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Sample Economics Solved Problems
Question 1: Ray Bond, from Problem 115, is trying to find a new supplier that will reduce his variable cost of production to $15 per unit. If he was able to succeed in reducing this cost, what would the breakeven point be?
Solution: In this case we have
\[C\left( x \right)=150+15x\]
and the revenue is again
\[R\left( x \right)=50x\]
So, we need to solve:
\[C\left( x \right)=R\left( x \right)\,\,\,\Rightarrow \,\,\,150+15x=50x\,\,\,\Rightarrow \,\,\,35x = 150\,\,\,\Rightarrow \,\,\,x=4.29\text{ units}\]
Question 2: Let the demand for pineapples in Sweden be given by the equation: Q=10003P. Suppose that Sweden does not produce any pineapples, but can import them from Indonesia according to the supply curve: Q=4P
a. Solve for the free market price and quantity. Calculate the consumer surplus (Your answer should be a number). Suppose that Sweden decides to put a tariff on pineapples of T per unit.
b. Calculate the quantity of pineapples bought by Sweden as a function of the tariff, T(Your answer should be an equation).
c. Solve for the price of pineapples to Swedish consumers a s a function of the tariff.
d. Solve for the tariff revenue to the Swedish government as a function of the tariff level, T
e. Write an equation that gives the total surplus to Sweden (both to the government and consumers) as a function of the tariff. Now, using calculus, find the tariff level that maximizes Swedish welfare.
f. What is the quantity and consumer price when the tariff is set optimally? What is the total Swedish welfare surplus? How does this compare to your answer in part (a)?
Solution: (a) We get:
\[10003P=4P\,\,\,\Rightarrow \,\,\,\,7P=1000\,\,\,\Rightarrow \,\,\,P*=\frac{1000}{7}\approx \$142.86\]
and the equilibrium quantity is \(Q*=4\left( 142.86 \right)=571.43\text{ units}\). The consumer surplus is CS = 0.5*(1000/3 – 1000/7)*571.43 = 54,421.9, and the producer surplus is PS = 0.5*(1000/7 – 0)*571.43 = 40,816.4
(b)(c) The new supply function is \(Q=4\left( PT \right)=4P4T\), so we solve now
\[10003P=4P4T\,\,\,\Rightarrow \,\,\,\,7P=10004T\,\,\,\Rightarrow \,\,\,P*=\frac{1000+4T}{7}\]
The equilibrium quantity after the tariff is:
\[Q*=10003\left( \frac{1000+4T}{7} \right)=\frac{4000}{7}\frac{12T}{7}\]
(d) The tariff revenue is
\[{{R}_{T}}=Q*\times T=\frac{4000T}{7}\frac{12{{T}^{2}}}{7}\]
(e) The total surplus to Sweden is
\[TS={{R}_{T}}+CS=\frac{4000T}{7}\frac{12{{T}^{2}}}{7}+0.5\left( \frac{1000}{3}\frac{1000+4T}{7} \right)\left( \frac{4000}{7}\frac{12T}{7} \right)\]
\[=\frac{8000000}{147}+\frac{12000T}{49}\frac{60{{T}^{2}}}{49}\]
so then: \(\frac{dTS}{dT}=\frac{120}{49}(100+T)=0\), which means that the optimal tariff is $100.
(f) We get that
\[P*=\frac{1000+4\left( 100 \right)}{7}=\$200\]
and the quantity is
\[Q*=\frac{4000}{7}\frac{12\left( 100 \right)}{7}=400\]
Now the total surplus for Sweden is
\[TS=400\left( 100 \right)+0.5\left( \frac{1000}{3}200 \right)400=\text{66,666}\text{.7}\]
which is greater than the consumer surplus found in (a).
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