Economics: Tutorial Solutions

TUTORIAL 10 (27 – 31 OCTOBER)
SOLUTION GUIDE
Question 1
Henry, Carla and Fred are housepainters. Henry and Carla can paint 100 square metres per hour using a standard paintbrush, while Fred can paint 80 square metres per hour. Any of the three can paint 200 square metres per hour using a roller.
(i) If Henry, Carla and Fred have only paintbrushes at their disposal, what is the average productivity
(in terms of square metres per painter hour) for the three painters taken as a team? Assume the three painters always work the same number of hours.
As a team the three painters can paint 280 square metres in 3 painter hours (note this equates to one hour of time, but there are 3 painters). So average productivity is just total output divided by the total labour input; 280/3 = 93.3 square metres per painter-hour.
(ii) Repeat part (i) for the cases in which the team has one, two, three and four rollers available. Are there diminishing returns to capital?
The first roller should be given to Fred, as it increases his production by the largest amount. The second and third rollers should be given to Harrison and Carla (in either order). There is no one to use the fourth roller. With this information we can calculate total output for the team per hour and total output per painter-hour as follows:

No. of rollers

Output by team

0
1
2
3
4

280
400
500
600
600

Output per painter-hour 280/3 = 93.3
400/3 = 133.3
500/3 = 167.7
600/3 = 200.0
600/3 = 200.0

The first roller increases the team’s output by 120, the second increases it by 100, the third by 100, and the fourth by zero. Since the extra output produced by an extra roller declines with more rollers (except for the third roller, which has the same addition to output as the second), we observe diminishing returns to capital.

1

(iii) An improvement in paint quality increases the area that can be covered per hour (either by brushes or rollers) by 20 per cent. How does this technological improvement affect your answers to part (ii)? Are there diminishing returns to capital? Does the technological improvement increase or reduce the economic value of an additional roller?
With the technological improvement, the table in part (ii) becomes:
No. of rollers Output by team Output per painter-hour 0
336
336/3 = 112
1
480
480/3 = 160
2
600
600/3 = 200
3
720
720/3 = 240
4
720
720/3 = 240
After the technological improvement, the first roller increases the team’s output by 144, the second by 120, the third by 120, and the fourth by zero. So we still observe diminishing returns to capital. However the economic value of additional rollers has increased, since each roller adds more output than before.
Question 2
(i) Explain what is meant by an aggregate production function?
A production function is a mathematical relationship summarising the process by which primary and secondary factors of production are combined to produce output.
(
)
Primary factors of production are labour (l) and capital (k) —physical, tangible items directly used by firms to produce output. Secondary factors of production refer to anything else that affects firms’ ability to produce output, such as entrepreneurial expertise and the skills of the workforce.
These are often described as the level of technology and are represented by the term A.
(ii) Write out the form of the Cobb-Douglas aggregate production function.
The Cobb–Douglas production function is a specific type of production function where the relationship between the output and the primary and secondary factors of production is given by:

(iii) What predictions does it make about (a) the marginal product of capital (b) the marginal product of labour and (c) returns to scale?
(a)The marginal product of capital is the first partial derivative of the Cobb Douglas production function, with respect to the capital input. The Cobb–Douglas production function implies the marginal products of capital is diminishing in the capital input:

2

(b)The marginal product of labour is the first partial derivative of the Cobb Douglas production function, with respect to the labour input. The Cobb–Douglas production function predicts that the marginal product of labour is diminishing in the labour input:
(
)

(c)The Cobb–Douglas production function predicts that production will be characterised by constant returns to scale (this last prediction relies on the exponents on capital and labour summing to one). A doubling of primary inputs leads to a doubling of output.
(

) (

)

Question 3
Suppose an economy’s production function is given by
. If the annual growth rate of output is 3.5 percent and labour and capital are growing at 2 percent annually, what contribution to output growth is made by total factor productivity?
We can use the growth accounting framework to write (where I am just ignoring the time subscripts form lectures/textbook)

Since we need to calculate TFP we can rearrange as

and substitute in the numbers
( )

TFP must be growing at 1.5 percent per annum.

3

( )

Question 4
In 2010 the New Zealand city of Christchurch was severely damaged by an earthquake. Draw a production function and show the capital stock of Christchurch before and after the earthquake.
What does the model imply about the marginal product of capital following the natural disaster?
We can illustrate the production function for Christchurch as follows.

Output (y) y=Af(k,l) y(before) y(after) k(after)

k(before)
Capital (k)

The earthquake destroyed a large proportion of Christchurch’s physical capital stock, but is assumed not to have had a negligible impact on technology or the labour input. Thus we move along a given production function.
The model implies that the marginal product of the capital stock in Christchurch that was not destroyed would have increased. (Note that the slope of a line tangent to the production function at k(after) is steeper than one at k(before)).
Question 5
(i) What is technology and why does its growth contribute to improved living standards?
Technology is defined as anything that raises the amount of real GDP that can be produced with a given amount of labour and capital. It includes the stock of knowledge needed to produce goods and services and a country’s ability to develop and apply new methods of production that improve productivity. As new technologies are developed, they increase productivity in industries other than the one in which they are produced. For example, the advent of motorised transport has enabled primary industries producers to transport their produce farther afield than their local communities. This has improved our standard of living since if you live in a temperate climate you can now consume products from tropical climates such as bananas and pineapples.
Technological change is the change in the existing technology between two dates. Technological progress is an improvement, or increase in technology which does usually increase, but it is possible for there to be a negative shock to technology (oil crisis). Technological progress includes things that increase the diversity and quality of goods and services. Technology can be embodied in a piece of capital (computers), or disembodied, where productivity is enhanced by the methods of production.

4

(ii) Why do economists claim that technology played a large role in the economic growth of the past
200 years?
Technology is able to pivot up the production function, which increases real GDP for a given level of capital and labour, and raises the living standards from year to year. With persistent growth in technology, per capita living standards can keep rising. This is what economists observed during the past 200 years.
(iii) Why does technology include different ways in which to organise a firm?
Technology also includes how firms are organised. Better organisation can mean a smaller bureaucracy and more output per hour of work without the addition of capital. Better flows of information can raise productivity. Better incentive programs that encourage workers to communicate their ideas to management, for example, also increase productivity.
(iv) What is wrong with a growth policy that focuses on capital formation rather than technology?
The biggest problem with a growth policy that focuses on capital formation (rather than on growth of technology) is that it can eventually reach the state of diminishing returns to capital, thus slowing down economic growth.
(v) How do property rights affect economic growth?
Intellectual property laws cover copyrights, trademarks, and patents. These laws help exclude those not involved in the development of a technology from using it without adequate compensation. If intellectual property rights are not protected, then the development of technology may be stunted or may fail to occur. Property rights in general offer an incentive for individuals to produce and prosper, as Adam Smith stated:
“Commerce and manufactures can seldom flourish long in any state which does not enjoy a regular administration of justice, in which the people do not feel themselves secure in the possession of their property…”
Adam Smith, The Wealth of Nations, book 5, chapter 3, paragraph 7.

5