Problem Set 4.doc

  • December 4, 2020

Introduction to Macroeconomics                                                                        New York University

Marc Lieberman                                                                                                Fall, 2014

Problem Set #4

Note: To learn more about solving these types of problems, attend one of the recitations Friday Oct 17 or Monday Oct 20.

The following five equations provide a numerical version of the classical / long-run model in a fictional economy:

labor market:                            LS = 2 + 0.7 (W/P)

LD = 62 – 0.8 (W/P)

production function:     Y = 10 √(L x K x R)  

[Note: On some computers, the square root sign might not show up.   This should say that output is ten times the square root of (L x K x R). If you’re curious, this equation for the production function creates a curve that gets continually flatter, exhibiting “diminishing returns to labor” as we discussed in lecture.]

            Loanable funds market:

Household Supply of funds (Saving): S = 200 + 500r

                        Business demand for funds (Planned investment): IP = 900 – 3000r

                        Government demand for funds: Depends on budget deficit (see below)


  • LS  and LD  are labor supply and demand (equations giving us the number of millions of workers supplied and demanded at each real wage rate.
    • W/P is the real wage rate ($ per hour in purchasing power)
    • L is employment (number of millions of workers) 

[So, for example, if L = 10, that would represent 10 million workers, and you use L = 10 whenever the model requires a value for L]

  • Y is real output and income ($ billions per year)

[So, for example, if Y = 10, that would represent $10 billion dollars per year, and you would use Y = 10 whenever the model requires a value for Y; and the same idea for other variables measured in billions of dollars]

  • K is the real capital stock ($ trillions)

[So, for example, if K = 10, that would represent $10 trillion dollars worth of capital (e.g. buildings, machines, software, etc.),  and you use K = 10 whenever the model requires a value for K]

  • R is the quantity of land  (billions of acres)

[So, for example, if R = 10, that would represent 10 billion acres, and you use R = 10 whenever the model requires a value for R]

  • S is real saving ($ billions per year)
    • T is real net tax revenue ($ billions per year)
    • IP is real planned investment ($ billions per year)
    • r is the real interest rate (in decimal form)

[So, for example, if r = .10, that would represent an interest rate of 10%, and you use r = 0.10 whenever the model requires a value for r]

  • G = government purchases ($ billions per year)
    • T = net taxes ($ billions per year)

Initially in this economy:

  • Capital stock is $10 trillion (so “K” = 10)
  • Land is 33⅓ billion acres (so “R” = 33⅓)
  • G = $200 billion per year (so “G” = 200)
  • T = $200 billion per year (so “T” = 200)


1. In this problem (as in class), we keep the model simply by assuming we are dealing with a closed economy – one with no international trade (no exports or imports). 

2. In this problem, we assume that saving depends only on the interest rate, and is not also dependent on disposable income.  (Look at the equation for saving to see this; you can see that disposable income is not part of the equation.)  This keeps the model simpler, but won’t change any of our important conclusions.

3. In the steps to get your answers, carry out decimals to at least 2 or 3 places.

Part I. Finding the initial equilibrium:

  1. Solve for the equilibrium real wage (W/P) and employment (L).

[Hint: use LS = LD.   More specifically, set the expression for LS (which contains the real wage rate) equal to the expression for LD (which also contains the wage rate) and solve for the real wage rate, W/P.  Next, substitute the numerical value for W/P into either the labor supply or labor demand equation to get equilibrium employment (L)  (You should get the same answer for employment with either equation.]

  • Find equilibrium real GDP (Y)

[Hint: substitute this value for equilibrium employment, as well as the values for K and R that are given in the problem, into the production function to get equilibrium output (and income) Y.]

  • Find the equilibrium interest rate (r), household saving (S) and planned investment spending (IP)

[Hint: Set the supply of loanable funds equation equal to the demand for loanable funds equation, and solve for the equilibrium interest rate, r.  (Remember: with the initial values of G and T given in the problem, there is no budget deficit, so the government isn’t borrowing.)  Finally, substitute equilibrium r into the savings and planned investment equations to get equilibrium S and IP.]

  • What is the equilibrium value of consumption spending in this economy? 

[Hint: Consumption is total income for households (Y) that is neither taxed (T) nor saved (S), so C= Y – S – T.]

  • Check if Say’s Law holds for the equilibrium output level you found in 2. above. (i.e., check whether total spending in this economy (C + IP + G) is equal to total output (Y), except for rounding differences)
  • Illustrate the classical macro equilibrium with three roughly drawn graphs: the labor market, the production function, and the loanable funds market. (Don’t worry about drawing the graphs to scale. But do identify on your graphs the equilibrium values of w/p, L, Y, r, and S and IP , by just showing where on the vertical and horizontal axes they would be found.)

Part II. An Increase in Government Purchases

Suppose that, starting from the initial situation in Part I, the government increases its purchases by $50 billion, with no change in taxes.

  1. Find the equilibrium values of the same variables you found questions 1 through 4 in Part I (w/p, L, Y, r, S, IP, and C). 

[Hint #1: some of these variables will be unaffected by the rise in G.  Hint #2: when solving for r, remember that now, the demand for loanable funds will include not just investment spending, but also the budget deficit.]

  • Check whether Say’s Law still holds after the increase in government purchases.
  • Illustrate the effects of government purchases on the classical equilibrium. Do this by adding to the graph you drew in Part I, showing any shifts in curves and any changes in equilibrium values in your graph, and indicate any new numbers for the equilibrium values.
  • Calculate the changes in C, I and G that occurred as a result of the increase in G from $200 billion to $250 billion.
  • Has there been “complete crowding out” in this example?  Explain.

Part III. A Tax Cut

Suppose that, starting from the initial situation in Part I, the government cuts net taxes by $50 billion, with no change in government purchases.  Assume the tax cut has no supply side effects (it does not affect the supply or demand for labor and has no direct impact on planned investment or the growth in the capital stock).

  1. Find the equilibrium values of the same variables you found questions 1 through 4 in Part I (w/p, L, Y, r, S, IP, and C). 

[Note: In the numerical version of the classical model in this problem set, saving depends only on the interest rate, and not on disposable income. So the tax cut—even though it would increase disposable income—has no direct impact on saving in this example.  This tells us that initially, households must want to spend the entire increase in their disposable income due to the tax cut (that is, untilthe interest rate rises and makes them want to save a bit more too, which means increase their spending by a bit less than initially). So the tax cut – while it will have no direct impact on saving, will indirectly affect saving through its effect on the interest rate.]

  • Check whether Say’s Law still holds after the tax cut purchases increase (i.e., check whether total spending is equal to total output)
  • Has there been “complete crowding out” in this example?  Explain.
  • How would the tax cut affect your graph in Part II?  Would it be the same?  Different?  Explain briefly. Illustrate
  • Consider the following statement:  “As economists well know, in the long run, permanently greater government spending or permanently lower taxes will increase total spendingin the economy, which in turn will increase output and employment over the long run.” Is this statement true?  False?  Explain.

Hint: compare the answers in parts I, II and III