(a.) The equation for the IS curve is found by substituting expressions for aggregate demand into the equation for Y:

• Y = C + I + G = 100 + (2/3)(Y - 600) + 800 - (50/3)r + 500 = (2/3)Y + 1,000 - (50/3)r, so
• (1/3)Y = 1,000 - (50/3)r
• Y = 3,000 - 50r.

The IS curve plots Y against r in an inverse relationship. It shows all combinations of aggregate output (Y) and r for which the real sector is in equilibrium--that is, all combinations for which aggregate demand for output equals aggregate supply of output. The position of the IS curve is determined by the autonomous component (ie., 3,000). When r is 0, Y is 3,000. This can be confirmed from the full expression for Y:

• Y = 100 + (2/3)(3,000 - 600) + 800 + 500 = 100 + 1,600 + 800 + 500 = 3,000.

For different values of r, there will be different levels of equilibrium Y.

It is possible to express the IS curve with G and T (as well as r) identified separately as independent variables as follows:

• Y = 2,900 - (2/3)T + G - 50r.

This shows explicitly the effect of the two government policy variables on the combinations of Y and r that are compatible with equilibrium in the real sector of the economy.

(b.) The LM curve shows combinations of Y and r for which the monetary sector is in equilibrium--that is, for which the real money supply equals real money demand. We are told that:

• Ms/P = Md/P.

Because of this equality, we can simply use M/P and write:

• M/P = 0.5Y - 50r
• 50r = 0.5Y - M/P
• r = 0.5Y/50 - M/(50P) = 0.01Y - M/(50P),

which is the LM equation.

The LM equation shows that the rate of interest is positively related to Y and negatively related to the real quantity of money. The positive relationship with Y occurs because of the transactions demand for money--that is, as Y increases, consumers and firms need larger quantities to facilitate transactions. The negative relationship with r occurs because as interest rates increase, the opportunity cost of holding non-interest-bearing cash increases so that consumers and firms economize on cash balances and switch part of their holdings to such interest-bearing instruments as savings deposits and bonds.

(c.) Using the LM equation, we can derive an expression for r in terms of Y:

• r = 0.01Y - M/(50P) = 0.01Y - 1,200/((50)(1.0)) = 0.01Y - 24.

Using this equation to substitute for r in the IS equation, we can solve for equilibrium Y:

• Y = 3,000 - 50r = 3,000 - (50)(0.01Y - 24) = 3,000 - 0.5Y + 1,200
• 1.5Y = 4,200
• Y = 2,800.

So, the equilibrium level of Y is 2,800.

Plugging this equilibrium level of Y into the LM curve and using the values given for P (1.0) and M (1,200), we can solve for equilibrium r:

• r = 0.01Y - M/(50P) = (0.01)(2,800) - 1,200/((50)(1.0)) = 28 - 24 = 4.

So, the equilibrium rate of interest is 4%.

We can check on the validity of the values for equilibrium Y (2,800) and r (4%) by applying them in the equation for aggregate output:

• Y = C + I + G = 100 + (2/3)(2,800 - 600) + 800 - (50/3)(4) + 500 = 2,800.

So the values are confirmed.

If P is 2.0 instead of 1.0, then solving the LM equation for r gives:

• r = 0.01Y - M/(50P) = 0.01Y - 1,200/((50)(2.0)) = 0.01Y - 12.

Using this equation to substitute for r in the IS equation to solve for equilibrium Y:

• Y = 3,000 - 50r = 3,000 - (50)(0.01Y - 12) = 3,000 - 0.5Y + 600
• 1.5Y = 3,600
• Y = 2,400.

So, the new equilibrium level of Y is 2,400.

Plugging this into the LM curve and using the original value for M (1,200) and the new value for P (2.0), we can again solve for equilibrium r:

• r = 0.01Y - M/(50P) = (0.01)(2,400) - 1,200/((50)(2.0)) = 24 - 12 = 12.

So, the new equilibrium rate of interest is 12%. Doubling the price index halved the real value of the money supply and drove interest rates up dramatically.

We can again check on the validity of the values for equilibrium Y (2,400) and r (12%) by again applying them in the equation for aggregate output:

• Y = C + I + G = 100 + (2/3)(2,400 - 600) + 800 - (50/3)(12) + 500 = 2,400.

So the values are confirmed again in this case.

(d.) Assuming P still equals 2.0, to write the numerical aggregate demand equation for the economy, we solve both the IS and the Lm curves for r and then equate them and solve for Y as a function of M/P, G and T. From the IS curve:

• Y = 3,000 - 50r
• 50r = 3,000 - Y
• r = 60 - Y/50.

From the LM curve, we have:

• r = 0.01Y - M/(50P).

Equating the two equations and solving for Y:

• 60 - Y/50 = 0.01Y - M/(50P)
• 0.02 Y + 0.01Y = M/(50P) + 60
• Y = 2,000 + (2/3)(M/P).

Breaking the government policy instruments G and T out of the constant term gives:

• Y = 1,900 - (2/3)T + G + (2/3)(M/P)

which is the aggregate demand curve.

That it is correct can be confirmed by substituting values in for the variables:

• Y = 1,900 - (2/3)600 + 500 +(2/3)(1,200/2.0) = 2,400.