PAI 723 Economics for Public Decisions > Exercise 8

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).
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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 04/05/2019