Econ 2123 L1 Problem Set 2

1.  

(a)

In the short run, a decline in business confidence so that investment demand falls for any interest rate would shift the aggregate demand curve to the left from AD to AD’.  It leads to a decrease in output to Y’ which is below the natural level of output Yn.  The price level falls from P to P’.  As the output is below the natural level of output Yn and the expected price level falls with the price level, the aggregate supply curve shifts down from AS to AS’ over time until output has risen back to the natural level of output.  The adjustment ends once wage setters no longer have a reason to change their expectations.  

In the medium run, output returns to the natural level of output and the price level further falls from P’ to P’’.  

(b)  As the unemployment rate moves in opposite direction with the level of output, the unemployment rate increases in the short run but gradually returns to its original level which is the natural rate in the medium run.  

(c)  The Fed should increase the money supply, which would shift the aggregate demand curve to the right.  A monetary expansion of the proper size exactly offsets the effect of the decline in business confidence on the aggregate demand curve.  Therefore, the net effect is that the aggregate demand curve does not move in the short run or medium run, and neither does the aggregate supply curve.  

(d)  Under the policy of a monetary expansion in (c), output and the price level are higher than that in (a) in the short run.  

(e)  In the short run, the unemployment rate is lower than part (b).  The unemployment rate however is the same as part (b) in the medium run.  

2.  

(a)  When firms are extremely pessimistic about sales that they do not want to borrow at any interest rate, interest rate would then have no effect on investment.  

(b)  As the interest rate does not affect equilibrium output at all, the IS curve is vertical and has an undefined slope.  

(c)  The LM curve is not affected and remains unchanged.  

(d)  As the price level does not affect equilibrium output, the AD curve is vertical and has an undefined slope.  

(e)  As there is a shock to the variable z so that the AS curve shifts up, given the vertical AD curve, the equilibrium output does not change while the price level goes up in the short run.  

(f)  As equilibrium output is larger than the natural level of output (Y>Yn) and the price level is also higher than the expected price level (P>Pe), the expected price level goes up and the AS curve shifts up.  However, since interest rate has no effect on investment, output will remain unchanged at its original natural level and cannot adjust to its new lower natural level after the shock to z.  The price level then keeps going up forever and so does the expected price level.  

3.  

(a)  When K = 49 and N = 81,

Y = √K √N

Y = √49 √81

Y = (7) (9)

Y = 63

Therefore, the output is 63.

(b)  As both capital and labor double,

Y = √K √N

Y’ = √2K √2N

Y’ = (√K √N) (√2 √2)

Y’ = 2 (√K √N)

Y’ = 2Y

Therefore, the output doubles as well.  

(c)  Yes, this production function is characterized by constant returns to scale.  According to the definition of constant returns to scale, if the quantities of capital and labor are doubled then the output will also double.  The result from (b) proves the application of CRS.  

(d)  A relation between out-put per worker and capital per worker:  

Y = √K √N

Y / N = √K / √N

(e)  Let K / N = 4,

Y / N = √K / √N

Y/ N = √4

Y/ N = 2

When K / N is doubled to 8,

Y/ N = √8

Y / N = 2√2

No, Y / N does not double as a result.  

(f)  No, the relation between output per worker and capital per worker does not exhibit constant returns to scale.  

(g)  In (c), we are looking at the effect on output when we increase both capital and labor in equal proportion.  

However, in (f), we are alternatively looking at the effect on output when we increase capital only, not both capital and labor in equal proportion.  There are decreasing returns to labor instead.  

(h)  

4.  

(a)  

Y = Kα N1-α

Y’ = (aK)α (aN)1-α

Y’ = a (Kα N1-α)

Y’ = aY

According to the definition of constant returns to scale, multiplying all inputs by a constant will lead to a proportional change in output by that constant as well.  As shown above, this production function is characterized by constant returns to scale.  

(b)  Decreasing returns to capital refers to the property that increases in capital lead to smaller and smaller increases in the output as the level of capital increases.  

Y = Kα N1-α