# Solution

Here are notes on the solution. This is not a complete answer to the exercise but it will let you check your work.

### Part A

1. The present value of her income is:

PVI = I_0 + I_1/(1+r) = $200k + ($50k)/1.2 = $241,667 Her demand equations can be found from the general Cobb-Douglas equations by substituting C_0 for Q_x, C_1 for Q_y, PVI for M,$1 for P_x and ($1)/1.2 for P_y: C_0 = (0.75*PVI)/($1) = (0.75*$241667)/($1) = 181,250

C_1 = (0.25*PVI)/(($1)/1.2) = (0.75*$241667)/(($1)/1.2) = 72,500 Checking: C_0 + C_1/1.2 = 181250 + 72500/1.2 = 241,667 = PVI Since C_o is less than I_0, she is saving in period 1 with S =$18,750.
2.  The graph looks like this:

### Part B

1. The present value of his income is:

PVI = I_0 + I_1/(1+r) = $50k + ($400k)/1.1 = $413,636 His consumption in each period can be found by using his preferred consumption ratio and the intertemporal budget constraint: C_1/C_0 = 4/1 or C_1 = 4*C_0 (preferences) C_o + C_1/1.1 = PVI (budget constraint) C_0 + (4*C_0)/1.1 = PVI C_0*(1+4/1.1) = PVI C_0 = (PVI)/(1+4/1.1) C_0 = ($413636)/4.6363 = $89,216 C_1 = 4*C_0 =$356,863

Check: C_0 + C_1/1.1 = $89216 + ($356863)/1.1 = $413,636 = PVI Since C_0 is greater than I_0, he borrows in period 0 with B =$39,216.
2.  The graph looks like this:

### Part B

1. The present value of her income without training, PVI_n, is the following:

PVI_n = I_0 + I_1/(1+r) = $70000 + ($110000)/(1.1) = $170,000 With training, her net income in each period will be: I_o^text(net) =$70,000 - $20,000 =$50,000

I_1^text(net) = $110,000 +$33,000 = $143,000 Her PVI becomes PVI_t: PVI_t = I_0^text(net) + I_1^text(net)/(1+r) =$50000 + ($143000)/(1.1) =$180,000

Since PVI_t > PVI_n, she's better off taking the training. Her consumption in the two periods will be:

C_0 = (0.5*$180000)/($1) = 90,000

C_1 = (0.5*$180000)/(($1)/1.1) = 99,000

Her consumption in period 0, $90,000, is larger than her net income in that period,$50,000, so she's borrowing the difference: $40,000. 2. The graph is shown below: To describe what's going on in words: in period 0 she pays$20,000 in tuition (moving along the C_0 axis from $70,000 down to$50,000) and then borrows $40,000 (moving along the C_0 axis from$50,000 to up to $90,000). In period 1 she gets a$33,000 raise (along the C_1 axis from $110,000 to$143,000) and then repays $44,000 (from$143,000 down to \$99,000).
URL: http://wilcoxen.maxwell.insightworks.com/pages/4703.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 04/05/2019