222B Assignment 2 - Answers

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a. Total factor productivity is A. A= Y/(K0.5N0.5) so in 1999, A=6.02 and in 2000, A=5.72
Growth Rate of A

=(5.72 - 6.02)*100/6.02

b. In order to produce output of 2200 in 2001 (given A is constant from 2000 to 2001), the amount of K that is required is determined by the following equation:


c. The output in 10 years time, assuming productivity remains unchanged, will be:


d. To show that doubling the inputs K and N will exactly double output, we multiply each variable by 2 and calculate:


We can see that the production function is homogeneous of degree one because this property is true for any number, i.e. X=2, 3, 100, etc.

Q2. MPK is the increase in output from one unit increase in capital. In other words, it is the slope of the line tangent to the production function at any given K. Note: MPK decreases as K increases.

MPN is the increase in output resulting from one unit increase in labour input. In other words, it is the slope of the line tangent to the production function at any given N. Note: MPN decreases as N increases

a. Supposing the production function exhibits diminishing returns to scale then two properties of the MPK function are: (1) MPK decreases as K increases, (2) MPK is positive for all K.

b. Supposing the production function exhibits constant returns to scale (in N), then the properties of the MPN function are: (1) MPN remains constant as N increases, (2) MPN is positive for all N

Q3. a. Firms wish to set MPN=w and labour supply must equal labour demand in equilibrium, so we can find N and w in the following manner:

Demand for labour is found where MPN=w
so 400-0.2N = w

N=400/0.2 - w/0.2
set supply=demand:
so N=2000-5(150/(1.5-t))
and w(1-t)=150(1-t)/(1.5-t)

{other valid answers include: w=1500/[5+10(1-t)] or w=300/[1+2(1-t)] }

b. when t increases we see that:
(1.5-t) decreases so 150/(1.5-t) increases, in other words, w increases.
Also, 2000-150/(1.5-t) decreases, so N decreases.
To see the effects on w(1-t) is more complicated. We determine the effect on after tax wages by taking the derivative of w(1-t) with respect to t and obtain:

This number is less than 0 for all t.
{Other valid answers include a simple explanation of how increases in t will affect w(1-t), the derivative method is not necessary}
So we see that an increase in t decreases after tax real wages.

c. If t=0.5 then we can calculate w and N as follows:


If a minimum wage is set at 200 employment will decrease and there will exist involuntary unemployment:

Nd= 2000-5w = 2000-1000 = 1000
Ns= 500+10(1-t)(w) = 500+1000 = 1500
So employment will be 1000 and 500 people will be involuntarily unemployed.

d. If a technological shock changes MPN to 400-0.5N we can see that at any level of N the marginal productivity of labour is lower after this shock. Thus, it is an adverse technological shock.

Now the firm will recalculate MPN=w to get a labour demand curve as follows:
Nd = 400/0.5 - w/0.5 = 800-2w

given the minimum wage of 200 this means that:
Nd = 800-400 = 400
but since Ns=1500 still, we have an increase in involuntary unemployment resulting from the adverse technological shock coupled with a minimum wage. Employment now is 400 and unemployment is 1100.

Q4 Desired capital stock is that amount of capital stock that allows a firm to earn the highest amount of expected profit. Because a lag often occurs in obtaining and installing new capital, firms compare the expected future marginal product of capital MPKf with the user cost of capital, where user cost of capital is the expected real cost of using a unit of capital for one period. Firms maximize profit by buying capital until MPKf = User Cost; that is, as long as the benefits (MPKf) exceed the costs (User Cost) of purchasing one more unit of capital, firms will purchase that unit, until such time as the benefits exactly equal the costs. Graphically:

a. A technological improvement will increase MPKf for any given K, but will not change the user cost of capital; therefore, a technological improvement will increase the amount of capital stock desired. Graphically:

b. An increase in N due to increased immigration will increase MPKf for any given K, but will not increase the user cost of capital; therefore, an increase in immigration will increase the amount of capital stock desired. (note: there was a typo in the assignment, N was ommitted by accident so full marks are awarded where this caused confusion) Graphically:

c. An increase in the real interest rate will increase the user cost of capital. UC=Pk(d+r) where d=depreciation and r=expected real interest rate. Thus, an increase in r means and increase in UC. Therefore, an increase in r will decrease the amount of capital stock desired.

Q5. Given current and future MPK=10000-2K+N, current and future MPN=50-2N+K,Pk=$5000, r=0.1, d=0.15, and w=$15,

a.The user cost of capital is: UC

=5000(0.1+0.15) = 5000(.25) = $1250
b. The firm's optimal level of employment and desired capital stock can be found by: UC=MPK and w=MPN,
(2) w=15=50-2N+K

from (2)K=2N-35 sub this into (1) to get:

so 3N=8820


so K = 2(2940)-35 = 5845

Q6. The equilibrium condition in the goods market is:

or Y-Cd-G=Id
which is Sd=Id


Note that: Id is downward sloping. This slope is because as r increases the user cost of capital increases so the desired amount of capital stock decreases and therefore Id decreases.
Sd is upward sloping. This is because as r increases the rewards to saving increases so people wish to save more (a counter effect is that as r increases you need to save less in order to earn the same amount in the future, but empirically it is shown that this second effect is dominated by the first effect).

a. Ricardian equivalence implies that the timing of temporary tax changes doesn't affect desired consumption or desired national savings. Thus, an increase in the income tax rate will not shift Sd, nor will it shift Id, and as such, there will be no change in the goods marked equilibrium from and increased income tax rate.

b. A temporary decrease in government spending will cause Sd to shift right. Sd=Y-Cd-G, so for any given r, a decrease in G implies that Sd is higher. The goods marked equilibrium will shift right and down, yielding a lower equilibrium r and a higher equilibrium Sd,Id.

c. An increase in tax on a firm's revenue will increase the user cost of capital:
so MPKf=Pk(r+d)/(1-τ)
which increases as τ increases. Graphically:

(note: if you see τ in any of the above equations it means that your browser does not support the current code. Wherever you see τ there should be a lower case greek letter tau in its place. Thus Id is lower for any given r (the Id curve shifts down) and the goods market equilibrium shifts down and to the left. So the new equilibrium r is lower and the new equilibrium Sd, Id are lower.

Q7. Given full employment output = 5000, G=0, Cd=3000-2000r+(0.1)Y, and Id=2000-4000r

a. The equilibrium values of r, Id, and Cd can be found in the following manner:

Sd= Y-Cd-G = 5000-Cd-O

In equilibrium Sd=Id, so we have
Sd= 5000-3000-2000r-(0.1)(5000) = 2000-4000r = Id
r= 500/6000 = 0.083 or 8.3%

Id= 2000-(4000)(0.083)= 1666.67

Cd= 3000-2000(0.083)+(0.1)(5000) = 3333.33

b. If government spending, G, increased from 0 to 1000 we can calculate the new values of r, Cd and Id as follows:

= 5000-3000+2000r-(0.1)(5000)-1000
= 500+2000r

At equilibrium Sd=Id, so:
500+2000r = -4000r +2000
6000r = 1500
so r = 0.25

Id = 2000-(4000)(.25) = 1000

Cd = 3000-(2000)(.25)+500 = 3000

Intuitatively, as government spending incrases disposable income decreases which causes savings to decrease, because savings decreases then at any given r the amount desired to be saved will be lower, so in order to raise the necessary capital for investment, for any level of investment, firms will have to pay a higher r. At higher r the user cost of capital is higher and firms will desire a lower amount of capital stock; hence investment will be lower. Graphically