ME

 Mutual Industries owns 3 plants at which it produces exactly the same cars.

Plant 1 has cost function:

• $T{C}_1=300-10q+50q^2$ → (1)

Plant 2:

• $T{C}_2=50+10q^2$→ (2)

Plant 3:

• $T{C}_3=1000+20\mathrm{q3}$→ (3)

Mutual decides to produce 5 cars in the least costly way.
→ $Q=5$

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(1) If Mutual Industries decides to produce 5 cars in the most least costly manner in a single plant, which plant will be chosen?

Put $q=5$ in all 3 equations:

• $T{C}_1=300-10\times 5+50\times 5^2=300-50+1250=1500$

• $T{C}_2=50+10\times 5^2=50+250=300$

• $T{C}_3=1000+20\times 5=1100$

 Least cost → Plant 2

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(2) How many cars will be produced in each plant?

Let:

• $q_1$ cars produced in Plant 1

• $q_2$ cars produced in Plant 2

• $q_{\mathrm{3=}}$ cars produced in Plant 3

Then:

• $q_1+q_2+q_3=5$ → (i)

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Image 2:

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(ii) If the production takes place in different plants, how many cars will be produced in each plant? Find the cost?

No. of cars produced	Plant 1	Plant 2	Plant 3
1	240	60	1002
2	480	90	1040
3	720	140	1050
4	1500	210	1080
5	1500	300	1100

Cost of production is least when Plant 2 does it.

 If only no. of cars = 1 + x is scaled, 5 cars will be produced in Plant 2 only → at cost 300

→ 5th no. of cars produced in:

• $P_1,P_2,P_3=0,5,0$
  

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(3) Budget eqn is given by: $P_1{x}_1+P_2{x}_2=m$
The government decides to impose:

• a lump sum tax of t (on income),

• a unit tax on good 1 of τ (→ always add: + tax),

• and a per unit subsidy on good 2 of s (→ minus: – subsidy)

What is the formula for the new budget line?

New Budget Equation:

$$(P_1 + τ)x_1 + P_2x_2 = (m - t + s)$$