Question

В государстве Спортляндия открылось производство усовершенствованных беговых дорожек с системой охлаждения и вентиляции. Все фирмы, производящие спортивный инвентарь функционируют в Спортляндии в условиях совершенной конкуренции. Для того, чтобы граждане Спортляндии стали активнее заниматься спортом государство решило ввести субсидию для покупателей беговых дорожек по ставке S денежных единиц за каждый спортивный тренажер. Функция предложения беговых дорожек qs=4p-20, функция спроса на эти тренажеры qd=60-2p, где величина спроса и предложения измеряется в сотнях тысяч единиц товара, а цена в денежных единицах Спортляндии. (а) Какой должна быть ставка субсидии, чтобы количество продаваемых тренажеров возросло вдвое? (б) Если бы вместо введения субсидии на покупку тренажеров, сохранив при этом расходы госбюджета, правительство Спортляндии ввело субсидию на выручку производителей беговых дорожек по ставке s% (0 1) каков должен быть размер этой субсидии в %? 2) как изменилось бы количество приобретаемых беговых дорожек в Спортляндии в этом случае, по сравнению с пунктом (а)? Объясните полученный результат.

Answer 1

б).

Функция предложения после введения субсидии на выручку производителей беговых дорожек будет иметь вид:

Qs = -20 + 4*(Р - S)

И установится новое равновесие на рынке:

60 - 2Р = -20 + 4*(Р - S)

60-2Р=-20+4Р-4S

6Р = 80+4S

4S = 6Р-80

S = (6Р-80)/4

Откуда:

S = 1,5Р-20

Для определения общей суммы субсидированного товара найдём экстремум функции:

(1,5Р-20)*(60-2Р)

90Р-3Р2-1200+40Р

130Р-3Р2-1200

Приравняем к нулю первую производную данной функции:

130-6Р=0

Р=21,7

Размер субсидии составит:

S = (6Р-80)/4 = (6*21,7-80)/4 = 50,2/4 = 12,6.

 

 

Answer 2

Subsidy for Purchasers of Treadmills in Sportlandia

In order to encourage citizens of Sportlandia to engage in more physical activity, the government has decided to introduce a subsidy for purchasers of treadmills. The subsidy will be provided at a rate of S monetary units per treadmill. To determine the optimal subsidy rate, we need to analyze the supply and demand functions for treadmills in Sportlandia.

The supply function for treadmills is given by qs = 4p - 20, where qs represents the quantity supplied (measured in hundreds of thousands of units) and p represents the price (measured in monetary units).

The demand function for treadmills is given by qd = 60 - 2p, where qd represents the quantity demanded (measured in hundreds of thousands of units) and p represents the price (measured in monetary units).

(a) To determine the subsidy rate that would double the quantity of treadmills sold, we need to find the price at which the quantity demanded is twice the quantity supplied. Setting qd equal to 2 times qs, we have:

2(4p - 20) = 60 - 2p

Simplifying the equation, we get:

8p - 40 = 60 - 2p

10p = 100

p = 10

Therefore, the price at which the quantity demanded is twice the quantity supplied is 10 monetary units per treadmill. To calculate the subsidy rate, we need to substitute this price into the supply function:

qs = 4p - 20 qs = 4(10) - 20 qs = 40 - 20 qs = 20

So, the subsidy rate should be set at 20 monetary units per treadmill in order to double the quantity of treadmills sold.

(b) If the government of Sportlandia decided to introduce a subsidy on the revenue of treadmill manufacturers instead of a subsidy for purchasers, while keeping the budget expenditure the same, the subsidy rate would be expressed as a percentage of the revenue. Let's denote this subsidy rate as s% (0 < s < 1).

To determine the size of this subsidy, we need to calculate the revenue of treadmill manufacturers. Revenue is equal to the price multiplied by the quantity sold. In this case, the price is p and the quantity sold is qs. Therefore, the revenue function is given by:

R = p * qs

To find the subsidy rate that would keep the budget expenditure the same, we need to set the subsidy amount equal to the revenue multiplied by the subsidy rate:

S = s% * R

Substituting the revenue function into the equation, we get:

S = s% * (p * qs)

To determine the subsidy rate, we need to solve for s%:

s% = S / (p * qs)

Substituting the values from part (a), where p = 10 and qs = 20, we get:

s% = S / (10 * 20) s% = S / 200

Therefore, the subsidy rate as a percentage of revenue should be set at S / 200 in order to keep the budget expenditure the same.

2) If the government of Sportlandia introduced the subsidy on the revenue of treadmill manufacturers instead of a subsidy for purchasers, the quantity of treadmills purchased would change compared to part (a). In this case, the subsidy rate is expressed as a percentage of revenue, so the subsidy amount increases as the revenue increases.

Since the subsidy rate is S / 200, the subsidy amount increases as the revenue increases. As a result, the revenue of treadmill manufacturers would increase, leading to an increase in the quantity of treadmills purchased in Sportlandia.

Please note that the calculations provided are based on the given supply and demand functions and assumptions made in the question.