Question

Question 1: The boiler in the heating plant at arctic College is designed to burn either fuel oil or coal. Suppose that one ton of coal, C, produces as many BTUs as 100 gallons of fuel oil, F. (a) Draw a typical isoquant for BTU production. (b) What is the elasticity of substitution between fuel oil and coal? (c) Suppose that fuel oil is $1.00 per gallon and coal is $80 per ton. Draw the output expansion path on the diagram and describe the marginal and average cost curves. (d) Repeat (c). assuming that the price of coal has increased to $120 per ton.

Answer 1

a).

Consider the given problem here the 1 ton of coal produces as many BTU as 100 gallon of fuel oil. So, here the production function is perfect substitute types. The following fig shows the isoquant for BTU.

b).

Let’s assume the typical production function is, => Q = a*C + b*F. Here 1 ton of coal produces as many BTU as 100 gallon of fuel oil, => a*C = b*F, => a*1 = b*100, => a/b = 100. So, here the MRS of the given isoquant is “100 = constant”. So, the “In(MRS)” is also constant, => d[In(MRS)] = 0.

=> Elasticity of Substitution = d[In(K/L)]/ d[In(MRS)] = d[In(K/L)]/ 0 = undefined.

c).

Let’s assume the price of coal is $80 per ton and the price of fuel is $1 per gallon, => the input price ratio is “80/1 = 80”. Here MRS is more than input price ratio, => use of Coal is relatively cheaper compare to the use of the use of fuel. So, at the optimum the producer will use only coal and no fuel. So, the “output expansion path” is “F=0”. Consider the following fig shows the output expansion path.

Here if the iso-cost function is C1 then the optimum choice of the producer is A1, if the iso-cost function is C2 then the optimum choice of the producer is A2. Now, if we add all these points, we get the output expansion path, which is coincide with the horizontal axis.

Here “F=0”, => the production is “Q=a*C, => C=Q/a”. So, the cost function is given below.

TC = 80*C+1*F = 80*(Q/a), => TC = (80/a)*Q, => AC = 80/a, and MC = 80/a. Here the TC function is the ray through the origin, => the AC and MC are same.

d).

Let’s assume the price of coal increases to “120”, => the input price ratio is “120/1 = 120 > MRS”, => use of Fuel is relatively cheaper compare to the use of the use of coal. So, at the optimum the producer will use only fuel and no coal. So, the “output expansion path” is “C=0”. So, here the output expansion path is coincide with the vertical axis.

Here “C=0”, => the production is “Q=b*F, => F=Q/b”. So, the cost function is given below.

TC = 120*C+1*F = (Q/b), => TC = Q/b, => AC = 1/b, and MC = 1/b. Here the TC function is the ray through the origin, => the AC and MC are same.