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Unformatted text preview: . And this is clearly visible in the figure above. Therefore, firm A exhibits
diminishing marginal return. The analysis and results are identical for
.
Firm B:
Taking derivative of production function with respect to capital, yields
. Since this is a constant the firm B does
not exhibit diminishing marginal return. Similarly
, and thus identical to
.
(11.11) Ajax Cola, a soft drinks manufacturer, uses a technology that combines equal amounts of sweetener, labor ( ),
capital ( ), and materials ( ) as complementary inputs to produce soft drinks ( ). Currently, Ajax Cola uses sugar ( ) as
a sweetener.
Ajax purchases/hires/leases all inputs as a price taker. Denote the price of sugar by
the wage rate by
rental rate by , and the price of materials by . In this question, assume all inputs are variable. , the capital (a) Write down Ajax Cola’s production function and characterize its returns to scale. Show all calculations below.
Answer
Since inputs must be combined in fixed ratios, the production function is:
( ) To check for returns to scale, suppose we scale up all inputs by 2. The new output will be:
( )
( ) Note that: 18
ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Doubling all inputs doubles output, i.e. there are constant returns to scale. (b) Calculate the optimal inputs required to produce target output . Show all calculations below.
Answer
To produce target output we require:
( ) At the optimum: (c) What is Ajax Cola’s cost function? From this calculate the average and marginal cost functions. Show all calculations
below.
Answer
The cost of producing target output is:
()
From above: so that:
()
() () ( ) The marginal and average cost functions are:
()
() () () (
( )
)
19 ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Note that () ( ), as must be the case when there are constant returns to scale. (d) What is the elasticity of total cost with respect to sugar prices? Show all calculations below.
Answer:
We want: It’s best to compute this via the point elasticity method: Recall:
() ( ) This implies: ( ) ( ) (e) Due to greater volatility in sugar prices, Ajax Cola has reconfigured its technology to enable the use of sugar ( )
and/or fructose ( ) as sweeteners, where 2 units of fructose can be perfectly substituted for a unit of sugar (for
constructing production function of sweetener, assume to be good 1 and to be good 2) . Denote the price of
fructose as . Write down Ajax Cola’s production function.
Answer
The production function was:
( ) Now Ajax can use sugar or fructose as perfect substitutes. Holding all other inputs constant, the “sub” production
function for sweeteners is: The slope of the isoquants is: If sugar and fructose are on the x and y axes respectively, then since 2 units of fructose can be perfectly substituted for a
unit of sugar, it must be that:
20
ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO...
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