consumer consuming two commodities X and Y has the following utility function .If the price of the two commodities are Birr 1 and 4 respectively and his/her budget is birr 10.
Find the quantities of good X and Y which will maximize utility.
Total utility at optimum point
Find the at optimum point
Show optimum point graphically
To find the quantities of goods X and Y that will maximize utility, we need to set up the consumer's optimization problem.
Let's denote the quantity of good X as x and the quantity of good Y as y. The utility function can be represented as U(x, y) = x^0.5 * y^0.5.
The consumer's budget constraint is given by the equation 1x + 4y = 10.
To solve for the optimum point, we can use the Lagrange multiplier method.
Let L(x, y, λ) = U(x, y) - λ(1x + 4y - 10).
We take the partial derivatives of L(x, y, λ) with respect to x, y, and λ, and set them equal to zero to find the critical points:
∂L/∂x = 0.5x^(-0.5) * y^0.5 - λ = 0
∂L/∂y = 0.5x^0.5 * y^(-0.5) - 4λ = 0
∂L/∂λ = 1x + 4y - 10 = 0
From the first equation, we can rewrite it as y/x = 2λ. Plugging this into the second equation, we get:
0.5x^0.5 * (2λx) = 4λ
x^0.5 = 8λ
x = (8λ)^2
x = 64λ^2
Substituting this value of x back into the budget constraint equation, we get:
64λ^2 + 4y = 10
y = (10 - 64λ^2)/4
y = (5 - 32λ^2)/2
To find the values of λ and hence x and y, we solve this system of equations simultaneously.
Substituting the equation y/x = 2λ into the utility function U(x, y), we can simplify it as U(x) = x^0.5 * (2λx)^0.5 = 2λx.
Now, we can express the budget constraint equation as y = (10 - x)/4 and substitute it into the utility function:
U(x) = 2λx
U(x) = 2λx = 2λ(10 - x)/4
U(x) = λ(10x - x^2)/2
To maximize U(x), we find the derivative of U(x) with respect to x and set it equal to zero:
dU/dx = λ(10 - 2x) = 0
10 - 2x = 0
x = 5
Substituting this value of x back into the budget constraint equation, we get:
y = (10 - x)/4
y = (10 - 5)/4
y = 1.25
Therefore, the quantities of good X and Y that will maximize utility are x = 5 and y = 1.25, respectively.
To find the total utility at the optimum point (x = 5, y = 1.25), we substitute these values into the utility function:
U(5, 1.25) = 5^0.5 * 1.25^0.5
U(5, 1.25) = 2.236 * 1.118
U(5, 1.25) = 2.5
Thus, the total utility at the optimum point is 2.5.
Graphically, the optimum point can be shown by plotting the budget constraint line (1x + 4y = 10) and the level curves of the utility function (U(x, y) = constant), then identifying the point where the budget constraint is tangent to one of the level curves.