positive, reflecting the fact that we didn't spend all our money on consumption.

01:48 And then we'll get money back to finance further consumption of c2.

01:51 If savings is negative, it means that the expenditure on c1 was greater than y1.

01:55 We've got to deduct that off plus interest charges from how much we can do.

02:00 On c2.

02:01 Now, if you want to, and this is highly recommended, right, you can substitute one of these things into the other, right? this gives me the combined budget constraint of 2c2 equals to y2 plus 1 plus r outside of s is equal to y1 minus 2c1, right? if you want to group terms, you need to divide everything by 1 plus r and then once i do that i'll bring the 2c1 to the other side so i get 2c1 plus 2c2 over 1 plus r is equal to y1 plus y2 over 1 plus r sometimes this is called the um the budget the lifetime income budget constraint now i want to differentiate uh you with respect to c1 right get my first uh get my first order of condition.

02:58 So the key thing is here is note that c2 is a function of c1, right? however much i spend on c1 affects how much i can spend on c2.

03:10 So i can't treat c2.

03:11 I'm not doing a partial derivative here...

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