Economics Tutorial 6

No need to go through Q 1 and 2.

3.[pic 1];                      [pic 2]

Profit maximising conditions:

p * MPK =  r;                                                            p*MPL= w

[pic 3];                      [pic 4]

Solve the two equations above to get the solution (you can allow students to try)

[pic 5]➔   L1/3=(3rK 2/3)/p ➔  L = 27r3K2/p3

Substitute into [pic 6],[pic 7]➔[pic 8]

L=27r3(p3/27r2w)2/p3➔[pic 9]

[pic 10]➔[pic 11]

[pic 12]

;    ;     [pic 13][pic 14][pic 15]

;    ;     [pic 16][pic 17][pic 18]

Students should be able to do the comparative statics with calculas. Another important thing is that they should provide a brief intuition – opportunity for discussion.

Demand for input (i.e. the optimum level of inputs increase with the price of output. This is because the marginal revenue from selling additional units would be higher, so the optimum level of production (to sell) will be higher. In order to produce more, you would have to buy more of both types of inputs.When the price of an input increases, the marginal cost of that input goes up. So the optimum amount of that input which you would use would reduce. Since this going down, the firm will also reduce the other input to produce less. (remember, in this case you optimally choose not only the inputs, but also the output).

4.

1.  MP1 = ;      MP2 = [pic 19][pic 20]

1. To maximise, 4* MP1 = w1  ;                  4*MP2 = w2

;      [pic 21][pic 22]

Solve the above to get:

;      ➔[pic 23][pic 24]

Can you notice what will happen when the prices of inputs change?

1. ;      .[pic 25][pic 26]

Optimum level of output is also 1.

Can substitute some other numbers to check if the comparative statics you analysed can be observed. If prices of inputs increase, input demand reduces and output would also decrease.

5.

[pic 27]

Shows constant returns to scale.

I have not asked this, but started typing. So just leaving it here in case you have time!

If all inputs are increased by a proportion k,   ((kx1) 1/2 + 3(kx2)1/2) 2=

                                                                             =  (k1/2(x1) 1/2 +3 k1/2(x2)1/2) 2

= =  {k1/2[(x1) 1/2 + 3(x2)1/2]} 2

= ky

Technical rate of substitution is ….. (check lecture notes and book for definition)

[pic 28]

[pic 29]

Technical rate of substation = -[pic 30]

This is the slope of production function (isoquant) at the pint x1, x2. If the firm wants to increase x1 by a unit, x2 should be reduced by to maintain the same level of output. [pic 31]

We keep x2 on the y-axis.

 which is the iso-cost line[pic 32]

Slope of the isocost function is –w1/w2

At the cheapest input bundle to produce 16 units, the slope of the isoquant is equal to the slope of the isocost line (production function and iso-cost will be tangent to each other).

➔=[pic 33][pic 34][pic 35]

Substitute into the production function producing 16,

[pic 36]

[pic 37]

[pic 38]

[pic 39]

Can do some comparative statistics about what happens to the demand for the inputs when the prices change.

Therefore the total cost = w1x1*+w2x2*