Optimal Aggregate Consumption and Saving
Student Instruction Sheet

Introduction
In this exercise, we will seek the saving rate that maximizes long run consumption in Solow’s long run growth model.  In this simple model, output (Y) is either saved and turned into investment, or consumed.  If the saving rate is s, then consumption (C) is:
						C = Y – sY					(1)
Denote the saving rate by s, the growth rate of the labor supply by n, the growth rate of labor productivity by g, the capital depreciation rate by d, and the effective labor supply by LE.  The effective labor supply (LE) is the labor supply adjusted for productivity.  Thus, the effective labor supply grows at rate (n + g).  Solow shows that in the long run, capital (K) and output (Y) will be defined by the above parameters in the following way:
						K/Y = s/(n+g+d)				(2)
						DK/K = DY/Y = (n + g)				(3)
Equations (2) and (3) define the “balanced growth path.”  We define production according to the Cobb-Douglas production function:
						Y = AKaLE1-a					(4)
Let’s rewrite equation (2) to solve for K:
						K = sY/(n+g+d)					(5)
This new equation (5), along with equation (4) forms a system of two equations with two unknowns (K, Y); the rest of the terms are parameters.  We should be able to solve (5) and (4) for K and Y as a function of parameters, but because Y is a nonlinear equation, we cannot solve it by hand.  We will use Excel to solve some numerical examples of the model.

It should be clear that the level of output depends on s, the saving rate.  Specifically, as s rises, the level of capital rises, and output rises.  However, as s rises, consumption may rise or fall overall; a rising s leads to more output (part of which may be consumed), but a lower proportion of output is consumed.  It should also be clear that neither s = 0 or s = 1 is optimal.  When s = 0, all output is consumed, and capital depreciates until nothing is produced, and consumption goes to zero.  When s = 1, no consumption goods are produced, and consumption is zero. This exercise will search for the optimal tradeoff between consumption and investment, called the “golden rule” saving rate in the context of numerical examples of the Solow growth model.

Instructions

1.	The next several steps will guide you to set up a spreadsheet to contain the information above.  In the first column, write down the parameter and variable names.  To keep the sheet easy to read, put the title “Parameters” in cell A1; in cells A2 through A8 list the parameters s, n, g, d, A, a, and LE.  In the next row (9) put the title “Variables” and list the three variables; K, Y, and C.
2.	Assume the following values for the parameters: s = 0.000, n = 0.015, g = 0.010, d = 0.050, A = 1.000, a = 0.300, and LE = 10.000.  Type these values into column B.  You can change the number of decimal places shown in any cell by clicking on the cell, then on the “Increase decimal” button, which is next to the comma button near the top of the screen.
3.	In cell B10, write a formula that defines capital (K) according to equation (5).  When writing the formula, refer to the appropriate cell addresses rather than using numbers.  Make sure to use an asterisk (*) for multiplication, and make sure to use appropriate parentheses.  Your formula should be:  =B2*B11/(B3+B4+B5)
4.	In cell B11, write a formula that defines output according to equation (4).  Make sure to use a carrot (^) for exponents, and parentheses to keep exponent terms together.  You will likely get an error that says that Microsoft Excel cannot calculate a formula because of circular references.  Click cancel.  Click on the Tools menu, then options, then the Calculation tab.  Click (check) the box next to Iteration.  Click OK.  You just allowed Excel to solve the equations simultaneously.  Excel uses a numerical method to solve the equations:  essentially, it substitutes in different values for K and Y until both equations are satisfied.  The options for the Iteration tool tell Excel how many numbers it should try (Maximum iterations), and how close to the correct answer it should be (Maximum change).  If your computer is slow, you may wish to limit the number of iterations to say, 25, and press F9 (recalculate spreadsheet) as is necessary.  If you have set up the iteration correctly, you should get K = 0 and Y = 0.
5.	In cell B12, write a formula for consumption that corresponds to equation (1).  You have now determined the long run levels of Y, K, and C for a saving rate of zero.  Hopefully, this is not the best saving rate!
6.	In columns C through L, we will compute the economy for saving rates up to 1.0, in increments of 0.1.  Copy column B to column C.  Replace the parameter values for n though LE with equations that define the value equal to the previous column’s value.  For example, for cell C3 (the value for n), the formula should be:  =B3
7.	For s, we want to increase its value by 0.1 in each column.  Write a formula in cell C2 that takes the previous column’s value and increases it by 0.1.  Your formula should read = B2+0.1  The values for K, Y and C have likely not changed.  This is because K=0, Y=0 is also a solution to the two equations.  Since we are looking at economies with some capital, we must force Excel from 0 to find the relevant solution.  In cell C10 (the formula for K), replace the formula with 1.  Then copy cell B10 to cell C10 to restore the formula.  This procedure forces Excel to look for a solution to the two equations with a non-zero capital.  Your answers should change to 15.083, 11.312, and 10.181, respectively if you did this correctly.
8.	Copy column C to columns D though L.  The parameter s should increase by 0.1 in each column, and the value for s in column L should be 1.000.  Where is consumption maximized?  It should appear to be around a value of s equal to 0.3.
9.	To show this maximum graphically, we will plot consumption against the saving rate.  First, we must select the data to plot.  Highlight (select) cells B2 through L2 (the s data) by clicking on cell B2 and moving the mouse through to L2 while holding down the mouse button.  Release the mouse button, then press and hold down the Ctrl button on the keyboard.  Now highlight cells B12 through H12 (the C data) while holding down Ctrl.  When done, you should see B2 through L2 and B12 through L12 highlighted.  Click on the Insert menu, then “Chart…”  This starts the Chart Wizard that will lead you through the steps to graph the data.  On the left of the first screen, select the “XY Scatter” option.  On the right side, select the bottom left option.  This option plots points and lines, but doesn’t turn the lines connecting the points into smooth curves.  Click Next, then Next again.  Type in titles, and change any other options you desire, then click Next.  Click the “As new sheet” option, and type in a name of the graph sheet if you like.  Click finish.  You should see a plot of the data, with s on the X‑axis, and C on the Y‑axis.  Where is consumption maximized?  Again, it should be around s = 0.3.
10.	We will now use the Excel Solver tool to find the exact value of s that maximizes consumption.  Go back to the original data spreadsheet.  Copy column B to column N.  Change s to 0.5 in column N, to start the Solver somewhere in the middle of the range of s values (this is called “seeding”).  If K, Y, and C do not change from 0 (as in step 7), change K to 1, then copy cell L10 to N10 as in step 7 above.  This should give you the same answer as in column G.  In this new column (N), click on cell N12 which contains the value for consumption.  Click on the Tools menu, and click on “Solver…”  A window will appear with several options.  Under “Set Target Cell”  should be written $N$12 (which is what you want).  Because you wish to maximize, make sure that “Max” is selected next to the “Equal To” option.  “Under By Changing Cells” you want the cell that contains the s data, cell N2.  Click Solve.  After a short time, a window should appear telling you that Solver found an optimal solution (if not, you did something wrong, likely in typing in the formulas).  How does this answer compare to your approximation earlier?  It should be the same.
11.	In the original block of the model (columns B through L), experiment with different parameter values to find what affects the golden rule saving rate.  You should find that the Cobb-Douglas parameter a affects the golden rule rate, because the golden rule saving rate is that saving rate that sets the marginal product of capital equal to the depreciation rate, and a affects the marginal product of capital.  Changing LE will not affect the marginal product of capital because capital automatically increases with labor in a balanced growth model.