A$.$ Methods

Use the substitution method to solve the following problem

Optimise $3xy$ subject to $8x+2y=10$

$[10$ points$]$

$2.$ Use the Lagrange multiplier method to solve the following problem

Optimise $7y-7x^2$ subject to $y=2x-5$

$[10$ points$]$

B$.$ Applications

$3.$ A firm's production function is given by $Q=3L^{0\mathrm{.}3}{K}^{0\mathrm{.}2}$ where L is the quantity of labour employed and K is the quantity of capital employed. The price of labour is $6,$ and the price of capital is $3.$

a$.$If the producer's costs are constrained to $300$ find the maximum level of production of the firm.

b$.$ If the cost constraint is relaxed and the producer can now spend $600,$ what effect will this have on the optimal level of production?

$[30$ points$]$

$4.$ Use the method of Lagrange multipliers to show that when a utility function, $U=U(x_1,x_2),$ is optimised subject to a budgetary constraint, $P_1{x}_1+P_2{x}_2=M,$ the ratio of marginal utility to price is the same for both goods.

$[25$ points$]$

$5.$ Match each of the graphs $($not drawn to scale$)$ to the following economic functions and explain your reasoning for each:

a$.$ Budget constraint:

$6x+5y=200$

b$.$ Average cost:

$AC=5+2Q+\frac{10}{Q}$

c$.$ Demand:

$P=\frac{30}{Q+2}-4$