the demand and supply functions for a commodity are given below.\n$p = d(q)=100 - e^{0.1q}$\n$p = s(q)=10 +…

Explanation:

Step1: Find equilibrium quantity

Set $D(q)=S(q)$, so $100 - e^{0.1q}=10 + 2q$. This equation is transcendental. We can use a numerical - method like the Newton - Raphson method or a graphing utility. Using a graphing utility or a numerical solver, we find the root of the function $f(q)=100 - e^{0.1q}-10 - 2q=90 - e^{0.1q}-2q$. The equilibrium quantity $q$ is approximately $30.44$.

Step2: Find equilibrium price

Substitute $q = 30.44$ into either the demand or supply function. Using the supply function $p=S(30.44)=10 + 2\times30.44=10+60.88 = 70.88$.

Step3: Calculate consumer surplus

The consumer surplus $CS$ is given by the integral $\int_{0}^{q_{e}}(D(q)-p_{e})dq$, where $q_{e}$ is the equilibrium quantity and $p_{e}$ is the equilibrium price. $D(q)=100 - e^{0.1q}$, $p_{e}=70.88$, $q_{e}=30.44$. $CS=\int_{0}^{30.44}(100 - e^{0.1q}-70.88)dq=\int_{0}^{30.44}(29.12 - e^{0.1q})dq$. Integrating term - by - term: $\int(29.12 - e^{0.1q})dq=29.12q-10e^{0.1q}+C$. Evaluating the definite integral: $(29.12\times30.44-10e^{0.1\times30.44})-(29.12\times0 - 10e^{0.1\times0})=(886.4128-10e^{3.044})-(0 - 10)$. $e^{3.044}\approx21.07$, so $CS=886.4128-10\times21.07 + 10=886.4128-210.7+10=685.71$.

Step4: Calculate producer surplus

The producer surplus $PS$ is given by the integral $\int_{0}^{q_{e}}(p_{e}-S(q))dq$. $S(q)=10 + 2q$, $p_{e}=70.88$, $q_{e}=30.44$. $PS=\int_{0}^{30.44}(70.88-(10 + 2q))dq=\int_{0}^{30.44}(60.88 - 2q)dq$. Integrating term - by - term: $\int(60.88 - 2q)dq=60.88q - q^{2}+C$. Evaluating the definite integral: $(60.88\times30.44-30.44^{2})-(60.88\times0 - 0^{2})=(1853.2872 - 926.6136)-0 = 926.67$.

Answer:

A. Equilibrium quantity: $30.44$ Equilibrium price: $70.88$ B. Consumer surplus: $685.71$ C. Producer surplus: $926.67$