Instructors: Lawrence McDonough; Ryan Davies

1. A more extensive aggregate production function makes sense when studying a number of problems. The Cobb-Douglas production function in the text is expanded to include 4 factors of production (capital, labour, energy and materials). It can be written as:
Y = AK^@1L^@2E^@3M^@4 The data for the Rubble Economy are given below. Fill in the missing cells, denoted "????" (a spreadsheet might be fastest).

Data:

-	y	A	K	L	E	M
Alphas	-	-	0.12	0.4	0.1	0.38
1950	668.384	????	166	5.9	30.3	492
1960	1067.321	????	260	7.9	42.4	621
1970	1570.826	????	368	10.7	58.7	789.3
1980	2003.762	????	479	12.6	78.2	901.3

 

Growth Rates:

1950-60	0.596868	????	????	????	????	????
1960-70	????	????	????	????	????	????
1970-80	????	????	????	????	????	????

 

Marginal Products:

1950	-	-	0.483169	????	????	????
1960	-	-	????	????	????	????
1970	-	-	????	????	????	????
1980	-	-	????	????	????	0.844813

 
2. What is the "marginal revenue product" of an input? Why do economists equate this to the price of the input? What happens to the MRP of labour if we increase the quantity of labour used? Relate this to the demand for labour curve. What happens to the labo ur demand curve if the price of the output increases? What happens to the labour demand curve if there is an increase in any other input?

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3. Identify the three components of unemployment rate and their relationship to the natural rate of unemployment. Suppose that the rate of growth of potential output in Canada is about 2.5% and that the expected 1998 rate is 3.2%. Look up Canada's most recent monthly unemployment rate, and use Okun's Law (Appendix 3.A in the text) to calculate the change expected in the unemployment rate.

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4. Answer the following questions before and after looking up annual employment data for 1993-1997 (Statistics Canada on the Web).

a. The labour force has been increasing. (Y/N) ________

b. Full-time jobs have been increasing. (Y/N) ________

c. Part-time jobs have been increasing. (Y/N) ________

d. The ratio of part-time to full-time jobs has increased. (Y/N) ________

e. There are approximately ________ part-time jobs for every 100 full-time jobs.

f. The participation rate is increasing. (Y/N) ________

g. The employment to population ratio is increasing. (Y/N) _________

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5. You have $100,000 to invest. Assume that you are in the 26% federal income tax bracket and that provincial taxes are 45% of the federal amount. Inflation is 2%. Assume the following:

 
• Interest income is taxed as regular income. (click here for additional details)
 
• 75% of capital gains are taxable as regular income. (click here for additional details)
 
• Dividend income is taxed as follows:

(i) Multiply the amount of the dividend by 1.25. This is the "taxable amount of the dividend".
(ii) Calculate federal tax based on the taxable amount of the dividend.
(iii) Subtract from the federal tax a tax credit equal to 13.33% of the taxable amount of the dividend.
(iv) Calculate provincial taxes as usual. (click here for additional details)

Using the above information, calculate real after-tax returns on the following investments:

(1) $100,000 Guaranteed Income Certificate (GIC), paying 5% interest.

(2) $100,000 invested in a blue-chip stock paying 4.5% dividend yield (assume no opportunity for capital gains)

(3) $100,000 invested in a growth stock paying no dividends but expected to be worth $104,500 next year.

Where should you put your money?

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6. Large institutions in the public and private sectors purchase fleets of cars which are treated as capital equipment. New technologies have increased the lifespan of corporate autos. How should this affect the user cost of this type of capital? How wo uld it affect the demand for autos? Discuss your answer.

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7. The economy has been operating at less than the trend rate of growth but forecasters are now promising a better than average few years' growth in income. Explain how this might affect desired savings and the Savings-Investment equilibrium. Might investment be affected by such announcements? Explain.

END
 
 

Answers:

1. Answer Table:

   Data:

   -	y	A	K	L	E	M
   Alphas	-	-	0.12	0.4	0.1	0.38
   1950	668.384	12	166	5.9	30.3	492
   1960	1067.321	14.3	260	7.9	42.4	621
   1970	1570.826	15.8	368	10.7	58.7	789.3
   1980	2003.762	16.9	479	12.6	78.2	901.3

    

   Growth Rates:

   1950-60	0.596868	0.191667	0.566265	0.338983	0.39934	0.262195
   1960-70	0.471747	0.104895	0.415385	0.35443	0.384434	0.271014
   1970-80	0.27561	0.06962	0.30163	0.17757	0.332198	0.141898

    

   Marginal Products:

   1950	-	-	0.483169	45.31417	2.205888	0.516232
   1960	-	-	0.49261	54.04157	2.517267	0.653111
   1970	-	-	0.512226	58.72248	2.676024	0.756257
   1980	-	-	0.501986	63.6115>	2.562356	0.844813

2. The MRP=Price of output times increase in output for a one unit increase of the input:
   

   MRP = P _output * MP_input
   The MRP is the value in dollars to the firm from employing one more unit of the input. Consider labour inputs at a wage of w. If w > MRP, the cost to the firm is larger than the benefit for that unit of labour. If w < MRP, the firm is better off to hire a unit of labour since the (marginal) benefits are greater than the (marginal ) costs. The firm will therefore choose to operate where w = MRP. This implies,

   w = P * MP_N

   MP_N = w = p = Real Wage

   One of the properties of the production function is diminishing marginal product: as N increases, MP_n falls. If the real wage falls (nominal wage falls, price of output rises) then MP_n > w=p and firms will hire more labour which decreases MP_n . Hiring stops when MP_n = w = p once again.

   An increase in any other input will increase the marginal products of all other inputs. This means that more can be produced with the same level of other inputs. The production function twists up and the slope of the production function increases, which in turn implies that the MP_nincreases. To see this, consider a Cobb-Douglas production function as in #1,

   Y1 = AK^a1 N^a2 E^a3 M^a4

   Y2 = AK^a1(N + dN)^a2 E^a3 M^a4

   MPN = Y2 - Y1 = AK^a1 E^a3 M^a4 [(N + dN)^a2 - N ^a2]

   For any given dN, the marginal product will increase if A, K, E, or M is increased.

   Total unemployment consists of frictional, structural and cyclical unemployment. Frictional plus structural unemployment is the natural rate. Frictional unemployment refers to that portion of the unemployed who are searching for a suitable job. Not all vacancies are desireable to all individuals looking for a job, nor are all applicants considered suitable by searching firms. Appropriate matching takes some time. This unemployment is expected for any economy as some sectors grow and other contract. Structural unemployment refers to those who experience long spells broken briefly by short periods of work. The business cycle tends to have little or no effect on this portion of unemployment. Cyclical unemployment is that portion of unemployment which is attributable to the business cycle and is measured as the difference between tha actual and natural rates of unemployment.

   From the text, we can write OKUN's Law as

   change in u = - 0.5*[(change in Y / Y) - (change in pot. Y / pot. Y)]

   change in u = - 0.5*[3.2 - 2.5] = -0.35

   u₉₈ = 8.4% - 0.35% = 8.05%

   Answer the following questions before and after looking up annual employment data for 1993-1997 (from www.statcan.ca).

   (a) The labour force has been increasing (Y,N)?______Y

   (b) Full time jobs have been increasing (Y,N)?________Y

   (c) Part time jobs have been increasing (Y,N)?________Y

   (d) The ratio of part time to full time jobs has increased (Y,N)?_______N

   From: www.statcan.ca/english/Pgdb/Economy/Economic/econ10.htm

   Year	1993	1994	1995	1996	1997
   Part	2480	2493	2508	2589	2649
   Full	10534	10798	10996	11087	11291
   Ratio	0.235428	0.230876	0.228083	0.233517	0.234612

   e. There are approximately _____23_____ part time jobs for every 100 full time jobs.

   f. The participation rate is increasing (Y,N)?_____N_____(65.5 to 64.8)

   g. The employment to population ratio is increasing (Y,N)?_____Y________(58.2 to 58.9)

   The relevant calculations are:

   (1) Guaranteed Income Certificate (GIC)

   Interest earned (0.05 x $100,000) = $5,000

   Federal tax (0.26 x $5,000) = $1,300

   Provincial tax (0.45 x $1,300) = $585

   Total tax ($1,300 + $585) = $1,885

   After tax nominal rate of return
   = [($5,000 - $1,885) / $100,000] x 100 = 3.115 %

   After tax real rate of return
   = 3.115% - 2.00% = 1.115%

   (2) Blue-chip stock

   Dividend income (0.045 x $100,000) = $4,500

   Taxable amount of dividend (1.25 x $4,500) = $5,625

   Gross federal tax (0.26 x $5,625) = $1,462.50

   Federal dividend tax credit (0.1333 x $5,625) = $749.81

   Total (net) federal tax ($1,462.50 - $749.81) = $712.69

   Provincial tax (0.45 x $712.69) = $320.71

   Total tax payable ($712.69 + $320.71) = $1033.40

   After tax nominal rate of return
   = [($4,500 - $1033.40)/$100,000] x 100 = 3.4666%

   After tax real rate of return
   = 3.4666% - 2.00% = 1.4666%

   (3) Growth stock

   Capital gain = $4,500

   Taxable amount of capital gain (0.75 x $4,500) = $3,375

   Federal tax (0.26 x $3,375) = $877.50

   Provincial tax (0.45 x $877.50) = $394.88

   Total tax ($877.50 + $394.88) = $1,272.38

   After tax nominal rate of return
   = [($4,500 - $1,272.38)/$100,000] x 100 = 3.2276%

   After tax real rate of return
   = 3.2276% - 2.00% = 1.2276%

   Therefore, Blue-chip stock has highest after tax real rate of return.

   If cars last longer, then the depreciation rate is smaller and the user cost (below) is decreased:
   

   uc = rp_K + dp_K

   The demand for capital is defined by the MPK curve which slopes down. The lower user cost is associated with a larger quantity demanded. This is odd. A firm that requires 50 vehicles will now use 55 to produce the same level of output? Perhaps they can reduce the size of the repair shop and increase the number of people on the road. Note that there may be measurement problem here: the quality problem seen earlier. If the quality change is reflected in the price then the the price of the cars would rise. The user cost may rise or fall. More information is required to give a definitive answer on quantity demanded.

   Expected income is increasing so more CURRENT desired consumption will take place, which implies less current savings. Using the I-S diagram, S shifts left, and the real rate of interest increases which decreases desired investment! What a drag (literally) -- increased consumption is crowding out investment. If the good news forecast means higher incomes for consumers then it must also mean increased demand for producers (demand shift), and hence higher prices for their goods. This implies a higher marginal revenue product for capital and increased demand. In short, higher expected incomes lead to both decreased desired current savings and expected increases in MPK, hence increased desired investment. Thus in the short run, one can expect increased real interest rates. (To be shown in a diagram.) In the long run, one can expect consumption,savings and investment to rise with actual increases in income.

   END